See Theorem 16.4 in Chapter 16 for a more general version. 12 that is, α does not belong to K and satisfies a quadratic equation with coefficients in K 13 that is, the other root of the polynomial defining α 19 19/12/2016

Theorem 1.15. Let G be a finite index subgroup of GL 2 pRq. Let α 0 P p K be a quadratic irrational over K. For the weak-star convergence of measures on p K, we have lim sÑ`8

pln qq pq `1q m 0 rGL 2 pRq : Gs 2 q 2 pq ´1q 3 ˇˇln | tr g 0 | ˇˇs ´1 ÿ αPG¨α 0 , hpαqďs

where g 0 P G fixes α 0 with | tr g 0 | ą 1, and m 0 is the index of g Z 0 in G α 0 . Another equidistribution result of an orbit of quadratic irrationals under PGL 2 pRq is obtained by taking another complexity, constructed using crossratios with a fixed quadratic irrational. We denote by ra, b, c, ds " pc´aqpd´bq pc´bqpd´aq the crossratio of four pairwise distinct elements in p K. If α, β P p K are two quadratic irrationals over K such that α R tβ, β σ u, 14 let

which is also an appropriate complexity when α varies in a given orbit of quadratic irrationals by homographies under PGL 2 pRq. See Section 18.1 and for instance [PaP14b, §4] for motivations and results in the Archimedean case.

Theorem 1.16. Let G be a finite index subgroup of GL 2 pRq. Let α 0 , β P p K be two quadratic irrationals over K. For the weak-star convergence of measures on p K ´tβ, β σ u, we have, with g 0 and m 0 as in the statement of Theorem 1.15,

1 that is, commutes with α 26 19/12/2016

Proof. We only give a proof of Claim (4) and refer to for instance [BridH] for proofs of the first three classical assertions.

Since the action of Γ on X is properly discontinuous, and by the compactness of γ Z z Ax γ , there exists N ě 1 such that for all x P Ax γ , the cardinality of tβ P Γ : dpx, βxq ď 2ru is at most N . Let L " AN . For every loxodromic element α P Γ with λpαq " λpγq ď A, assume that rx, ys and rx 1 , y 1 s are segments in Ax γ and Ax α respectively, with length exactly L such that dpx, x 1 q, dpy, y 1 q ď r. We may assume, up to replacing them by their inverses, that γ translates from x towards y and α translates from x 1 towards y 1 . In particular for k " 0, . . . , N , we have dpα ´kγ k x, xq ď dpγ k x, α k x 1 q `dpx 1 , xq ď 2r since γ k x and α k x 1 are respectively the points at distance kλpγq ď kA ď L from x and x 1 on the segments rx, ys and rx 1 , y 1 s. Hence by the definition of N , there exists k ‰ k 1 such that α ´kγ k " α ´k1 γ k 1 . Therefore γ k´k 1 " α k´k 1 , which implies by Assertion (3) that Ax γ " Ax α . l

For every x P X, recall that the Gromov-Bourdon visual distance d x on B 8 X seen from x (see [Bou]) is defined by

ηq " lim tÑ`8 e 1 2 pdpξt, ηtq´dpx, ξtq´dpx, ηtqq , (2.1) where ξ, η P B 8 X and t Þ Ñ ξ t , η t are any geodesic rays ending at ξ, η respectively. By the triangle inequality, for all x, y P Ă M and ξ, η P B 8 Ă M , we have e ´dpx, yq ď d x pξ, ηq d y pξ, ηq ď e dpx, yq . (2.2)

In particular, the identity map from pB 8 X, d x q to pB 8 X, d y q is a bilipschitz homeomorphism. Under our assumptions, pB 8 X, d x 0 q is hence a compact metric space, on which IsompXq acts by bilipschitz homeomorphisms. The following well-known result compares shadows of balls to balls for the visual distance.

Lemma 2.2. For every geodesic ray ρ in X, starting from x P X and ending at ξ P B 8 X, for all R ě 0 and t P sR, `8r , we have

Proof. The lower bound is for instance the lower bound in [HeP2, Lem. 3.1] (which only uses the CATp´1q property). In order to prove the upper bound, let ξ 1 P O x Bpρptq, Rq and let ρ 1 be the geodesic ray from x to ξ 1 . The closest point p to ρptq on the image of ρ 1 satisfies dpp, ρptqq ď R, hence dpx, pq ě t ´R. Then

It is easy to check that this distance is indeed Hausdorff, hence that T 1 X is locally compact, and that it induces on T 1 X the quotient topology of the compact-open topology of G X. The map Þ Ñ v is 1-Lipschitz.

The action of IsompXq on G X induces an isometric action of IsompXq on T 1 X. The antipodal map and the footpoint projection restricted to G X respectively induce an IsompXqequivariant isometric map ι : T 1 X Ñ T 1 X and an IsompXq-equivariant 1 2 -Hölder-continuous map π : T 1 X Ñ X called the antipodal map and footpoint projection of T 1 X. The canonical projection from G X to T 1 X is IsompXq-equivariant and commutes with the antipodal map: For all γ P IsompXq and P G X, we have γv " v γ , ιv " v ι and πpv q " πp q. We denote again by ι : ΓzT 1 X Ñ ΓzT 1 X the quotient map of ι.

Let B 2

8 X be the subset of B 8 X ˆB8 X which consists of pairs of distinct points at infinity of X. Hopf 's parametrisation of G X is the homeomorphism which identifies G X with B 2 8 X ˆR, by the map Þ Ñ p ´, `, tq, where t is the signed distance from the closest point to the basepoint x 0 on the geodesic line to p0q. 3 We have g s p ´, `, tq " p ´, `, t `sq for all s P R, and for all γ P Γ, we have γp ´, `, tq " pγ ´, γ `, t `tγ, ´, `q where t γ, ´, `P R depends only on γ, ´and `. In Hopf's parametrisation, the restriction of the antipodal map to G X is the map p ´, `, tq Þ Ñ p `, ´, ´tq.

The strong stable leaf of w P G `X is W `pwq " P G X : lim tÑ`8 dp ptq, wptqq " 0 ( , and the strong unstable leaf of w P G ´X is W ´pwq " ιW `pιwq " P G X : lim tÑ´8 dp ptq, wptqq " 0 ( .

For every w P G ˘X , let d W ˘pwq be Hamenstädt's distance on W ˘pwq defined as follows: 4 for all , 1 P W ˘pwq, let d W ˘pwq p , 1 q " lim tÑ`8 e 1 2 dp p¯tq, 1 p¯tqq´t .

Introduction

In this book, we study equidistribution and counting problems concerning locally geodesic arcs in negatively curved spaces endowed with potentials, and we deduce, from the special case of tree quotients, various arithmetic applications to equidistribution and counting problems in non-Archimedean local fields.

For several decades, tools in ergodic theory and dynamical systems have been used to obtain geometric equidistribution and counting results on manifolds, both inspired by and with applications to arithmetic and number theoretic problems, in particular in Diophantine approximation. Especially pioneered by Margulis, this field has produced a huge corpus of works, by Bowen, Cosentino, Clozel, Dani, Einseidler, Eskin, Gorodnik, Ghosh, Guivarc'h, Kim, Kleinbock, Kontorovich, Lindenstraus, Margulis, McMullen, Michel, Mohammadi, Mozes, Nevo, Oh, Pollicott, Roblin, Shah, Sharp, Sullivan, Ullmo, Weiss and the last two authors, just to mention a few contributors. We refer for now to the surveys [START_REF] Babillot | Points entiers et groupes discrets : de l'analyse aux systèmes dynamiques[END_REF][START_REF] Oh | Orbital counting via mixing and unipotent flows[END_REF][START_REF] Parkkonen | Counting arcs in negative curvature[END_REF][START_REF] Parkkonen | A survey of some arithmetic applications of ergodic theory in negative curvature[END_REF] and we will explain in more details in this introduction the relation of our work with previous works.

In this text, we consider geometric equidistribution and counting problems weighted with a potential function in quotient spaces of CATp´1q spaces by discrete groups of isometries. The CATp´1q spaces form a huge class of metric spaces that contains (but is not restricted to) metric trees, hyperbolic buildings and simply connected Riemannian manifolds with sectional curvature bounded above by ´1. See [BridH] and Chapter 2 for a review of some basic properties of these spaces. Although some of the equidistribution and counting results with potentials on negatively curved manifolds are known (see for instance [PauPS]), as well as some of such results on CATp´1q spaces without potential (see for instance [START_REF] Roblin | Ergodicité et équidistribution en courbure négative[END_REF]), bringing together these two aspects and producing new results and applications is one of the goals of this book.

We extend the theory of Patterson-Sullivan, Bowen-Margulis and skinning measures to CATp´1q spaces with potentials, with a special emphasis on trees endowed with a system of conductances. We prove that under natural nondegeneracy, mixing and finiteness assumptions, the pushforward under the geodesic flow of the skinning measure of properly immersed locally convex closed subsets of CATp´1q spaces equidistributes to the Gibbs measure, generalising the main result of [START_REF] Parkkonen | Skinning measure in negative curvature and equidistribution of equidistant submanifolds[END_REF].

We also prove that the (appropriate generalisations of) the initial and terminal tangent vectors of the common perpendiculars to any two properly immersed locally convex closed subsets jointly equidistribute to the skinning measures when the lengths of the common per-7

19/12/2016 pendiculars tend to `8. This result is then used to prove asymptotic results on weighted counting functions of common perpendiculars whose lengths tend to `8. Common perpendiculars have been studied, in various particular cases, sometimes not explicitly, by Basmajian, Bridgeman, Bridgeman-Kahn, Eskin-McMullen, Herrmann, Huber, Kontorovich-Oh, Margulis, Martin-McKee-Wambach, Meyerhoff, Mirzakhani, Oh-Shah, Pollicott, Roblin, Shah, the last two authors and many others. See the comments after Theorem 1.5 below, and the survey [START_REF] Parkkonen | Counting arcs in negative curvature[END_REF] for references.

In the Part III of this book, we apply the geometric results obtained for trees to deduce arithmetic applications in non-Archimedean local fields. In particular, we prove equidistribution and counting results for rationals and quadratic irrationals in any completion of any function field over a finite field.

Let us now describe more precisely the content of this book, restricted to special cases for the sake of the exposition.

Geometric and dynamical tools

Let Y be a geodesically complete connected proper locally CATp´1q space (or good orbispace), such that the fundamental group of Y is not virtually nilpotent. In this introduction, we will mainly concentrate on the cases where Y is either a metric graph (or graph of finite groups in the sense of Bass and Serre, see [START_REF] Serre | Arbres, amalgames, SL 2 . 3ème éd. corr[END_REF]) or a Riemannian manifold (or good orbifold) of dimension at least 2 with sectional curvature at most ´1. Let G Y be the space of locally geodesic lines of Y , on which the geodesic flow pg t q tPR acts by real translations on the source. When Y is a simplicial 1 graph (of finite groups), we consider the discrete time geodesic flow pg t q tPZ , see Section 2.7. If Y is a Riemannian manifold, then G Y is naturally identified with the unit tangent bundle T 1 Y by the map that associates to a locally geodesic line its tangent vector at time 0. In general, we define T 1 Y as the space of germs of locally geodesic lines in Y , and G Y maps onto T 1 Y with possibly uncountable fibers.

Let F : T 1 Y Ñ R be a continuous map, called a potential, which plays the same role in the construction of Gibbs measures/equilibrium states as the energy function in Bowen's treatment of the thermodynamic formalism of symbolic dynamical systems in [START_REF] Bowen | Equilibrium states and the ergodic theory of Anosov diffeomorphisms[END_REF]Sect. 1]. In this introduction, we assume that F is bounded in order to simplify the statements. We define in Section 3.3 the critical exponent δ F associated with F , which describes the logarithmic growth of an orbit of the fundamental group on the universal cover of Y weighted by the (lifted) potential F , and which coincides with the classical critical exponent when F " 0. When Y is a metric graph, we associate in Section 3.5 a potential F c to a system of conductances c (that is, a map from the set of edges of Y to R), in such a way that the correspondence c Þ Ñ F c is bijective at the level of cohomology classes, and we denote δ Fc by δ c . We assume in the remainder of this introduction that δ F is finite and positive.

We say that the pair pY, F q satisfies the HC-property if the integral of F on compact locally geodesic segments of Y varies in a Hölder-continuous way on its extremities (see Definition 3.4). The pairs which have the HC-property include Riemannian manifolds with pinched sectional curvature at most ´1 and Hölder-continuous potentials, and metric graphs with any potential. This HC-property is the new technical idea compared to [PauPS] which allows the extensions to our very general framework. See also [ConLT], under the strong assumption that Y is compact.

1 that is, if its edges all have lengths 1 8 19/12/2016

In Chapter 4, building on the works of [START_REF] Roblin | Ergodicité et équidistribution en courbure négative[END_REF] when F " 0 and of [PauPS] 2 when Y is a Riemannian manifold, we generalise, to locally CATp´1q spaces Y endowed with a potential F satisfying the HC-property, the construction and basic properties of the Patterson densities at infinity of the universal cover of Y associated with F and the Gibbs measure m F on G Y associated with F .

Using the Patterson-Sullivan-Bowen-Margulis approach, the Patterson densities are limits of renormalised measures on the orbit points of the fundamental group, weighted by the potential, and the Gibbs measures on G Y are local products of Patterson densities on the endpoints of the geodesic line, with the Lebesgue measure on the time parameter, weighted by the Gibbs cocycle defined by the potential.

Generalizing a result of [START_REF] Coornaert | Récurrence de marches aléatoires et ergodicité du flot géodésique sur les graphes réguliers[END_REF], we prove in Section 6.2 that when Y is a regular simplicial graph and c is an anti-reversible system of conductances, then the Patterson measures, normalized to be probability measures, are harmonic measures (or hitting measures) on B 8 Y for a transient random walk on the vertices, whose transition probabilities are constructed using the total mass of the Patterson measures.

Gibbs measures were first introduced in statistical mechanics, and are naturally associated via the thermodynamic formalism3 with symbolic dynamics. We prove in Section 4.2 that our Gibbs measures satisfy a Gibbs property analogous to the one in symbolic dynamics. If F " 0, the Gibbs measure m F is the Bowen-Margulis measure m BM . If Y is a compact Riemannian manifold and F is the strong unstable Jacobian v Þ Ñ ´d dt |t"0 ln Jac `gt |W ´pvq ˘pvq, then m F is the Liouville measure and δ F " 0 (see [START_REF] Paulin | Equilibrium states in negative curvature[END_REF]Chap. 7] for more general assumptions on Y ). Thus, one interesting aspect of Gibbs measure is that they form a natural family of measures invariant under the geodesic flow that interpolates between the Liouville measure and the Bowen-Margulis measure (which in variable curvature are in general not in the same measure class). Another interesting point is that such measures are plentiful: a recent result of Belarif [Bel] proves that when Y is a geometrically finite Riemannian manifold with pinched negative curvature and topologically mixing geodesic flow, the finite and mixing Gibbs measures associated with bounded Hölder-continuous potentials are, once normalised, dense (for the weak-star topology) in the whole space of probability measures invariant under the geodesic flow.

The Gibbs measures are remarkable measures for CATp´1q spaces endowed with potentials due to their unique ergodic-theoretic properties. Let pZ, pφ t q tPR q be a topological space endowed with a continuous one-parameter group of homeomorphisms and let ψ : Z Ñ R be a bounded continuous map. Let M be the set of Borel probability measures m on Z invariant under the flow pφ t q tPR . Let h m pφ 1 q be the (metric) entropy of the geodesic flow with respect to m P M . The metric pressure for ψ of a measure m P M and the pressure of ψ are respectively P ψ pmq " h m pφ 1 q `żZ ψ dm and P ψ " sup mPM P ψ pmq .

An element m P M is an equilibrium state for ψ if the least upper bound defining P ψ is attained on m. Let F 7 : G Y Ñ R be the composition of the canonical map G Y Ñ T 1 Y with F , and note that F 7 " F if Y is a Riemannian manifold. When F " 0 and Y is a Riemannian manifold, whose sectional curvatures and their first derivatives are bounded, by [START_REF] Otal | Principe variationnel et groupes kleiniens[END_REF]Thm. 2], the pressure P F coincides with the entropy of the geodesic flow, it is equal to the critical exponent of the fundamental group of Y , and the Bowen-Margulis measure m F " m BM , normalised to be a probability measure, is the measure of maximal entropy. When Y is a Riemannian manifold whose sectional curvatures and their first derivatives are bounded and F is Höldercontinuous, by [START_REF] Paulin | Equilibrium states in negative curvature[END_REF]Thm. 6.1], we have P F " δ F . If furthermore the Gibbs measure m F is finite and normalised to be a probability measure, then m F is an equilibrium state for F .

We prove an analog of these results for the potential F 7 when Y is a metric graph of groups. The case when Y is a finite simplicial graph4 is classical by the work of Bowen [START_REF] Bowen | Equilibrium states and the ergodic theory of Anosov diffeomorphisms[END_REF], as it reduces to arguments of subshifts of finite type (see for instance [START_REF] Coornaert | Symbolic dynamics and hyperbolic groups[END_REF]). When Y is a compact5 locally CATp´1q-space,6 a complete statement about existence, uniqueness and Gibbs property of equilibrium states for any Hölder-continuous potential is given in [ConLT].

Theorem 1.1 (The variational principle for metric graphs of groups). Assume that Y is a metric graph of finite groups, with a positive lower bound and finite upper bound on the lengths of edges. If the critical exponent δ F is finite, if the Gibbs measure m F is finite, then P F 7 " δ F and the Gibbs measure normalised to be a probability measure is the unique equilibrium state for F 7 .

The main tool is a natural coding of the discrete time geodesic flow by a topological Markov shift (see Section 5.1). This coding is delicate when the vertex stabilisers are nontrivial, as in particular it does not satisfy in general the Markovian property of dependence only on the immediate past (see Section 5.2). We then apply results of Buzzi and Sarig in symbolic dynamics over a countable alphabet (see Appendix A written by J. Buzzi), and suspension techniques introduced in Section 5.3.

In Chapter 7, we generalise for nonconstant potentials on any geodesically complete connected proper locally CATp´1q space Y the construction of the skinning measures σ D and σ D on the outer and inner unit normal bundles of a connected proper nonempty properly immersed closed locally convex subset D of Y . By definition, D is the image, by the universal covering map, of a proper nonempty closed convex subset of the universal cover of Y , whose family of images under the universal covering group is locally finite. We refer to Section 2.5 for the appropriate definition of the outer and inner unit normal bundles of D when the boundary of D is not smooth. We construct these measures σ D and σ D as the induced measures on Y of appropriate pushforwards of the Patterson densities associated with the potential F to the outer and inner unit normal bundles of the lift of D in the universal cover of Y . This construction generalises the one in [START_REF] Parkkonen | Skinning measure in negative curvature and equidistribution of equidistant submanifolds[END_REF] when F " 0, which itself generalises the one in [START_REF] Oh | The asymptotic distribution of circles in the orbits of Kleinian groups[END_REF][START_REF] Oh | Equidistribution and counting for orbits of geometrically finite hyperbolic groups[END_REF] when M has constant curvature and D is a ball, a horoball or a totally geodesic submanifold.

In Section 10.1, we prove the following result on the equidistribution of equidistant hypersurfaces in CATp´1q spaces. This result is a generalisation of [START_REF] Parkkonen | Skinning measure in negative curvature and equidistribution of equidistant submanifolds[END_REF]Theo. 1] (valid in Riemannian manifolds with zero potential) which itself generalised the ones in [START_REF] Margulis | On some aspects of the theory of Anosov systems[END_REF][START_REF] Eskin | Mixing, counting, and equidistribution in Lie groups[END_REF][START_REF] Parkkonen | Équidistribution, comptage et approximation par irrationnels quadratiques[END_REF] when Y has constant curvature, F " 0 and D is a ball, a horoball or a totally geodesic submanifold. See also [START_REF] Roblin | Ergodicité et équidistribution en courbure négative[END_REF] when Y is a CATp´1q space, F " 0 and D is a ball or a horoball.

Theorem 1.2. Let pY, F q be a locally CATp´1q space endowed with a potential satisfying the HC-property. Assume that the Gibbs measure m F on G Y is finite and mixing for the geodesic flow pg t q tPR , and that the skinning measure σ D is finite and nonzero. Then, as t tends to `8, the pushforwards pg t q ˚σD of the skinning measure of D by the geodesic flow weak-star converges towards the Gibbs measure m F (after normalisation as probability measures).

We prove in Theorem 10.4 an analog of Theorem 1.2 for the discrete time geodesic flow on simplicial graphs and, more generally, simplicial graphs of groups. As a special case, we recover known results on nonbacktracking simple random walks on regular graphs. The equidistribution of the pushforward of the skinning measure of a subgraph is a weighted version of the following classical result, see for instance [ABLS], which under further assumptions on the spectral properties on the graph gives precise rates of convergence.

Corollary 1.3. Let Y be a finite regular graph which is not bipartite. Let Y 1 be a nonempty connected subgraph. Then the n-th vertex of the non-backtracking simple random walk on Y starting transversally to Y 1 converges in distribution to the uniform distribution as n Ñ `8.

See Chapter 10 for more details and for the extensions to nonzero potential and to graphs of groups, as well as Section 10.4 for error terms.

The distribution of common perpendiculars

Let D ´and D `be connected proper nonempty properly immersed locally convex closed subsets of Y . A common perpendicular from D ´to D `is a locally geodesic path in Y starting perpendicularly from D ´and arriving perpendicularly to D `.7 We denote the length of a common perpendicular α from D ´to D `by λpαq, and its initial and terminal unit tangent vectors by v ά and v ὰ . In the general CATp´1q case, v α are two different parametrisations (by ¯r0, λpαqs) of α, considered as elements of the space p G Y of generalised locally geodesic lines in Y , see [BartL] or Section 2.3. For all t ą 0, we denote by PerppD ´, D `, tq the set of common perpendiculars from D ´to D `with length at most t (considered with multiplicities), and we define the counting function with weights by

N D ´, D `, F ptq " ÿ αPPerppD ´, D `, tq e ş α F ,
where ş α F " ş λpαq 0 F pg t v ά q dt. We refer to Section 12.1 for the definition of the multiplicities in the manifold case, which are equal to 1 if D ´and D `are embedded and disjoint. Higher multiplicities for common perpendiculars α can occur for instance when D ´is a non-simple closed geodesic and the initial point of α is a multiple point of D

´.

Let PerppD ´, D `q be the set of all common perpendiculars from D ´to D `(considered with multiplicities). The family pλpαqq αPPerppD ´, D `q is called the marked ortholength spectrum from D ´to D `. The set of lengths (with multiplicities) of elements of PerppD ´, D `q is called the ortholength spectrum of D ´, D `. This second set has been introduced by Basmajian [Basm] (under the name "full orthogonal spectrum") when M has constant curvature, and D ´and D `are disjoint or equal embedded totally geodesic hypersurfaces or embedded horospherical cusp neighbourhoods or embedded balls. We refer to the paper [BridK] which 7 See Section 2.6 for explanations when the boundary of D ´or D `is not smooth.

11

19/12/2016 proves that the ortholength spectrum with D ˘" BM determines the volume of a compact hyperbolic manifold M with totally geodesic boundary (see also [Cal] and [MasM]). We prove in Chapter 12 that the critical exponent δ F of F is the exponential growth rate of N D ´, D `, F ptq, and we give an asymptotic formula of the form N D ´, D `, F ptq " c e δ F t as t Ñ `8, with error term estimates in appropriate situations. The constants c that will appear in such asymptotic formulas will be explicit, in terms of the measures naturally associated with the (normalised) potential F : the Gibbs measure m F and the skinning measures of D ánd D `.

When F " 0 and Y is a Riemannian manifold with pinched sectional curvature and finite and mixing Bowen-Margulis measure, the asymptotics of the counting function N D ´, D `, 0 ptq are described in [START_REF] Parkkonen | Counting common perpendicular arcs in negative curvature[END_REF]Thm. 1]. The only restriction on the two convex sets D ˘is that their skinning measures are finite. Here, we generalise that result by allowing for nonzero potential and more general CATp´1q spaces than just manifolds.

The counting function N D ´, D `, 0 ptq has been studied in negatively curved manifolds since the 1950's and in a number of more recent works, sometimes in a different guise. A number of special cases (all with F " 0 and covered by the results of [START_REF] Parkkonen | Counting common perpendicular arcs in negative curvature[END_REF]) were known: • D ´and D `are reduced to points, by for instance [START_REF] Huber | Zur analytischen Theorie hyperbolischen Raumformen und Bewegungsgruppen[END_REF], [START_REF] Margulis | Applications of ergodic theory for the investigation of manifolds of negative curvature[END_REF] and [START_REF] Roblin | Ergodicité et équidistribution en courbure négative[END_REF],

• D ´and D `are horoballs, by [BeHP], [START_REF] Hersonsky | Counting orbit points in coverings of negatively curved manifolds and Hausdorff dimension of cusp excursions[END_REF], [Cos] and [START_REF] Roblin | Ergodicité et équidistribution en courbure négative[END_REF] without an explicit form of the constant in the asymptotic expression, • D ´is a point and D `is a totally geodesic submanifold, by [Her], [EM] and [START_REF] Oh | Counting visible circles on the sphere and Kleinian groups[END_REF] in constant curvature, • D ´is a point and D `is a horoball, by [Kon] and [KonO] in constant curvature, and [Kim] in rank one symmetric spaces, • D ´is a horoball and D `is a totally geodesic submanifold, by [START_REF] Oh | The asymptotic distribution of circles in the orbits of Kleinian groups[END_REF] and [START_REF] Parkkonen | Équidistribution, comptage et approximation par irrationnels quadratiques[END_REF] in constant curvature, and • D ´and D `are (properly immersed) locally geodesic lines in constant curvature and dimension 3, by [START_REF] Pollicott | The Schottky-Klein prime function and counting functions for Fenchel double crosses[END_REF]. We refer to the survey [START_REF] Parkkonen | Counting arcs in negative curvature[END_REF] for more details on the manifold case.

When X is a compact metric or simplicial graph and D ˘are points, the asymptotics of N D ´, D `, 0 ptq as t Ñ `8 is treated in [Gui], as well as [START_REF] Roblin | Ergodicité et équidistribution en courbure négative[END_REF]. Under the same setting, see also the work of Kiro-Smilansky-Smilansky announced in [KiSS] for a counting result of paths (not assumed to be locally geodesic) in finite metric graphs with rationally independent edge lengths and vanishing potential.

The proofs of the asymptotic results on the counting function N D ´, D `, F are based on the following simultaneous equidistribution result that shows that the initial and terminal tangent vectors of the common perpendiculars equidistribute to the skinning measures of D ánd D `. We denote the unit Dirac mass at a point z by ∆ z and the total mass of any measure m by }m}.

Theorem 1.4. Assume that Y is a nonelementary Riemannian manifold with pinched sectional curvature at most ´1 or a metric graph. Let F : T 1 Y Ñ R be a potential, with finite and positive critical exponent δ F , which is bounded and Hölder-continuous when Y is a manifold. Let D ˘be as above. Assume that the Gibbs measure m F is finite and mixing for the geodesic flow. For the weak-star convergence of measures on

19/12/2016

There is a similar statement for nonbipartite simplicial graphs and for more general graphs of groups on which the discrete time geodesic flow is mixing for the Gibbs measure, see the end of Chapter 11 and Section 12.4. Again, the results can then be interpreted in terms of nonbacktracking random walk.

In Section 12.2, we deduce our counting results for common perpendiculars of the subsets D ´and D `from the above simultaneous equidistribution theorem.

Theorem 1.5. (1) Let Y, F, D ˘be as in Theorem 1.4. Assume that the Gibbs measure m F is finite and mixing for the continuous time geodesic flow and that the skinning measures σ Dá nd σ D`a re finite and nonzero. Then, as s Ñ `8,

N D ´, D `, F psq " }σ D´} }σ D`} }m F } e δ F s δ F .
(2) If Y is a finite nonbipartite simplicial graph, then

N D ´, D `,F pnq " e δ F }σ D´} }σ D`} pe δ F ´1q }m F } e δ F n .
The above Assertion ( 1) is valid when Y is a good orbifold instead of a manifold or a metric graph of finite groups instead of a metric graph (for the appropriate notion of multiplicities), and when D ´and D `are replaced by locally finite families. See Section 12.4 for generalisations of Assertion (2) to (possibly infinite) simplicial graphs of finite groups and Sections 12.3 and 12.6 for error terms.

We avoid any compactness assumption on Y , we only assume that the Gibbs measure m F of F is finite and that it is mixing for the geodesic flow. By Babillot's theorem [START_REF] Babillot | On the mixing property for hyperbolic systems[END_REF], if the length spectrum of Y is not contained in a discrete subgroup of R, then m F is mixing if finite. If Y is a Riemannian manifold, this condition is satisfied for instance if the limit set of a fundamental group of Y is not totally disconnected, see for instance [START_REF] Dal | Remarques sur le spectre des longueurs d'une surface et comptage[END_REF][START_REF] Dal | Topologie du feuilletage fortement stable[END_REF]. When Y is a metric graph, Babillot's mixing condition is in particular satisfied if the lengths of the edges of Y are rationally independent.

As in [START_REF] Parkkonen | Counting common perpendicular arcs in negative curvature[END_REF], we have very weak finiteness and curvature assumptions on the space and the convex subsets we consider. Furthermore, the space Y is no longer required to be a manifold and we extend the theory to non-constant weights using equilibrium states. Such weighted counting has only been used in the orbit-counting problem in manifolds with pinched negative curvature in [PauPS]. The approach is based on ideas from Margulis's thesis to use the mixing of the geodesic flow. Our measures are much more general. As in [START_REF] Parkkonen | Counting common perpendicular arcs in negative curvature[END_REF], we push simultaneously the unit normal vectors to the two convex sets D ´and D `in opposite directions.

Classically, an important characterization of the Bowen-Margulis measure on closed negatively curved Riemannian manifolds (F " 0) is that it coincides with the weak-star limit of properly normalised sums of Lebesgue measures supported on periodic orbits. The result was extended to CATp´1q spaces with zero potential in [START_REF] Roblin | Ergodicité et équidistribution en courbure négative[END_REF] and to Gibbs measures in the manifold case in [START_REF] Paulin | Equilibrium states in negative curvature[END_REF]Thm. 9.11]. As a corollary of the simultaneous equidistribution result Theorem 1.4, we obtain a weighted version for simplicial and metric graphs of groups.

The following is a simplified version of such a result for Gibbs measures of metric graphs.

Let Per 1 ptq be the set of prime periodic orbits of the geodesic flow on Y . Let λpgq denote the length of a closed orbit g. Let L g be the Lebesgue measure along g and let L g pF 7 q be the period of g for the potential F .

13

19/12/2016

Theorem 1.6. Assume that Y is a finite metric graph, that the critical exponent δ F is positive and that the Gibbs measure m F is mixing for the (continuous time) geodesic flow. As t Ñ `8, the measures δ F e δ F t ÿ gPPer 1 ptq e LgpF q L g and δ F t e δ F t ÿ gPPer 1 ptq e LgpF q L g λpgq converge to m F }m F } for the weak-star convergence of measures. See Section 13.2 for the proof of the full result and for a similar statement for (possibly infinite) simplicial graphs of finite groups. As a corollary, we obtain counting results of simple loops in metric and simplicial graphs, generalising results of [ParP], [Gui].

Corollary 1.7. Assume that Y is a finite metric graph with all vertices of degree at least 3 such that the critical exponent δ F is positive.

(1) If the Gibbs measure is mixing for the geodesic flow, then ÿ gPPer 1 ptq e LgpF q " e δ F t δ F t as t Ñ `8.

(2) If Y is simplicial and if the Gibbs measure is mixing for the discrete time geodesic flow, then ÿ gPPer 1 ptq e LgpF q " e δ F e δ F ´1 e δ F t t as t Ñ `8.

In the cases when error bounds are known for the mixing property of the continuous time or discrete time geodesic flow on G Y , we obtain corresponding error terms in the equidistribution result of Theorem 1.2 generalising [START_REF] Parkkonen | Skinning measure in negative curvature and equidistribution of equidistant submanifolds[END_REF]Theo. 20] (where F " 0) and in the approximation of the counting function N D ´, D `, 0 by the expression introduced in Theorem 1.5. In the manifold case, see [START_REF] Kleinbock | Bounded orbits of nonquasiunipotent flows on homogeneous spaces[END_REF], [Clo], [START_REF] Dolgopyat | On decay of correlation in Anosov flows[END_REF], [Sto], [Live], [GLP], and Section 12.3 for definitions and precise references. Here is an example of such a result in the manifold case.

Theorem 1.8. Assume that Y is a compact Riemannian manifold and m F is exponentially mixing under the geodesic flow for the Hölder regularity, or that Y is a locally symmetric space, the boundary of D ˘is smooth, m F is finite, smooth, and exponentially mixing under the geodesic flow for the Sobolev regularity. Assume that the strong stable/unstable ball masses by the conditionals of m F are Hölder-continuous in their radius.

(1) As t tends to `8, the pushforwards pg t q ˚σD ´of the skinning measure of D ´by the geodesic flow equidistribute towards the Gibbs measure m F with exponential speed.

(2) There exists κ ą 0 such that, as t Ñ `8,

N D ´, D `, F ptq " }σ D´} }σ D`} δ F }m F } e δ F t `1 `Ope ´κt q ˘.
14 19/12/2016 See Section 12.3 for a discussion of the assumptions and the dependence of Op¨q on the data. Similar (sometimes more precise) error estimates were known earlier for the counting function in special cases of D ˘in constant curvature geometrically finite manifolds (often in small dimension) through the work of Huber, Selberg, Patterson, Lax and Phillips [LaxP], Cosentino [Cos], Kontorovich and Oh [KonO], Lee and Oh [LeO].

When Y is a finite volume hyperbolic manifold and the potential F is constant 0, the Gibbs measure is proportional to the Liouville measure and the skinning measures of totally geodesic submanifolds, balls and horoballs are proportional to the induced Riemannian measures of the unit normal bundles of their boundaries. In this situation, there are very explicit forms of the counting results in finite-volume hyperbolic manifolds, see [START_REF] Parkkonen | Counting common perpendicular arcs in negative curvature[END_REF]Cor.21], [START_REF] Parkkonen | Counting arcs in negative curvature[END_REF]. These results are extended to complex hyperbolic space in [START_REF] Parkkonen | Counting and equidistribution in Heisenberg groups[END_REF].

As an example of this result, if D ´and D `are closed geodesics of Y of lengths ´and `, respectively, then the number N psq of common perpendiculars (counted with multiplicity) from D ´to D `of length at most s satisfies, as s Ñ `8,

N D ´, D `, 0 psq " π n 2 ´1Γp n´1
2 q 2 2 n´2 pn ´1qΓp n 2 q

´ VolpY q e pn´1qs .

(1.1)

Counting in weighted graphs of groups

From now on in this introduction, we only consider metric or simplicial graphs or graphs of groups.

Let Y be a connected finite graph with set of vertices V Y and set of edges EY (see [START_REF] Serre | Arbres, amalgames, SL 2 . 3ème éd. corr[END_REF] for the conventions). We assume that the degree of the graph at each vertex is at least 3. Let λ : EY Ñ s0, `8r with λpeq " λpeq for every e P EYq be an edge length map, let Y " |Y| λ be the geometric realisation of Y where the geometric realisation of every edge e P EY has length λpeq, and let c : EY Ñ R be a map, called a (logarithmic) system of conductances in the analogy between graphs and electrical networks, see for instance [Zem].

Let Y ˘be proper nonempty subgraphs of Y. For every t ě 0, we denote by PerppY ´, Y `, tq the set of edge paths α " pe 1 , . . . , e k q in Y without backtracking, of length λpαq " ř k i"1 λpe i q at most t, of conductance cpαq " ř k i"1 cpe i q, starting from a vertex of Y ´but not by an edge of Y ´, ending at a vertex of Y `but not by an edge of Y `. Let

N Y ´,Y `ptq " ÿ αPPerppY ´,Y `, tq
e cpαq be the number of paths without backtracking from Y ´to Y `of length at most t, counted with weights defined by the system of conductances.

Recall that a real number x is Diophantine if it is badly approximable by rational numbers, that is, if there exist α, β ą 0 such that |x ´p q | ě α q ´β for all p, q P Z with q ą 0. We obtain the following result, which is a very simplified version of our results for the sake of this introduction.

Theorem 1.9. (1) If Y has two cycles whose ratio of lengths is Diophantine, then there exists C ą 0 such that for every k P N ´t0u, as t Ñ `8, N Y ´,Y `ptq " C e δc t `1 `Opt ´kq ˘.

15 19/12/2016

(2) If λ " 1, then there exists C 1 , κ ą 0 such that, as n P N tends to `8, N Y ´,Y `pnq " C 1 e δc n `1 `Ope ´κ n q ˘.

Note that the Diophantine assumption on Y in Theorem 1.9 (1) is standard in the theory of quantum graphs (see for instance [BerK]).

The constants C " C Y ˘, c, λ ą 0 and C 1 " C 1 Y ˘, c ą 0 in the above asymptotic formulas are explicit. When c " 0 and λ " 1, the constants can often be determined concretely, as indicated in the two examples below. 8 Among the ingredients in these computations are the explicit expressions of the total mass of many Bowen-Margulis measures and skinning measures obtained in Chapter 8.

See Sections 12.4, 12.5 and 12.6 for generalisations of Theorem 1.9 when the graphs Y ȃre not embedded in Y, and for versions in (possibly infinite) metric graphs of finite groups. In particular, Assertion (2) remains valid if Y is the quotient of a uniform simplicial tree by a geometrically finite lattice in the sense of [START_REF] Paulin | Groupes géométriquement finis d'automorphismes d'arbres et approximation diophantienne dans les arbres[END_REF], such as an arithmetic lattice in PGL 2 over a non-Archimedian local field, see [START_REF] Lubotzky | Lattices in rank one Lie groups over local fields[END_REF]. Recall that a locally finite metric tree X is uniform if it admits a discrete and cocompact group of isometries, and that a lattice Γ of X is a lattice in the locally compact group of isometries of X preserving without inversions the simplicial structure. We refer for instance to [BasK, BasL] for uncountably many examples of tree lattices.

Example 1.10. (1) When Y is a pq `1q-regular finite graph with constant edge length map λ " 1 and vanishing system of conductances c " 0, then δ c " ln q, and if furthermore Y `and Y ´are vertices, then (see Equation (12.10))

C 1 " q `1 pq ´1q CardpV Yq .

(2) When Y is biregular of degrees p `1 and q `1 with p, q ě 2, when λ " 1 and c " 0, then δ c " ln ? pq , and if furthermore the subgraphs Y ˘are simple cycles of lengths L ˘, then (see Equation (12.11))

C 1 " p ? q `?pq 2 L ´L2 ppq ´1q CardpEYq .

The main tool in order to obtain the error terms in Theorem 1.9 and its more general versions is to study the error terms in the mixing property of the geodesic flow. Using the already mentioned coding (given in Section 5.2) of the discrete time geodesic flow by a twosided topological Markov shift, classical reduction to one-sided topological Markov shift, and results of Young [START_REF] Young | Statistical properties of dynamical systems with some hyperbolicity[END_REF] on the decay of correlations for Young towers with exponentially small tails, we in particular obtain the following simple criteria for the exponential decay of correlation of the discrete time geodesic flow. See Theorem 9.1 for the complete result. In particular, we do not assume Y to be finite.

Theorem 1.11. Assume that the Gibbs measure m F is finite and mixing for the discrete time geodesic flow on Y. Assume moreover that there exist a finite subset E of V Y and C 1 , κ 1 ą 0 such that for all n P N, we have m F `t P G Y : p0q P E and @ k P t1, . . . , nu, pkq R Eu ˘ď C 1 e ´κ1 n .

Then the discrete time geodesic flow has exponential decay of Hölder correlations for m F .

8 See Section 12.4 for more examples.

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The assumption of having exponentially small mass of geodesic lines which have a big return time to a given finite subset of V Y is in particular satisfied (see Section 9.2) if Y is the quotient of a uniform simplicial tree by a geometrically finite lattice in the sense of [START_REF] Paulin | Groupes géométriquement finis d'automorphismes d'arbres et approximation diophantienne dans les arbres[END_REF], such as an arithmetic lattice in PGL 2 over a non-Archimedian local field, see [START_REF] Lubotzky | Lattices in rank one Lie groups over local fields[END_REF], but also by many other examples of Y.

These results allow to prove in Section 9.3, under Diophantine assumptions, the rapid mixing property for the continuous time geodesic flow, that leads to Assertion (1) of Theorem 1.9, see Section 12.6. The proof uses suspension techniques due to Dolgopyat [START_REF] Dolgopyat | Prevalence of rapid mixing in hyperbolic flows[END_REF] when Y is a compact metric tree, and to Melbourne [START_REF] Melbourne | Rapid decay of correlations for nonuniformly hyperbolic flows[END_REF] otherwise.

As a corollary of the general version of the counting result Theorem 1.5, we have the following asymptotic for the orbital counting function in conjugacy classes for groups acting on trees. Given x 0 P X and a nontrivial conjugacy class K in a discrete group Γ of isometries of X, we consider the counting function N K, x 0 ptq " Cardtγ P K : dpx 0 , γx 0 q ď tu , introduced by Huber [START_REF] Huber | Über eine neue Klasse automorpher Funktionen und ein Gitterpunktproblem in der hyperbolischen Ebene[END_REF] when X is replaced by the real hyperbolic plane and Γ is a lattice. We refer to [START_REF] Parkkonen | On the hyperbolic orbital counting problem in conjugacy classes[END_REF] for many results on the asymptotic growth of such orbital counting functions in conjugacy classes, when X is replaced by a finitely generated group with a word metric, or a complete simply connected pinched negatively curved Riemannian manifold. See also [ChaP, ArCT].

Theorem 1.12. Let X be a uniform metric tree with vertices of degree ě 3, let δ be the Hausdorff dimension of B 8 X, let Γ be a discrete group of isometries of X, let x 0 be a vertex of X with trivial stabiliser in Γ, and let K be a loxodromic conjugacy class in Γ.

(1) If the metric graph ΓzX is compact and has two cycles whose ratio of lengths is Diophantine, then there exists C ą 0 such that for every k P N ´t0u, as t Ñ `8, N K, x 0 ptq " C e δ 2 t `1 `Opt ´kq ˘.

(2) If X is simplicial and Γ is a geometrically finite lattice of X, then there exist C 1 , κ ą 0 such that, as n P N tends to `8, N K, x 0 pnq " C 1 e δ tpn´λpγqq{2u `1 `Ope ´κ n q ˘.

We refer to Theorem 13.1 for a more general version, including a version with a system of conductances in the counting function, and when K is elliptic. When ΓzX is compact and Γ is torsion free, 9 Assertion (1) of this result is due to Kenison and Sharp [KeS], who proved it using transfer operator techniques for suspensions of subshifts of finite type. Up to strengthening the Diophantine assumption, using work of Melbourne [START_REF] Melbourne | Rapid decay of correlations for nonuniformly hyperbolic flows[END_REF] on the decay of correlations of suspensions of Young towers, we are able to extend Assertion (1) to all geometrically finite lattices Γ of X in Chapter 13.1.

The constants C " C K,x 0 and C 1 " C 1 K,x 0 are explicit. For instance in Assertion (2), if X is the geometric realisation of a regular simplicial tree X of degree q `1, if x 0 is a vertex of X, if K is the conjugacy class of γ 0 with translation length λpγ 0 q on X, if VolpΓz zXq "

ÿ rxsPΓzV X 1 |Γ x |
9 in particular Γ then has the very restricted structure of a free group 17 19/12/2016 is the volume 10 of the quotient graph of groups Γz zX , then C 1 " λpγ 0 q rZ Γ pγ 0 q : γ Z 0 s VolpΓz zXq , where Z Γ pγ 0 q is the centraliser of γ 0 in Γ. When furthermore Γ is torsion free and ΓzX is finite, as δ " ln q, we get N K, x 0 pnq " λpγ 0 q CardpΓzXq q tpn´λpγ 0 qq{2u `Opq p1´κ 1 q n{2 q as n P N tends to `8, thus recovering the result of [Dou] who used the spectral theory of the discrete Laplacian.

Selected arithmetic applications

We end this introduction by giving a sample of our arithmetic applications (see Part III of this book) of the ergodic and dynamical results on the discrete time geodesic flow on simplicial trees described in Part II of this book, as summarized above. Our equidistribution and counting results of common perpendiculars between subtrees indeed produce equidistribution and counting results of rationals and quadratic irrationals in non-Archimedean local fields. We refer to [BrPP] for an announcement of the results of Part III, with a presentation different from the one in this introduction.

To motivate what follows, consider R " Z the ring of integers, K " Q its field of fractions, p K " R the completion of Q for the usual Archimedean absolute value | ¨|, and Haar p K the Lebesgue measure of R (which is the Haar measure of the additive group R normalised so that Haar p K pr0, 1sq " 1).

The following equidistribution result of rationals, due to Neville [Nev], is a quantitative statement on the density of K in p K: For the weak-star convergence of measures on p K, as s Ñ `8, we have lim Furthermore, there exists P N such that for every smooth function ψ : p K Ñ C with compact support, there is an error term in the above equidistribution claim evaluated on ψ, of the form Opspln sq}ψ} q where }ψ} is the Sobolev norm of ψ. The following counting result due to Mertens on the asymptotic behaviour of the average of Euler's totient function ϕ : k Þ Ñ CardpR{kRq ˆ, follows from the above equidistribution one:

n ÿ k"1 ϕpkq " 3 π n 2 `Opn ln nq .
See [START_REF] Parkkonen | On the arithmetic of crossratios and generalised Mertens' formulas. Numéro Spécial "Aux croisements de la géométrie hyperbolique et de l'arithmétique[END_REF] for an approach using methods similar to the ones in this text, and for instance [START_REF] Hardy | An introduction to the theory of numbers[END_REF]Th. 330] for a more traditional proof, as well as [Wal] for a better error term.

Let us now switch to a non-Archimedean setting, restricting to positive characteristic in this introduction. See Part III for analogous applications in characteristic zero.

Let F q be a finite field of order q. Let R " F q rY s be the ring of polynomials in one variable Y with coefficients in F q . Let K " F q pY q be the field of rational fractions in Y with coefficients in F q , which is the field of fractions of R. Let p K " F q ppY ´1qq be the field of formal Laurent series in the variable Y ´1 with coefficients in F q , which is the completion of K for the (ultrametric) absolute value | P Q | " q deg P ´deg Q . Let O " F q rrY ´1ss be the ring of formal power series in Y ´1 with coefficients in F q , which is the ball of centre 0 and radius 1 in p K for this absolute value.

Note that p K is locally compact, and we endow the additive group p K with the Haar measure Haar p K normalised so that Haar p K pOq " 1. The following results extend (with appropriate constants) when K is replaced by any function field of a nonsingular projective curve over F q and p K any completion of K, see Part III.

The following equidistribution result 11 of elements of K in p K gives an analog of Neville's equidistribution results for function fields. Note that when G " GL 2 pRq, we have pP, Qq P Gp1, 0q if and only if xP, Qy " R. We denote by H x the stabiliser of any element x of any set endowed with any action of any group H. Theorem 1.13. Let G be any finite index subgroup of GL 2 pRq. For the weak-star convergence of measures on p K, we have

lim tÑ`8
pq `1q rGL 2 pRq : Gs pq ´1q q 2 rGL 2 pRq p1,0q : G p1,0q s q ´2 t ÿ pP,QqPGp1,0q, deg Qďt

∆ P Q " Haar p K .
We emphazise the fact that we are not assuming G to be a congruence subgroup of GL 2 pRq. This is made possible by our geometric and ergodic methods.

The following variation of this result is more interesting when the class number of the function field K is larger than 1 (see Corollary 16.7 in Chapter 16).

Theorem 1.14. Let m be a nonzero fractional ideal of R with norm Npmq. For the weak-star convergence of measures on p K, we have lim tÑ`8 q `1 pq ´1q q 2 s ´2 ÿ px,yqPmˆm Npmq ´1 Npyqďs, Rx`Ry"m ∆ x y " Haar p K .

In the next two statements, we assume that the characteristic of K is different from 2. If α P p K is quadratic irrational over K, 12 let α σ be the Galois conjugate of α, 13 let trpαq " α`α σ and npαq " αα σ , and let hpαq "

1 |α ´ασ | .
This is an appropriate complexity for quadratic irrationals in a given orbit by homographies under PGL 2 pRq. See Section 17.2 and for instance [START_REF] Hersonsky | On the almost sure spiraling of geodesics in negatively curved manifolds[END_REF]§6] for motivations and results. Note that although there are only finitely many orbits by homographies of PGL 2 pRq on K (and exactly one in the particular case of this introduction), there are infinitely many orbits of PGL 2 pRq in the set of quadratic irrationals in p K over K. The following result gives in particular that any orbit of quadratic irrationals under PGL 2 pRq equidistributes in p K, when the complexity tends to infinity. See Theorem 17.5 in Section 17.2 for a more general version. We denote by ¨the action by homographies of GL 2 p p Kq on P 1 p p Kq " p K Y t8 " r1 : 0su.

The fact that the measure towards which we have an equidistribution is only absolutely continuous with respect to the Haar measure is explained by the invariance of α Þ Ñ h β pαq under the (infinite) stabiliser of β in PGL 2 pRq. See Theorem 18.4 in Section 18.1 for a more general version.

The last statement of this introduction is an equidistribution result for the integral representations of quadratic norm forms px, yq Þ Ñ npx ´yαq on K ˆK, where α P p K is a quadratic irrational over K. See Theorem 19.1 in Section 19 for a more general version, and for instance [START_REF] Parkkonen | On the arithmetic of crossratios and generalised Mertens' formulas. Numéro Spécial "Aux croisements de la géométrie hyperbolique et de l'arithmétique[END_REF]§5.3] for motivations and results in the Archimedean case.

There is an extensive bibliography on the integral representation of norm forms and more generally decomposable forms over function fields, we only refer to [START_REF] Schmidt | Thue's equation over function fields[END_REF][START_REF] Mason | On Thue's equation over function fields[END_REF][START_REF] Győry | Bounds for the solutions of norm form, discriminant form and index form equations in finitely generated integral domains[END_REF][START_REF] Mason | Norm form equations I[END_REF]. These references mostly consider higher degrees, with an algebraically closed ground field of characteristic 0, instead of F q .

Theorem 1.17. Let G be a finite index subgroup of GL 2 pRq and let β P K v be a quadratic irrational over K. For the weak-star convergence of measures on p K ´tβ, β σ u, we have lim sÑ`8 pq `1q rGL 2 pR v q : Gs q 2 pq ´1q 3 rGL 2 pR v q p1,0q : G p1,0q s s 14 See Section 18.1 when this condition is not satisfied.

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Furthermore, we have error estimates in the arithmetic applications: There exists κ ą 0 such that for every locally constant function with compact support ψ : p K Ñ C in Theorems 1.13, 1.14 and 1.15, or ψ : p K ´tβ, β σ u Ñ C in Theorems 1.16 and 1.17, there are error terms in the above equidistribution claims evaluated on ψ, of the form Ops ´κq where s " q t in Theorem 1.13, with for each result an explicit control on the test function ψ involving only some norm of ψ, see in particular Section 15.4.

The link between the geometry described in the first part of this introduction and the above arithmetic statements is provided by the Bruhat-Tits tree of pPGL 2 , p Kq, see [START_REF] Serre | Arbres, amalgames, SL 2 . 3ème éd. corr[END_REF] and Section 15.1 for background. We refer to Part III for more general arithmetic applications. geodesics) if any isometric map from an interval in R to X extends to at least one isometric map from R to X. We will put a special emphasis on the case when X is a (proper, geodesically complete) R-tree, that is, a uniquely arcwise connected geodesic metric space. In the Introduction, we have denoted by Y the geodesically complete proper locally CATp´1q good orbispace ΓzX, see for instance [START_REF] Ghys | Sur les groupes hyperboliques d'après Mikhael Gromov[END_REF]Ch. 11] for the terminology.

We denote by B 8 X the space at infinity of X, which consists of the asymptotic classes of geodesic rays in X. It coincides with the space of (Freudenthal's) ends of X when X is an R-tree. We denote by ΛΓ the limit set of Γ and by C ΛΓ the convex hull in X of ΛΓ.

When X is an R-tree, then a subset D of X is convex if and only if it is connected, and we will call it a subtree. In particular, if X is an R-tree, then C ΛΓ is equal to the union of the geodesic lines between pairs of distinct points in ΛΓ, since this union is connected and contained in C ΛΓ.

A point ξ P B 8 X is called a conical limit point if there exists a sequence of orbit points of x 0 under Γ converging to ξ while staying at bounded distance from a geodesic ray ending at ξ. The set of conical limit points is the conical limit set Λ c Γ of Γ.

A point p P ΛΓ is a bounded parabolic limit point of Γ if its stabiliser Γ p in Γ acts properly discontinuously with compact quotient on ΛΓ ´tpu. The discrete nonelementary group of isometries Γ of X is said to be geometrically finite if every element of ΛΓ is either a conical limit point or a bounded parabolic limit point of Γ (see for instance [Bowd], as well as [START_REF] Paulin | Groupes géométriquement finis d'automorphismes d'arbres et approximation diophantienne dans les arbres[END_REF] when X is an R-tree, and [DaSU] for a very interesting study of equivalent conditions in an even greater generality).

For all x P X Y B 8 X and A Ă X, the shadow of A seen from x is the subset O x A of B 8 X consisting of the positive endpoints of the geodesic rays starting at x and meeting A if x P X, and of the geodesic lines starting at x and meeting A if x P B 8 X.

We denote by IsompXq the isometry group of X. The translation length of an isometry γ P IsompXq is λpγq " inf xPX dpx, γxq .

Recall that γ P IsompXq is loxodromic if λpγq ą 0, and that then Ax γ " tx P X : dpx, γxq " λpγqu is (the image of) a geodesic line in X, called the translation axis of γ.

We will need the following well-known lemma later on. An element of Γ is primitive if there is no γ 0 P Γ and k P N ´t0, 1u such that γ " γ k 0 .

Lemma 2.1.

(1) For every loxodromic element γ P Γ, there exists a primitive loxodromic element γ 0 P Γ and k P N ´t0u such that γ " γ k 0 .

(2) If α, γ P Γ are loxodromic with Ax γ " Ax α , then there exists p, q P Z and a primitive loxodromic element γ 0 P Γ such that α " γ p 0 and γ " γ q 0 .

(3) For all α, γ P Γ such that α is loxodromic, if γ centralises 1 α, then either γ pointwise fixes the translation axis of α, or γ is loxodromic, with Ax γ " Ax α .

(4) For every loxodromic element γ P Γ, for all A ą 0 and r ą 0, there exists L ą 0 such that for every loxodromic element α P Γ, if λpαq " λpγq ď A and if Ax γ and Ax α have segments of length at least L at Hausdorff distance at most r, then Ax γ " Ax α .

" e R e ´t .

Therefore ξ 1 P B dx pξ, e R e ´tq. l

The Busemann cocycle of X is the map β : B 8 X ˆX ˆX Ñ R defined by where t Þ Ñ ξ t is any geodesic ray ending at ξ. The above limit exists and is independent of x 0 , and we have | β ξ px, yq | ď dpx, yq .

(2.

3)

The horosphere with centre ξ P B 8 X through x P X is ty P X : β ξ px, yq " 0u, and ty P X : β ξ px, yq ď 0u is the (closed) horoball centred at ξ bounded by this horosphere. Horoballs are nonempty proper closed (strictly) convex subsets of X. Given a horoball H and t ě 0, we denote by H rts " tx P H : dpx, BH q ě tu the horoball contained in H (hence centred at the same point at infinity as H ) whose boundary is at distance t from the boundary of H

Generalised geodesic lines

Let p G X be the space of 1-Lipschitz maps w : R Ñ X which are isometric on a closed interval and locally constant outside it. 2 This space has been introduced by Bartels and Lück in [BartL], to which we refer for the following basic properties. The elements of p G X are called the generalised geodesic lines of X. We endow p G X with the distance d " d p G X defined by @ w, w 1 P p G X, dpw, w 1 q " ż `8 ´8 dpwptq, w 1 ptqq e ´2|t| dt .

(2.4)

The group IsompXq acts isometrically on p G X by postcomposition. The distance d induces the topology of uniform convergence on compact subsets on p G X, and p G X is a proper metric space.

The geodesic flow pg t q tPR on p G X is the one-parameter group of homeomorphisms of the space p G X defined by g t w : s Þ Ñ wps `tq for all w P p G X and t P R. It commutes with the action of IsompXq. If w is isometric exactly on the interval I, then g ´tw is isometric exactly on the interval t `I. Note that for all w P p G X and s P R, we have dpw, g s wq ď |s| ,

(2.5)

with equality if w P G X.

The footpoint projection is the IsompXq-equivariant 1 2 -Hölder-continuous map π : p G X Ñ X defined by πpwq " wp0q for all w P p G X. The antipodal map of p G X is the IsompXqequivariant isometric map ι : p G X Ñ p G X defined by ιw : s Þ Ñ wp´sq for all w P p G X, which satisfies ι ˝gt " g ´t ˝ι for every t P R and π ˝ι " π.

The positive and negative endpoint maps are the continuous maps from

p G X to X Y B 8 X defined by w Þ Ñ w ˘" lim tÑ˘8 wptq .
The space G X of geodesic lines in X is the IsompXq-invariant closed subspace of p G X consisting of the elements P p G X with ˘P B 8 X. Note that the distances on G X considered in [BartL] and [PauPS] are topologically equivalent, although slightly different from the restriction to G X of the distance defined in Equation (2.4). The factor e ´2|t| in this equation, 2 that is, constant on each complementary component 28 19/12/2016 sufficient in order to deal with Hölder-continuity issues, is replaced by e ´t2 { ? π in [PauPS] and by e ´|t| {2 in [BartL] (so that the above 1 2 -Hölder-continuity claim of π does follow from the one in [BartL]).

We will also consider the IsompXq-invariant closed subspaces

G ˘X " tw P p G X : w ˘P B 8 Xu ,
and their IsompXq-invariant closed subspaces G ˘, 0 X consisting of the elements ρ P G ˘X which are isometric exactly on ˘r0, `8r.

The subspaces G X and G ˘X satisfy G ´X X G `X " G X and they are invariant under the geodesic flow. The antipodal map ι preserves G X, and maps G ˘X to G ¯X as well as G ˘, 0 X to G ¯, 0 X. We denote again by ι : Γz p G X Ñ Γz p G X and by g t : Γz p G X Ñ Γz p G X the quotient maps of ι and g t , for every t P R.

Let w P p G X be isometric exactly on an interval I of R. If I is compact then w is a (generalised) geodesic segment, and if I " s´8, as or I " ra, `8r for some a P R, then w is a (generalised) (negative or positive) geodesic ray in X. Any geodesic line p w P G X such that p w| I " w| I is an extension of w. Note that p w is an extension of w if and only if γ p w is an extension of γw for any γ P IsompXq, if and only if ι p w is an extension of ιw, and if and only if g s p w is an extension of g s w for any s P R. For any subset Ω of p G X and any subset A of R, let Ω| A " tw| A : w P Ωu .

Remark 2.3. Let p i q iPN be a sequence of generalised geodesic lines such that rt í , t ì s is the maximal segment on which i is isometric. Let ps i q iPN be a sequence in R such that t ȋ ´si Ñ ˘8 as i Ñ `8 and i ps i q stays in a compact subset of X, then dp i , G Xq Ñ 0 as i Ñ `8. Furthermore if ps i q iPN is bounded, then up to extracting a subsequence, p i q iPN converges to an element in G X.

This conceptually important observation explains how it is conceivable that long common perpendicular segments may equidistribute towards measures supported on geodesic lines. See Chapter 11 for further developments of these ideas.

The unit tangent bundle

In this work, we define the unit tangent bundle T 1 X of X as the space of germs at 0 of the geodesic lines in X, that is the quotient space

T 1 X " G X{ "
where " 1 if and only if there exists ą 0 such that | r´ , s " 1 | r´ , s . The canonical projection from G X to T 1 X will be denoted by Þ Ñ v . When X is a Riemannian manifold, the spaces G X and T 1 X canonically identify with the usual unit tangent bundle of X, but in general, the map Þ Ñ v has infinite fibers.

We endow T 1 X with the quotient distance d " d T 1 X of the distance of G X, defined by:

The above limits exist, and Hamenstädt's distances are distances inducing the original topology on W ˘pwq. For all , 1 P W ˘pwq and γ P IsompXq, we have

d W ˘pγwq pγ , γ 1 q " d W ˘pwq p , 1 q " d W ¯pιwq pι , ι 1 q .
Furthermore, for every s P R, we have for all , 1 P W ˘pwq d W ˘pg s wq pg s , g s 1 q " e ¯sd W ˘pwq p , 1 q .

(2.7)

If X is an R-tree, for all w P G `X and , 1 P W `pwq, if rs, `8r is the maximal interval on which and 1 agree, then d W `pwq p , 1 q " e s .

The following lemma compares the distance in G X with Hamenstädt's distance for two leaves in the same strong (un)stable leaf.

Lemma 2.4. There exists a universal constant c ą 0 such that for all w P G ˘X and , 1 P W ˘pwq, wa have dp , 1 q ď c d W ˘pwq p , 1 q . 30 19/12/2016

Proof. We refer to [START_REF] Parkkonen | Skinning measure in negative curvature and equidistribution of equidistant submanifolds[END_REF]Lem. 3] for a proof of this result. Note that the distance on G X considered in loc. cit. is slightly different from the one in this text, but the proof adapts easily. l

Let H be a horoball in X, centred at ξ P B 8 X. The strong stable leaves W `pwq are equal for all geodesic rays w starting at time t " 0 from a point of BH and converging to ξ. Using the homeomorphism Þ Ñ ´from W `pwq to B 8 X ´tξu, Hamenstädt's distance on W `pwq defines a distance d H on B 8 X ´tξu that we also call Hamenstädt's distance. For all , 1 P W `pwq, we have

d H p ´, `q " d W `pwq p , 1 q ,
and for all η, η 1 P B 8 X ´tξu, we have

d H pη, η 1 q " lim tÑ`8 e 1 2 dp η p´tq, η 1 p´tqq´t , (2.8)
where η , η 1 are the geodesic lines starting from η, η 1 respectively, ending at ξ, and passing through the boundary of H at time t " 0. Note that for every t ě 0, if H rts is the horoball contained in H whose boundary is at distance t from the boundary of H , then we have

d H rts " e ´t d H .
(2.9)

Let w P G ˘X and η 1 ą 0. We define B ˘pw, η 1 q as the set of P W ˘pwq such that there exists an extension p w P G X of w with d W ˘pwq p , p wq ă η 1 . In particular, B ˘pw, η 1 q contains all the extensions of w, and is the union of the open balls centred at the extensions of w, of radius η 1 , for Hamenstädt's distance on W ˘pwq.

The union over t P R of the images under g t of the strong stable leaf of w P G `X is the stable leaf W 0`p wq " ď tPR g t W `pwq of w, which consists of the elements P G X with `" w `. Similarly, the unstable leaf of w P G ´X W 0´p wq " ď tPR g t W ´pwq , consists of the elements P G X with ´" w ´. Note that the (strong) (un)stable leaves are subsets of the space of geodesic lines G X.

The unstable horosphere H ´pwq of w P G ´X is the horosphere in X centred at w ´and passing through p wp0q for any extension p w P G X of w. The stable horosphere H `pwq of w P G `X is the horosphere in X centred at w `and passing through p wp0q for any extension p w P G X of w. These horospheres H ˘pwq do not depend on the chosen extensions p w of w P G ˘X . The unstable horoball HB ´pwq of w P G ´X and stable horoball HB `pwq of w P G `X are the horoballs bounded by these horospheres. Note that πpW ˘pwqq " H ˘pwq for every w P G ˘X , and that wp0q belongs to H ˘pwq if and only if w is isometric at least on ˘r0, `8r.

31

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Normal bundles and dynamical neighbourhoods

In this Section, adapting [START_REF] Parkkonen | Counting common perpendicular arcs in negative curvature[END_REF]§2.2] to the present context, we define spaces of geodesic rays that generalise the unit normal bundles of submanifolds of negatively curved Riemannian manifolds. When X is a manifold, these normal bundles are submanifolds of the unit tangent bundle of X, which identifies with G X. In general and in particular in trees, it is essential to use geodesic rays to define normal bundles, and not geodesic lines.

Let D be a nonempty proper (that is, different from X) closed convex subset in X. We denote by BD its boundary in X and by B 8 D its set of points at infinity. Let

P D : X Y pB 8 X ´B8 Dq Ñ D
be the (continuous) closest point map to D, defined on ξ P B 8 X ´B8 D by setting P D pξq to be the unique point in D that minimises the function y Þ Ñ β ξ py, x 0 q from D to R. The outer unit normal bundle B 1 `D of (the boundary of) D is

B 1 `D " tρ P G `, 0 X : P D pρ `q " ρp0qu .
The inner unit normal bundle B 1 ´D of (the boundary of) D is

B 1 ´D " ιB 1 `D " tρ P G ´, 0 X : P D pρ ´q " ρp0qu.
Note that B 1 `D and B 1 ´D are spaces of geodesic rays. If X is a smooth manifold, then these spaces have a natural identification with subsets of G X because every geodesic ray is the restriction of a unique geodesic line. But this does not hold in general.

Remark 2.5. As X is assumed to be proper with extendible geodesics, we have πpB 1 ˘Dq " BD. To see this, let x P BD and let px k q kPN be a sequence of points in the complement of D converging to x. For all k P N, let ρ k P B 1 `D be a geodesic ray with ρ k p0q " P D px k q and such that the image of ρ k contains x k . As the closest point map does not increase distances, the sequence pP D px k qq kPN converges to x. Since X is proper, the space B 8 X is compact and the sequence ppρ k q `qkPN has a subsequence that converges to a point ξ P B 8 X. The claim follows from the continuity of the closest point map.

The failure of the equality when X is not proper is easy to see for example, when X is the R-tree constructed by starting with the Euclidean line D " R and attaching a copy of the halfline r0, `8r to each x P D such that x ą 0. Then 0 P BD ´πpB 1 ˘Dq.

The restriction of the positive (respectively negative) endpoint map to B 1 `D (respectively B 1 ´D) is a homeomorphism to its image B 8 X ´B8 D. We denote its inverse map by P D . Note that P D " π ˝P D . For every isometry γ of X, we have B 1 ˘pγDq " γ B 1 ˘D and P γD ˝γ " γ ˝P D . In particular, B 1 ˘D is invariant under the isometries of X that preserve D. For every w P G ˘X , we have a canonical homeomorphism N w : W ˘pwq Ñ B 1 ¯HB ˘pwq, that associates to each geodesic line P W ˘pwq the unique geodesic ray ρ P B 1 ¯HB ˘pwq such that ¯" ρ ¯, or, equivalently, such that ptq " ρptq for every t P R with ¯t ą 0. It is easy to check that N γw ˝γ " γ ˝N w for every γ P IsompXq.

We define

U D " t P G X : ˘R B 8 Du .

(2.10)

Note that U D is an open subset of G X, invariant under the geodesic flow. We have U γD " γU D for every isometry γ of X and, in particular, U D is invariant under the isometries of X The continuous map f D takes P U D to the unique element ρ P B 1 ˘D such that ρ ˘" ˘. The fiber of ρ P B 1 `D for f D is exactly the stable leaf W 0`p ρq, and the fiber of ρ P B 1 ´D for f D is the unstable leaf W 0´p ρq. For all γ P IsompXq and t P R, we have

f γD ˝γ " γ ˝f D and f D ˝gt " f D .
(2.11)

Let w P G ˘X and η, η 1 ą 0. We define the dynamical pη, η 1 q-neighbourhood of w by

V w, η, η 1 " ď sPs´η, η r g s B ˘pw, η 1 q .
(2.12)

Example 2.6. If X is an R-tree, w P G `X and η ă ln η 1 , then V ẁ, η, η 1 is as in the following picture.

πpV ẁ, η, η 1 q w πpB `pw, η 1 qq p wpln η 1 q p wp0q w `" p w Clearly, B ˘pw, η 1 q " ιB ¯pιw, η 1 q, and hence we have V w, η, η 1 " ιV ῑ w, η, η 1 . Furthermore, for every s P R, g s B ˘pw, η 1 q " B ˘pg s w, e ¯sη 1 q hence g s V w, η, η 1 " V gs w, η, e ¯sη 1 .

(2.13)

For every γ P IsompXq, we have γB ˘pw, η 1 q " B ˘pγw, η 1 q and γV w, η, η 1 " V γw, η, η 1 . The map from s´η, ηr ˆB˘p w, η 1 q to V w, η, η 1 defined by ps, 1 q Þ Ñ g s 1 is a homeomorphism. For all subsets Ω ´of G `X and Ω `of G ´X , let

V η, η 1 pΩ ¯q " ď wPΩ ¯V w, η, η 1 , (2.14)
that we call the dynamical neighbourhoods of Ω ¯. Note that they are subsets of G X, not of G ˘X . The families pV η, η 1 pΩ ¯qq η,η 1 ą0 are nondecreasing in η and in η 1 . For every γ P IsompXq, we have γV η, η 1 pΩ ¯q " V η, η 1 pγΩ ¯q and for every t ě 0, we have

g ˘tV η, η 1 pΩ ¯q " V η, e ´tη 1 pg ˘tΩ ¯q . (2.15) Note that ď η , η 1 ą0 V η, η 1 pB 1 ˘Dq " U D ,
and that

Ş η, η 1 ą0 V η, η 1 pB 1 ˘Dq is the set of all extensions in G X of the elements of B 1 ˘D. Assume that Ω ¯is a subset of B 1 ˘D. The restriction of f D to V η, η 1 pΩ ¯q is a continuous map onto Ω ¯,
with fiber over w P Ω ¯the open subset V w, η, η 1 of W 0˘p wq.

We will need the following elementary lemma in Section 10.4.

19/12/2016

Lemma 2.7. There exists a universal constant c 1 ą 0 such that for every w P G `X which is isometric on rs w , `8r and every P V ẁ, η, η 1 , we have dp , wq ď c 1 pη `η1 `esw q .

Proof. By Equation (2.12) and by the definition of B `pw, η 1 q in Section 2.3, there exist s P s ´η, `ηr and an extension p w P G X of w such that d W `pwq pg s , p wq ď η 1 . By Equation (2.5), we have dp , g s q ď |s| ď η. Therefore the result follows from the triangle inequality dp , wq ď dp , g s q `dpg s , p wq `dp p w, wq . l

Creating common perpendiculars

Let D ´and D `be two nonempty proper closed convex subsets of X, where X is as in the beginning of Section 2.2. A geodesic arc α : r0, T s Ñ X, where T ą 0, is a common perpendicular of length T from D ´to D `if there exist w ¯P B 1 ˘D¯s uch that w ´|r0, T s " g ´T w `|r0, T s " α. Since X is CATp´1q, this geodesic arc α is the unique shortest geodesic segment from a point of D ´to a point of D `. There is a common perpendicular from D ´to D `if and only if the closures of D ´and D `in X Y B 8 X are disjoint. When X is an R-tree, then two closed subtrees of X have a common perpendicular if and only if they are nonempty and disjoint.

One of the aims of this text is to count orbits of common perpendiculars between two equivariant families of closed convex subsets of X. The crucial remark is that two nonempty proper closed convex subsets D ´and D `of X have a common perpendicular α of length a given T ą 0 if and only if the subsets g

T {2 B 1 `D´| r´T 2 , T 2 s and g ´T {2 B 1 ´D`| r´T 2 , T 2 s of p G X
intersect. This intersection then consists of the common perpendicular from D ´to D reparametrised by r´T 2 , T 2 s. As a controlled perturbation of this remark, we now give an effective creation result of common perpendiculars in R-trees. It has a version satisfied for X in the generality of Section 2.2, see the end of this Section.

Lemma 2.8. Assume that X is an R-tree. For all R ą 1, η P s0, 1s and t ě 2 ln R `4, for all nonempty closed connected subsets D ´, D `in X, and for every geodesic line P g t{2 V ὴ, R pB 1 `D´q X g ´t{2 V ή, R pB 1 ´D`q , there exist s P s´2η, 2ηr and a common perpendicular r c from D ´to D `such that • the length of r c is t `s,

• the endpoint of r c in D ¯is w ¯p0q where w ¯" f D¯p q,

• the footpoint p0q of lies on r c, and `s`q are in this order on . In particular, the segment rw ´p0q, x ´s Y rx ´, x `s Y rx `, w `p0qs is a nontrivial geodesic segment from a point of D ´to a point of D `that meets D ¯only at an endpoint. Hence, D ´and D àre disjoint, and rw ´p0q, w `p0qs is the image of the common perpendicular from D ´to D `.

max ! dpw ´`t 2 ˘, p0qq, dpw ``´t 2 ˘, p0qq
) ď η . 34 19/12/2016 p0q w ´p t 2 q w `p´t 2 q ď ln R ď ln R x ´xD ´w`p 0q D ` p t 2 `s`q w ´p0q p´t 2 ´s´q Proof. Let R, η, t, D
Let s " s ´`s `. The length of r c is p t 2 `s`q ´p´t 2 ´s´q " t `s. The point p0q lies on r c, we have w ¯" f D¯p q and the endpoints of r c are w ˘p0q. Furthermore, dpw ¯p˘t 2 q, p0qq " ˇˇdpw ¯p˘t 2 q, w ¯p0qq ´dp p0q, p¯t 2 ¯s¯q q ˇˇ" |s ¯| ď η . l

When X is as in the beginning of Section 2.2, the statement and the proof of the following analog of Lemma 2.8 is slightly more technical. We refer to [START_REF] Parkkonen | Counting common perpendicular arcs in negative curvature[END_REF]Lem. 7] for a proof in the Riemannian case, and we leave the extension to the reader, since we will not need it in this book.

Lemma 2.9. Let X be as in the beginning of Section 2.2. For every R ą 0, there exist t 0 , c 0 ą 0 such that for all η P s0, 1s and all t P rt 0 , `8r , for all nonempty closed convex subsets D ´, D `in X, and for all w P g t{2 V ὴ, R pB 1 `D´q X g ´t{2 V ή, R pB 1 ´D`q , there exist s P s ´2η, 2ηr and a common perpendicular r c from D ´to D `such that • the length of r c is contained in rt `s ´c0 e ´t 2 , t `s `c0 e ´t 2 s, • if w ¯" f D¯p wq and if p ˘is the endpoint of r c in D ˘, then dpπpw ˘q, p ˘q ď c 0 e ´t 2 , • the basepoint πpwq of w is at distance at most c 0 e ´t 2 from a point of r c, and maxt dpπpg t 2 w ´q, πpwqq, dpπpg ´t 2 w `q, πpwqq u ď η `c0 e ´t 2 . l 2.7 Metric and simplicial trees, and graphs of groups We use Serre's definitions in [Ser3, §2.1] concerning a graph X, with V X and EX its set of vertices and edges, and opeq, tpeq and e the initial vertex, terminal vertex and opposite edge of an edge e P EX. Recall that a connected graph is bipartite if it is endowed with a partition of its set of vertices into two nonempty subsets such that any two elements of either subset are not related by an edge.

The degree of a vertex x P V X is the cardinality of the set te P EX : opeq " xu. For all j, k P N, a graph X is k-regular if the degree of each vertex x P V X is k, and it is pj, kq-biregular if it is bipartite with the partition of its vertices into the two subsets consisting of vertices with degree j and with degree k respectively.

A metric graph pX, λq is a pair consisting of a graph X and a map λ : EX Ñ s0, `8r with a positive lower bound5 such that λpeq " λpeq, called its edge length map. A simplicial graph X is a metric graph whose edge length map is constant equal to 1.

The topological realisation of a graph X is the topological space obtained from the collection pI e q ePEX of closed unit intervals I e by the finest equivalence relation that identifies intervals corresponding to an edge and its opposite edge by the map t Þ Ñ 1 ´t and identifies the origins of the intervals I e 1 and I e 2 if and only if ope 1 q " ope 2 q, see [ [START_REF] Serre | Arbres, amalgames, SL 2 . 3ème éd. corr[END_REF]Sect. 2.1].

The geometric realisation of a metric tree pX, λq is the topological realisation of X endowed with the maximal geodesic metric that gives length λpeq to the topological realisation of each edge e P EX, and we denote it by X " |X| λ . We identify V X with its image in X. The metric space X determines pX, λq up to subdivision of edges, hence we will often not make a strict distinction between X and pX, λq. In particular, we will refer to convex subsets of pX, λq as convex subsets of X, etc.

If X is a tree, the metric space X is an R-tree, hence it is a CATp´1q space. Since λ is bounded from below by a positive constant, the R-tree X is geodesically complete if and only if X has no terminal vertex (that is, no vertex of degree 1).

We will denote by AutpX, λq, and Aut X in the simplicial case, the group of edge-preserving isometries of X that have no inversions. 6 Since the edge length map has a positive lower bound, the metric space X is proper if and only if X is locally finite. In this case, the nonelementary discrete subgroups Γ of isometries of X we will consider will always be edge-preserving and without inversion.

A locally finite metric tree pX 1 , λq is uniform if there exists some discrete subgroup Γ 1 of AutpX 1 , λq such that Γ 1 zX 1 is a finite graph. See [BasK, BasL] for characterisations of this property in the case of simplicial trees.

The space of generalised discrete geodesic lines of a locally finite simplicial tree X is the locally compact space p G X of 1-Lipschitz mappings w from R to its geometric realisation X " |X| 1 which are isometric on a closed interval with endpoints in Z Y t´8, `8u and locally constant outside it, such that wp0q P V X (or equivalently wpZq Ă V X). Note that p G X is hence a proper subset of p G X. By restriction to p G X, or intersection with p G X, of the objects defined in Sections 2.3 and 2.5 for p G X, we define the distance d on p G X, the subspaces G ˘X, G X, G ˘,0 X, the strong stable/unstable leaves W ˘pwq of w P G ˘X and their Hamenstädt distances d W ˘pwq , the stable/unstable leaves W 0˘p wq of w P G ˘X, the outer and inner unit normal bundles B 1 ˘D of a nonempty proper simplicial subtree D of X, the dynamical neighbourhoods V η, η 1 pΩ ¯q of 36 19/12/2016

subsets Ω ¯of B 1 ˘D as well as the fibrations

f D : U D " t P G X : ˘R B 8 Du Ñ B 1 ˘D ,
whose fiber over ρ P B 1 ˘D is W 0˘p ρq. Note that some definitions actually simplify when considering generalised discrete geodesic lines. For instance, for all w P G ˘X, η 1 ą 0 and η P s0, 1r , the dynamical neighbourhood V w, η, η 1 is equal to B ˘pw, η 1 q, and is hence independent of η P s0, 1r .

Besides the map π : G X Ñ V X defined as in the continuous case by Þ Ñ p0q, we have another natural map T π : G X Ñ EX, which associates to the edge e with opeq " p0q and tpeq " p1q. This map is equivariant under the group of automorphisms (without inversions) AutpXq of X, and we also denote by T π : ΓzG X Ñ ΓzEX its quotient map, for every subgroup Γ of AutpXq.

If X has no terminal vertex, for every e P EX, let B e X " t `: P G X, T πp q " eu be the set of points at infinity of the geodesic rays whose initial (oriented) edge is e. Given x 0 P V X, the discrete Hopf parametrisation now identifies G X with B 2 8 X ˆZ by the map Þ Ñ p ´, `, tq where t P Z is the signed distance from the closest vertex to the basepoint x 0 on the geodesic line to the vertex p0q.

The discrete time geodesic flow pg t q tPZ on p G X is the one-(discrete-)parameter group of homeomorphisms of p G X consisting of (the restriction to p G X of) the integral time maps of the continuous time geodesic flow of the geometric realisation of X: we have g t w : s Þ Ñ wps `tq for all w P p G X and t P Z.

Recall (see for instance [START_REF] Serre | Arbres, amalgames, SL 2 . 3ème éd. corr[END_REF][START_REF] Bass | Tree lattices[END_REF]) that a graph of groups pY, G ˚q consists of ' a graph Y, which is connected unless otherwise stated, ' a group G v for every vertex v P V Y, ' a group G e for every edge e P EY such that G e " G e , ' an injective group morphism ρ e : G e Ñ G tpeq for every edge e P V Y.

We will identify G e with its image in G tpeq by ρ e , unless the meaning is not clear (which might be the case for instance if opeq " tpeq).

A subgraph of subgroups of pY, G ˚q is a graph of groups pY 1 , G 1 ˚q where ' Y 1 is a subgraph of Y, ' for every v P V Y 1 , the group G 1 v is a subgroup of G v , ' for every e P EY, the group G 1
e is a subgroup of G e , and the monomorphism G 1 e Ñ G 1 tpeq is the restriction of the monomorphism G e Ñ G tpeq , and G 1 tpeq X ρ e pG e q " ρ e pG 1 e q .

This condition, first introduced in [START_REF] Bass | Covering theory for graphs of groups[END_REF]Coro. 1.14], is equivalent to the injectivity of the natural map G 1 tpeq {ρ e pG 1 e q Ñ G tpeq {ρ e pG e q. It implies by [START_REF] Bass | Covering theory for graphs of groups[END_REF]2.15] that when the underlying basepoint is chosen in Y 1 , the fundamental group of pY 1 , G 1 ˚q injects in the fundamental group 37 19/12/2016 of pY, G ˚q, and the Bass-Serre tree X 1 of pY 1 , G 1 ˚q injects in an equivariant way in the Bass-Serre tree X of pY, G ˚q.

Note that the fundamental group of pY, G ˚q does not always act faithfully on its Bass-Serre tree X, that is, the kernel of its action might be nontrivial.

The edge-indexed graph pY, iq of the graph of groups pY, G ˚q is the graph Y endowed with the map i : EY Ñ N ´t0u defined by ipeq " rG opeq : G e s (see for instance [BasK, BasL]).

In Section 12.4, we will consider metric graphs of groups pY, G ˚, λq which are graphs of groups endowed with an edge length function λ : EY Ñ s0, `8r (with λpeq " λpeq for every e P EY).

Example 2.10. The main examples of graphs of groups that we will consider in this text are the following ones. Let X be a simplicial tree and let Γ be a subgroup of AutpXq. The quotient graph of groups Γz zX is the following graph of groups pY, G ˚q (having finite vertex groups if X is locally finite and Γ is discrete). Its underlying graph Y is the quotient graph ΓzX. Fix a lift r z P V X Y EX for every z P V Y Y EY. For every e P EY, assume that r e " r e, and fix an element g e P Γ such that g e Ą tpeq " tpr eq. For every y P V Y Y EY, take as G y the stabiliser Γ r y in Γ of the fixed lift r y. Take as monomorphism ρ e : G e Ñ G tpeq the restriction to Γ r e of the conjugation γ Þ Ñ g ´1 e γg e by g ´1 e .

The volume form of a graph of finite groups pY, G ˚q is the measure vol pY, G˚q on the discrete set V Y, such that for every y P V Y,

vol pY, G˚q ptyuq " 1 |G y | ,
where |G y | is the order of the finite group G y . Its total mass, called the volume of pY, G ˚q, is

VolpY, G ˚q " } vol pY, G˚q } "

ÿ yPV Y 1 |G y | .
We denote by L 2 pY, G ˚q " L 2 pV Y, vol pY,G˚q q the Hilbert space of square integrable maps V Y Ñ C for this measure vol pY,G˚q , and by f Þ Ñ }f } 2 and pf, gq Þ Ñ xf, gy 2 its norm and scalar product. Let

L 2 0 pY, G ˚q " tf P L 2 pY, G ˚q : ż f d vol pY, G˚q " 0u .
When VolpY, G ˚q is finite, L 2 0 pY, G ˚q is the orthogonal subspace to the constant functions. We also consider a (edge-)volume form Tvol pY, G˚q on the discrete set EY such that for every e P EY,

Tvol pY, G˚q pteuq " 1 |G e | ,
with total mass TVolpY, G ˚q " }Tvol pY, G˚q } "

ÿ ePEY 1 |G e | .
The (edge-)volume form of a metric graph of groups pY, G ˚, λq is given by For λ " 1, this total mass agrees with that of the discrete definition above.

Remark 2.11. Note that TVolpY, G ˚q " CardpEYq when the edge groups are trivial. We have

TVolpY, G ˚q " ÿ ePEY 1 |G e | " ÿ yPV Y 1 |G y | ÿ ePEY, opeq"y |G y | |G e | " ÿ yPV Y degpr yq |G y | ,
where r y is any lift of y in the Bass-Serre tree of pY, G ˚q. In particular, if X is a uniform simplicial tree and Γ is discrete subgroup of Aut X, then the finiteness of VolpΓzzXq and of TVolpΓzzXq are equivalent. Defining the volume form on V Y by tyu Þ Ñ degpr yq |Gy| sometimes makes formulas simpler, but we will follow the convention which occurs in the classical references (see for instance [BasL]).

If the Bass-Serre tree of pY, G ˚q is pq `1q-regular, then

π ˚Tvol Y, G˚" pq `1q vol Y, G˚a nd TVolpY, G ˚q " pq `1q VolpY, G ˚q . (2.16)
We say that a discrete group of isometries Γ of a locally finite metric or simplicial tree pX, λq is a (tree) lattice of pX, λq if the quotient graph of groups Γz zX has finite volume. If X is simplicial, then this implies that Γ is a lattice in the locally compact group AutpXq (hence that AutpXq is unimodular), the converse being true for instance if X is regular or biregular (see [BasK]). For instance, if Γ is a uniform lattice of X (or pX, λq), that is, if Γ is a discrete subgroup of AutpX, λq and if the quotient graph ΓzX is finite, then Γ is a lattice of pX, λq.

A graph of finite groups pY, G ˚q is a cuspidal ray if Y is a simplicial ray such that the homomorphisms G en Ñ G openq are surjective for its sequence of consecutive edges pe i q iPN oriented towards the unique end of Y. By [START_REF] Paulin | Groupe modulaire, fractions continues et approximation diophantienne en caractéristique p[END_REF], a discrete group Γ 1 of AutpXq (hence of Isomp|X| 1 q) is geometrically finite if and only if it is nonelementary and if the quotient graph of groups by Γ 1 of its minimal nonempty invariant subtree is the union of a finite graph of groups and a finite number of cuspidal rays attached to the finite graph at their finite endpoints.

Remark 2.12. If X is a locally finite simplicial tree and if Γ 1 is a geometrically finite discrete group of AutpXq such that the convex hull of its limit set C ΛΓ 1 is a uniform tree, then Γ 1 is a lattice of C ΛΓ 1 .

Proof. Since C ΛΓ 1 is uniform, there is a uniform upper bound on the length of an edge path in C ΛΓ 1 which injects in Γ 1 zC ΛΓ 1 such that the stabiliser of each edge of this edge path is equal to the stabilisers of both endpoints of this edge. It is hence easy to see that the volume of each of the (finitely many) cuspidal rays in Γ 1 z zC ΛΓ 1 is finite, by a geometric series argument. Hence the volume of Γ 1 z zC ΛΓ 1 is finite. l

Note that contrarily to the case of Riemannian manifolds, there are many more (tree) lattices than there are geometrically finite (tree) lattices, even in regular trees, see for instance [BasL].

39

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In Part III of this text, we will consider simplicial graphs of groups that arise from arithmetics of non-Archimedean local fields. We say that Γ is algebraic if there exists a non-Archimedean local field p K (a finite extension of Q p for some prime p or the field of formal Laurent series over a finite field) and a connected semi-simple algebraic group G with finite centre defined over p K, of p K-rank one, such that X identifies with the Bruhat-Tits tree of G in such a way that Γ identifies with a lattice of Gp p Kq. If Γ is algebraic, then Γ is geometrically finite by [START_REF] Lubotzky | Lattices in rank one Lie groups over local fields[END_REF]. Note that X is then bipartite, see Section 2 of op. cit. for a discussion and references. See Sections 14 and 15.1 for more details, and the subsequent Sections for arithmetic applications arising from algebraic lattices.

40

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Chapter 3

Potentials, critical exponents and Gibbs cocycles

Let X be a geodesically complete proper CATp´1q space, let x 0 P X be an arbitrary basepoint, and let Γ be a nonelementary discrete group of isometries of X.

In this Chapter, we define potentials on T 1 X, which are new data on X in addition to its geometry. We introduce the fundamental tools associated with potentials, and we give some of their basic properties. The development follows [PauPS] with modifications to fit the present more general context.

In Section 3.5, we introduce a natural method to associate a (Γ-invariant) potential r F c : T 1 X Ñ R to a Γ-invariant function c : EX Ñ R defined on the set of edges of a simplicial or metric tree X, with geometric realisation X, that we call a system of conductances on X. This construction gives a nonsymmetric generalisation of electric networks.

Background on (uniformly local) Hölder-continuity

In this preliminary Section, we recall the notion of Hölder-continuity we will use in this text, which needs to be defined appropriately in order to deal with noncompactness issues. The Hölder-continuity will be used on one hand for potentials when X is a Riemannian manifold in Section 3.2, and on the other hand for error term estimates in Chapters 9, 10 and 11.

As in [PauPS], we will use the following uniformly local definition of Hölder-continuous maps. Let E and E 1 be two metric spaces, and let α P s0, 1s. A map f : E Ñ E 1 is • α-Hölder-continuous if there exist c, ą 0 such that for all x, y P E with dpx, yq ď , we have dpf pxq, f pyqq ď c dpx, yq α .

• locally α-Hölder-continuous if for every x P E, there exists a neighbourhood U of x such that the restriction of f to U is α-Hölder-continuous; • Hölder-continuous (respectively locally Hölder-continuous) if there exists α P s0, 1s such that f is α-Hölder-continuous (respectively locally α-Hölder-continuous); • Lipschitz if it is 1-Hölder-continuous and locally Lipschitz if it is locally 1-Hölder-continuous.

Let E and E 1 be two metric spaces. We say that a map f : E Ñ E 1 has • at most linear growth if there exists a, b ě 0 such that dpf pxq, f pyqq ď a dpx, yq `b for all

x, y P E, 41 19/12/2016

• subexponential growth if for every a ą 0, there exists b ě 0 such that dpf pxq, f pyqq ď b e a dpx, yq for all x, y P E.

Remark 3.1. When E is a geodesic space, a consequence of the (uniformly local) Höldercontinuous property of f : E Ñ E1 is that f then has at most linear growth: the definition implies that dpf pxq, f pyqq ď c α´1 dpx, yq `c α for all x, y in X, by subdividing the geodesic segment in E from x to y in P dpx,yq T segments of equal lengths at most and using the triangle inequality in E 1 .

For any metric space Z and α P s0, 1s, the Hölder norm of a bounded α-Hölder-continuous

function f : Z Ñ R is ||f || α " }f } 8 `sup x, yPZ 0ădpx,yqď1
|f pxq ´f pyq| dpx, yq α .

When the diameter of Z is bounded by 1, 1 this coincides with the usual definition. Note that even if the constant in the above definition of a α-Hölder-continuous map is less than 1, this norm is finite, since

sup x, yPZ ďdpx,yqď1 |f pxq ´f pyq| dpx, yq α ď 2 ´α }f } 8 .
Note that for all bounded α-Hölder-continuous maps f, g : Z Ñ R, we have

}f g} α ď }f } α }g} 8 `}f } 8 }g} α . (3.1)
We denote by C α c pZq (respectively C α b pZq) the space of α-Hölder-continuous real-valued functions with compact support (respectively which are bounded) on Z, endowed with this norm. Note that C α b pZq is a Banach space.2

A stronger assumption than the Hölder regularity is the locally constant regularity, that we now define. Alhough it is only useful for totally disconnected metric spaces, several error terms estimates in the literature use this stronger regularity (see for instance [AtGP, KemaPS] and Part III of this text). Let ą 0. For every metric space E and every set E 1 , we say that a map f : E Ñ E 1 is -locally constant if f is constant on every closed ball of radius (or equivalently of radius at most ) in E. We say that f : E Ñ E 1 is locally constant if there exists ą 0 such that f is -locally constant. Note that if E is a geodesic metric space and f : E Ñ E 1 is locally constant, then f is constant. But when E is for instance an ultrametric space, since two distinct closed balls of the same radius are disjoint, the above definition turns out to be very interesting (and much used in representation theory in positive characteristic, for instance). For example, the characteristic function 1 A of a subset A of E is -locally constant if and only if for every x P A, the closed ball Bpx, q is contained in A. In particular, the characteristic function of a closed ball of radius in an ultrametric space is -locally constant.

The next result says that the Hölder regularity is indeed stronger than the locally constant one.

Remark 3.2. Let E and E 1 be two metric spaces. If a map f : E Ñ E 1 is -locally constant, then it is α-Hölder-continuous for every α P s0, 1s. Indeed, for all x, y P E, if dpx, yq ď then dpf pxq, f pyqq " 0 ď c dpx, yq α for all c ą 0. For all P s0, 1s and β ą 0, we denote by C lc, β b pEq the vector space3 of -locally constant functions f : E Ñ R endowed with the lc-norm of exponent β defined by }f } lc, β " ´β }f } 8 .

The above remark proves that if β P s0, 1s, the inclusion map from C lc, β b pEq into C β b pEq is continuous. We will only use the lc-norms in Section 15.4.

Potentials

In this text, a potential for Γ is a continuous Γ-invariant function r

F : T 1 X Ñ R. The quotient function F : ΓzT 1 X Ñ R of r
F is called a potential on ΓzT 1 X. The function r F defines a continuous Γ-invariant function from G X to R, also denoted by r F , by r F p q " r F pv q.

For all x, y P X, and any geodesic line P G X such that p0q " x and pdpx, yqq " y, let

ż y x r F " ż dpx,yq 0 r F pv g t q dt .
Note that for all t P s0, dpx, yqr, the germ v g t is independent on the choice of such a line , hence ş y x r F does not depend on the extension of the geodesic segment rx, ys. The following properties are easy to check using the Γ-invariance of r F and the basic properties of integrals: For all γ P Γ ż γy γx r F "

ż y x r F , for the antipodal map ι ż x y r F " ż y x r F ˝ι , (3.2)
and, for any z P rx, ys,

ż y x r F " ż z x r F `ż y z r F . (3.3)
The period of a loxodromic isometry γ of X for the potential r F is

Per F pγq " ż γx x r F
for any x in the translation axis of γ. Note that, for all α P Γ and n P N ´t0u, we have

Per F pαγα ´1q " Per F pγq, Per F pγ n q " n Per F pγq and Per F pγ ´1q " Per F ˝ιpγq . (3.4)

In trees, we have the following Lipschitz-type control on the integrals of the potentials along segments. 19/12/2016 Lemma 3.3. When r F is constant or when X is an R-tree, for all x, x 1 , y, y 1 P X, we have

ˇˇż y x r F ´ż y 1 x 1 r F ˇˇď dpx, x 1 q sup π ´1prx, x 1 sq | r F | `dpy, y 1 q sup π ´1pry, y 1 sq | r F | .
Proof. When r F is constant, the result follows from the triangle inequality. Assume that X is an R-tree. Consider the case x " x 1 . Let z P X be such that rx, zs " rx, ys X rx, y 1 s. Using Equation (3.3) and the fact that dpy, zq `dpz, y 1 q " dpy, y 1 q, the claim follows. The general case follows by combining this case x " x 1 and a similar estimate for the case y " y 1 . l Some form of uniform Hölder-type control of the potential, analogous to the Lipschitztype one in the previous lemma, will be crucial throughout the present work. The following Definition 3.4 formalises this (weaker) assumption.

Definition 3.4. The triple pX, Γ, r F q satisfies the HC-property (Hölder-type control) if r F has subexponential growth when X is not an R-tree and if there exists κ 1 ě 0 and κ 2 P s0, 1s such that for all x, y, x 1 , y 1 P X with dpx, x 1 q, dpy, y 1 q ď 1, we have

ˇˇż y x r F ´ż y 1 x 1 r F ˇˇ(HC) ď `κ1 `max π ´1pBpx, dpx,x 1 qqq | r F | ˘dpx, x 1 q κ 2 ``κ 1 `max π ´1pBpy, dpy,y 1 qqq | r F | ˘dpy, y 1 q κ 2 .
By Equation (3.2), pX, Γ, r F ˝ιq satisfies the HC-property if and only if pX, Γ, r F q does. By Equation (2.3), for every κ P R, the triple pX, Γ, r F `κq satisfies the HC-property (up to changing the constant κ 1 ) if and only if pX, Γ, r F q does. When X is assumed to be a Riemannian manifold with pinched sectional curvature, requiring the potentials to be Hölder-continuous as in [PauPS] is sufficient to have the HC-property, as we will see below.

Proposition 3.5. The triple pX, Γ, r F q satisfies the HC-property if one of the following conditions is satisfied:

• r F is constant, • X is an R-tree,
• X is a Riemannian manifold with pinched sectional curvature and r F is Hölder-continuous.

Proof. The first two cases are treated in Lemma 3.3, and we may take for them κ 1 " 0 and κ 2 " 1 in the definition of the HC-property.

The claim for Riemannian manifolds follows from the property of at most linear growth of the Hölder-continuous maps (see Remark 3.1) and from [START_REF] Paulin | Equilibrium states in negative curvature[END_REF]Lem. 3.2], with constants κ 1 ą 0 and κ 2 P s0, 1s depending only on the Hölder-continuity constants of r F and on the bounds on the sectional curvature of X. l Remark 3.6. (1) If X " Ă M is a Riemannian manifold, then T 1 X is naturally identified with the usual Riemannian unit tangent bundle of X. If the potential r

F : T 1 Ă M Ñ R is Hölder- continuous for Sasaki's Riemannian metric on T 1 Ă M
, it is a potential as defined in [START_REF] Ruelle | The pressure of the geodesic flow on a negatively curved manifold[END_REF] and [PauPS]. Furthermore, the definition of ş y x r F coincides with the one in these references.

(2) The quotient function F is Hölder-continuous when r F is Hölder-continuous.

19/12/2016

Let r F , r F ˚: T 1 X Ñ R be potentials for Γ. We say that r F ˚is cohomologous to r F (see for instance [Livš]) if there exists a continuous Γ-invariant function r G : T 1 X Ñ R, such that, for every P G X, the map t Þ Ñ r Gpv g t q is differentiable and r F ˚pv q ´r F pv q " d dt |t"0 r Gpv g t q .

(3.5)

A potential r F is said to be reversible if r F `and r F ´are cohomologous. When working with Hölder-continuous potentials, the regularity requirement is for r G to also be Hölder-continuous. Note that the right-hand side of Equation (3.5) does not depend on the choice of the representative of its germ v . In particular, Per F pγq " Per F ˚pγ q for any loxodromic isometry γ if r F and r F ˚are cohomologous potentials.

Poincaré series and critical exponents

Let us fix a potential r F : T 1 X Ñ R for Γ, and x, y P X. The critical exponent of pΓ, F q is the element δ " δ Γ, F of the extended real line r´8, `8s defined by

δ " lim sup nÑ`8 1 n ln ÿ γPΓ, n´1ădpx,γyqďn e ş γy x r F .
The Poincaré series of pΓ, F q is the map Q " Q Γ, F, x, y : R Ñ r0, `8s defined by

Q : s Þ Ñ ÿ γPΓ e ş γy x p r F ´sq .
If δ ă `8, we say that pΓ, F q is of divergence type if the series Q Γ, F, x, y pδq diverges, and of convergence type otherwise. When F " 0, the critical exponent δ Γ, 0 is the usual critical exponent δ Γ P s0, `8s of Γ, the Poincaré series Q Γ, 0, x, y is the usual Poincaré series of Γ, and we recover the usual notion of divergence or convergence type of Γ.

The Poincaré series of pΓ, F q and its critical exponent make sense even if Γ is elementary (see for instance Lemma 3.7 (10)). The following result collects some of the basic properties of the critical exponent. The proofs from [START_REF] Paulin | Equilibrium states in negative curvature[END_REF]Lem. 3.3] generalise to the current setting, replacing the use of [START_REF] Paulin | Equilibrium states in negative curvature[END_REF]Lem. 3.2] by the HC-property.

Lemma 3.7. Assume that pX, Γ, r F q satisfies the HC-property. Then,

(1) the critical exponent δ Γ, F and the divergence or convergence of Q Γ, F, x, y psq are independent of the points x, y P X; they depend only on the cohomology class of r F ;

(2) Q Γ, F ˝ι, x, y " Q Γ, F, y, x and δ Γ, F ˝ι " δ Γ, F ; in particular, pΓ, F q is of divergence type if and only if pΓ, F ˝ιq is of divergence type ;

(3) the Poincaré series Qpsq diverges if s ă δ Γ, F and converges if s ą δ Γ, F ;

(4) δ Γ, F `κ " δ Γ, F `κ for any κ P R, and pΓ, F q is of divergence type if and only if pΓ, F `κq is of divergence type ;

(5) if Γ 1 is a nonelementary subgroup of Γ, denoting by

F 1 : Γ 1 zT 1 X Ñ R the map induced by r F , then δ Γ 1 , F 1 ď δ Γ, F ; 45 19/12/2016 (6) if δ Γ ă `8, then δ Γ `inf π ´1pC ΛΓq r F ď δ Γ, F ď δ Γ `sup π ´1pC ΛΓq r F ; (7) δ Γ, F ą ´8 ; (8) the map r F Þ Ñ δ Γ, F
is convex, sub-additive, and 1-Lipschitz for the uniform norm on the vector space of real continuous maps on π ´1pC ΛΓq ; 4

(9) if Γ 2 is a discrete cocompact group of isometries of X such that r F is Γ 2 -invariant, denoting by F 2 : Γ 2 zT 1 X Ñ R the map induced by r F , then δ Γ, F ď δ Γ 2 , F 2 ;
(10) if Γ is infinite cyclic, generated by a loxodromic isometry γ of X, then pΓ, F q is of divergence type and

δ Γ, F " max ! Per F pγq λpγq , Per F ˝ιpγq λpγq ) . l
Examples 3.8. (1) If δ Γ is finite and r F is bounded, then the critical exponent δ is finite by Lemma 3.7 (6).

(2) If X is a Riemannian manifold with pinched negative curvature or when X has a compact quotient, then δ Γ is finite. See for instance [Bou].

(3) There are examples of pX, Γq with δ Γ " `8 (and hence δ " `8 if r F is constant), for instance when X is the complete ideal hyperbolic triangle complex with 3 ideal triangles along each edge, see [GaP], and Γ its isometry group. Hence the finiteness assumption of the critical exponent is nonempty in general. For the type of results treated in this book, it is however natural and essential.

We may replace upper limits by limits in the definition of the critical exponents, as follows.

Theorem 3.9. Assume that pX, Γ, r F q satisfies the HC-property. If c ą 0 is large enough, then δ " lim Proof. The proofs of [START_REF] Paulin | Equilibrium states in negative curvature[END_REF]Theo. 4.2], either using the original arguments of [START_REF] Roblin | Sur la fonction orbitale des groupes discrets en courbure négative[END_REF] valid when r F is constant, or the super-multiplicativity arguments of [DaPS], extend, using the HC-property (see Definition 3.4) instead of [START_REF] Paulin | Equilibrium states in negative curvature[END_REF]Lem. 3.2].

l

In what follows, we fix a potential r F for Γ such that pX, Γ, r F q satisfies the HC-property. We define r F `" r F and r F ´" r F ˝ι, we denote by F ˘: ΓzT 1 X Ñ R their induced maps, and we assume that δ " δ Γ, F `" δ Γ, F ´is finite.

Gibbs cocycles

The (normalised) Gibbs cocycle associated with the group Γ and the potential r

F ˘is the map C ˘" C Γ,F ˘: B 8 X ˆX ˆX Ñ R defined by pξ, x, yq Þ Ñ C ξ px, yq " lim tÑ`8 ż ξt y p r F ˘´δq ´ż ξt x p r F ˘´δq ,
where t Þ Ñ ξ t is any geodesic ray with endpoint ξ P B 8 X.

We will prove in Proposition 3.10 below that this map is well defined, that is, the above limits exist for all pξ, x, yq P B 8 X ˆX ˆX and they are independent of the choice of the geodesic rays

t Þ Ñ ξ t . If r F ˘" 0, then C ´" C `" δ Γ β, where β is the Busemann cocycle. If X is an R-tree, then C ξ px, yq " ż p y p r F ˘´δq ´ż p x p r F ˘´δq , (3.6)
where p P X is the point for which r p, ξr " rx, ξr X ry, ξr ; in particular, the map ξ Þ Ñ C ξ px, yq is locally constant on the totally discontinuous space B 8 X.

The Gibbs cocycles satisfy the following equivariance and cocycle properties: For all ξ P B 8 X and x, y, z P X, and for every isometry γ of X, we have C γξ pγx, γyq " C ξ px, yq and C ξ px, zq `Cξ pz, yq " C ξ px, yq .

(3.7)

For every P G X, for all x and y on the image of the geodesic line , with ´, x, y, `in this order on , we have

C ´ ´px, yq " C ` `py, xq " ´C` `px, yq " ż y x p r F `´δq . (3.8)
Proposition 3.10. Assume that pX, Γ, r F q satisfies the HC-property and that δ ă `8.

(1) The maps C ˘: B 8 X ˆX ˆX Ñ R are well-defined.

(2) For all x, y P X and ξ P B 8 X, if dpx, yq ď 1, then

| C ξ px, yq | ď pκ 1 `δ `max π ´1pBpx, dpx,yqqq | r F |q dpx, yq κ 2 ,
with the constants κ 1 , κ 2 of the HC-property. If X is an R-tree, then for all x, y P X and ξ P B 8 X, we have

| C ξ px, yq | ď dpx, yq max π ´1prx, ysq | r F ˘´δ| .
(3) The maps C ˘: B 8 X ˆX ˆX Ñ R are locally Hölder-continuous (and locally Lipschitz when X is an R-tree). In particular, they are continuous.

(4) For all r ą 0, x, y P X and ξ P B 8 X, if ξ belongs to the shadow O x Bpy, rq of the ball Bpy, rq seen from x, then with the constants κ 1 , κ 2 of the HC-property,

ˇˇC ξ px, yq `ż y x p r F ˘´δq ˇˇď 2pκ 1 `δ `max π ´1pBpy, rqq | r F |q r κ 2 .
If X is an R-tree, then

ˇˇC ξ px, yq `ż y x p r F ˘´δq ˇˇď 2r max π ´1pBpy, rqq | r F ˘´δ| . 47 19/12/2016
Proof.

(1) The fact that C ξ px, yq is well defined when X is an R-tree follows from Equation (3.6). When X is not an R-tree, let ρ : t Þ Ñ ξ t be any geodesic ray with endpoint ξ P B 8 X, let t Þ Ñ x t (respectively t Þ Ñ y t ) be the geodesic ray from x (respectively y) to ξ. Let t x " β ξ px, ξ 0 q and t y " β ξ py, ξ 0 q, so that the quantity β " t y ´tx is equal to β ξ py, xq (which is independent of ρ), and for every t big enough, we have β ξ pξ t , x t`tx q " β ξ pξ t , y t`ty q " 0.

ξ ξ 0 y t x x t`tx y t`ty x ξ t t t y
Since X is CATp´1q, if t is big enough, then the distances dpξ t , x t`tx q and dpξ t , y t`ty q are at most one, and converge, in a nonincreasing way, exponentially fast to 0 as t Ñ `8. | r F ´δ| q maxtdpξ t , x t`tx q, dpξ t , y t`ty qu κ 2 , which converges to 0 since r F has subexponential growth by the assumptions of the HCproperty. Hence in order to prove Assertion (1), we only have to prove that lim sÑ`8 a s exists.

For all s ě t ě |β|, we have, by the additivity of the integral along geodesics (see Equation (3.3)) and by using again Equation (HC),

|a t ´as | " ˇˇż ys yt p r F ˘´δq ´ż x s´β x t´β p r F ˘´δq ˇď pκ 1 `max π ´1pBpx t´β , 1qq | r F ´δ|q dpy t , x t´β q κ 2 `pκ 1 `max π ´1pBpys, 1qq | r F ´δ| q dpy s , x s`β q κ 2 .
Again by the subexponential growth of r F , the above expression converges to 0 as t Ñ `8 uniformly in s, hence lim sÑ`8 a s exists by a Cauchy type of argument.

(2) Let pξ, x, yq P B 8 X ˆX ˆX. Assertion (2) of Proposition 3.10 follows from Equation (3.6) when X is an R-tree, since rx, ys " rx, ps Y rp, ys where p be the closest point to y on rx, ξr. When X is not an R-tree, Assertion (2) follows immediately from the HC-property of pX, Γ, r F ˘´δq.

(3) Let pξ, x, yq, pξ 1 , x 1 , y 1 q P B 8 X ˆX ˆX. By the cocycle property (3.7), we have

|C ξ px, yq ´Cξ 1 px 1 , y 1 q| ď |C ξ px, yq ´Cξ 1 px, yq| `|C ξ1 px, x 1 q| `|C ξ1 py, y 1 q| . (3.9) 48 19/12/2016
First assume that X is an R-tree. Let K be a compact subset of X, and let

K " inf x,yPK
e ´dpx, x 0 q´dpx, yq ą 0 .

x ξ 1 ξ p q y Let p, q be the points in X such that rx, ξr X ry, ξr " rp, ξr and rx, ξr X rx, ξ 1 r " rx, qs. If d x 0 pξ, ξ 1 q ď K , then by the definition of the visual distance and by Equation (2.2), we have e ´dpx, qq " d x pξ, ξ 1 q ď e ´dpx, yq ď e ´dpx, pq .

In particular q P rp, ξr, so that rx, ξ 1 r X ry, ξ 1 r " rp, ξ 1 r. Thus by Equation (3.6), we have

| C ξ px, yq ´Cξ 1 px, yq | " 0 .
Therefore, by Equation (3.9) and by the R-tree case of Assertion (2), if d x 0 pξ, ξ 1 q ď K , if x, y P K and dpx, x 1 q, dpy, y 1 q ď 1, then

|C ξ px, yq ´Cξ 1 px 1 , y 1 q| ď dpx, x 1 q max π ´1prx, x 1 sq | r F ˘´δ| `dpy, y 1 q max π ´1pry, y 1 sq | r F ˘´δ| .
Since r F is bounded on compact subsets of T 1 X, this proves that C ˘is locally Lipschitz.

Let us now consider the case when X is general. For all distinct ξ, ξ 1 P B 8 X, let t Þ Ñ ξ t and t Þ Ñ ξ 1 t be the geodesic rays from x 0 to ξ and ξ 1 respectively. By the end of the proof of Assertion (1), for every compact subset K of X, there exists a 1 , a 2 ą 0 such that for every x, y P K, we have for η P tξ, ξ 1 u,

ˇˇC η px, yq ´´ż ηt y p r F ˘´δq ´ż ηt x p r F ˘´δq ¯ˇˇď a 1 e ´a2 t .
Let T " ´1 2 ln d x 0 pξ, ξ 1 q. If T ě 0, by the properties of CATp´1q-spaces, there exist constants a 3 , a 4 ą 0 such that dpξ 2T , ξ 1 2T q ď a 3 and dpξ T , ξ 1 T q ď a 4 e ´T . Hence by Assertion (2), if

d x 0 pξ, ξ 1 q ď mint 1 a 2 4
, 1u (so that T ě 0 and dpξ T , ξ 1 T q ď 1), we have

| C ξ px, yq ´Cξ 1 px, yq | ď ˇˇż ξ T y p r F ˘´δq ´ż ξ 1 T y p r F ˘´δq ˇˇ`ˇˇż ξ T x p r F ˘´δq ´ż ξ 1 T x p r F ˘´δq ˇˇ`2 a 1 e ´a2 T ď2 pκ 1 `max π ´1pBpξ T , a 4 qq | r F ˘´δ|q dpξ T , ξ 1 T q κ 2 `2 a 1 e ´a2 T .
By the subexponential growth of r F , there exists a 5 ą 0 such that

| C ξ px, yq ´Cξ 1 px, yq | ď a 5 e ´κ2 2 T `2 a 1 e ´a2 T ď pa 5 `2 a 1 q d x 0 pξ, ξ 1 q mint κ 2 4 , a 2 2 u .
We now conclude from Equation (3.9) and Assertion (2) as in the end of the above tree case that C ˘is locally Hölder-continuous. | r F ˘´δ| .

In the general case, the result then follows similarly from Equation (3.10) by using Assertion (2) and the HC-property. l

Systems of conductances on trees and generalised electrical networks

Let pX, λq be a locally finite metric tree without terminal vertices, let X " |X| λ be its geometric realisation, and let Γ be a nonelementary discrete subgroup of IsompX, λq.

Let r c : EX Ñ R be a Γ-invariant function, called a system of (logarithmic) conductances for Γ. We denote by c : ΓzEX Ñ R the function induced by r c : EX Ñ R, which we also call a system of conductances on ΓzX.

Classically, an electric network5 (without sources or reactive elements) is a pair pG, e c q, where G is a graph and c : EG Ñ R a function, such that c is reversible: cpeq " cpeq for all e P EG, see for example [NaW], [Zem]. In this text, we do not assume our system of conductances r c to be reversible. In Chapter 6, we will even sometimes assume that the system of conductances is anti-reversible, that is, satisfying cpeq " ´cpeq for every e P EX.

Two systems of conductances r c, r c 1 : EX Ñ R are said to be cohomologous, if there exists a Γ-invariant map f :

V X Ñ R such that r c 1 ´r c " df ,
where for all e P EX, we have df peq " f ptpeqq ´f popeqq λpeq .

Proposition 3.11. Let r c : EX Ñ R be a system of conductances for Γ. There exists a potential r F on T 1 X for Γ such that for all x, y P V X, if pe 1 , . . . , e n q is the edge path in X without backtracking such that x " ope 1 q and y " tpe n q, then

ż y x r F " n ÿ i"1 r cpe i q λpe i q .
Proof. Any germ v P T 1 X determines a unique edge e v of the tree X, the first one into which it enters: if is any geodesic line whose class in T 1 X is v, the edge e v is the unique edge of X containing πpvq whose terminal vertex is the first vertex of X encountered at a positive time by . The function r F : T 1 X Ñ R defined by r F pvq " 4 r cpe v q λpe v q min dpπpvq, ope v qq, dpπpvq, tpe v qq ( (3.11) is a potential on the R-tree X, with r F pvq " 0 if πpvq P V X. Let us now compute ş y x r F , for all x, y P X. For every λ ą 0, let ψ λ : r0, λs Ñ R be the continuous map defined by ψ λ ptq " t 2 2 if t P r0, λ 2 s and ψ λ ptq " λ 2 4 ´pλ´tq 2 2 if t P r λ 2 , λs. Let pe 0 , e 1 , . . . , e n q be the edge path in X without backtracking such that x P e 0 ´ttpe 0 qu and y P e n ´tope n qu. An easy computation shows that ż y x r F " n´1 ÿ i"0 r cpe i q λpe i q `4 r cpe n q λpe n q ψ λpenq `dpy, ope n qq ˘´4 r cpe 0 q λpe 0 q ψ λpe 0 q `dpx, ope 0 qq ˘.

If x and y are vertices, the expression simplifies to the sum of the lengths of the edges weighted by the conductances. l

We denote by r F c the potential defined by Equation (3.11) in the above proof, and by F c : ΓzT 1 X Ñ R the induced potential. Note that F c is bounded if c is bounded. We call r F c and F c the potentials associated with the system of conductances r c and c. This is by no means the unique potential with the property required in Proposition 3.11. The following result proves that the choice is unimportant.

Given a potential r F : T 1 X Ñ R for Γ, let us define a map r c F : EX Ñ R by

r c F : e Þ Ñ r c F peq " 1 λpeq ż tpeq opeq r F .
(3.12)

Note that r c F is Γ-invariant, hence it is a system of conductances for Γ. We denote by c F : ΓzEX Ñ R the function induced by r c F : EX Ñ R. Note that r c F `κ " r c F `κ for every constant κ P R, that r c F is bounded if r F is bounded, and that c Fc " c by the above proposition.

Proposition 3.12. (1) Every potential for Γ is cohomologous to a potential associated with a system of conductances for Γ.

(2) If two systems of conductances r c 1 and r c are cohomologous, then their associated potentials r F c 1 and r F c are cohomologous.

(3) If X has no vertex of degree 2, if two potentials r F ˚and r F for Γ are cohomologous, then the systems of conductances r c F ˚and r c F for Γ are cohomologous.

Hence if X has no vertex of degree 2, the map rF s Þ Ñ rc F s from the set of cohomology classes of potentials for Γ to the set of cohomology classes of systems of conductances for Γ is bijective, with inverse rcs Þ Ñ rF c s.

Proof. (1) Let r F be a potential for Γ, and let r F ˚" r F c F be the potential associated with the system of conductances r c F . For all e P EX and t P s0, λpeqr , let v e, t P T 1 X be the germ of any geodesic line passing at time 0 through the point of e at distance t from opeq. Let 51 19/12/2016 r G : T 1 X Ñ R be the map defined by Gpvq " 0 if πpvq P V X and such that for all e P EX and t P s0, λpeqr , r Gpv e, t q " ż t 0 p r F ˚pv e, s q ´r F pv e, s qq ds .

Since ş tpeq opeq r F " λpeq r c F peq by construction and λpeq r c F peq " ş tpeq opeq r F ˚by Proposition 3.11, the map r G : T 1 X Ñ R is continuous. Let be a geodesic line. The map t Þ Ñ r Gpv g t q is obviously differentiable at time t " 0 if πp q R V X, with derivative r F ˚pv q ´r F pv q. Since r F ˚pvq vanishes if πpvq P V X by Equation (3.11), and by continuity of r F at such a point, this is still true if πp q P V X. Hence r F ˚and r F are cohomologous, and this proves the first claim.

(2) Assume that r c 1 and r c are cohomologous systems of conductances for Γ, and let f : V X Ñ R be a Γ-invariant function such that r c 1 ´r c " df . For all e P EX and t P s0, λpeqr , define r Gpv e, t q " λpeq ż t 0 p r F c 1 pv e, s q ´r F c pv e, s qq ds `f popeqq , which is Γ-invariant, whose limit as t Ñ 0 is f popeqq (independent of the edge e with given origin), and whose limit as t Ñ λpeq is λpeq `r c 1 peq ´r cpeq ˘`f popeqq " λpeq df peq `f popeqq " f ptpeqq (independent of the edge e with given extremity). As above, this proves that r G is continuous, and that F c 1 and F c are cohomologous.

(3) In order to prove the third claim, assume that r F ˚and r F are two cohomologous potentials for Γ, and let r G : T 1 X Ñ R be as in the definition of cohomologous potentials, see Equation (3.5). By the continuity of r G, for all elements v and v 1 in T 1 X such that πpvq " πpv 1 q P V X, we have r Gpvq " r Gpv 1 q, since (by the assumption on the degrees of vertices) the two edges (possibly equal) into which v and v 1 enter can be extended to geodesic lines with a common negative subray. Hence for every x P V X, the value f pxq " r Gpv x q for every v x P T 1 X such that πpv x q " x does not depend on the choice of v x . The map f : V X Ñ R thus defined is Γ-invariant. With the above notation and by Equation (3.12), we hence have, for every e P EX, r c F ˚peq ´r c F peq " 1 λpeq ż λpeq 0 p r F ˚pv e, t q ´r F pv e, t q dt " 1 λpeq

ż λpeq 0 d dt r Gpv e, t q dt " 1 λpeq
`r Gpv tpeq q ´r Gpv opeq q ˘" f ptpeqq ´f popeqq λpeq " df peq .

Hence r c F ˚and r c F are cohomologous. l

Given a metric tree pX, λq, we define the critical exponent of a Γ-invariant system of conductances r c : EX Ñ R (or of the induced system of conductances c : ΓzEX Ñ R) as the critical exponent of pΓ, r F c q where r F c is the potential for Γ associated with r c :

δ c " δ Γ, Fc .
By Proposition 3.12 (2) and Lemma 3.7 (1), this does not depend on the choice of a potential Patterson-Sullivan and Bowen-Margulis measures with potential on CATp´1q spaces Let X, x 0 , Γ be as in the beginning of Section 2.2,1 and let r F be a potential for Γ. From now on, we assume that the triple pX, Γ, r F q satisfies the HC-property of Definition 3.4 and that the critical exponent δ " δ Γ, F ˘is finite.

In this chapter, we discuss geometrically and dynamically relevant measures on the boundary at infinity of X and on the space of geodesic lines G X. We extend the theory of Gibbs measures from the case of manifolds with pinched negative sectional curvature treated in [PauPS] 2 to CATp´1q spaces with the HC-property.

Patterson densities

A family pµ x q xPX of finite nonzero (positive Borel) measures on B 8 X, whose support is ΛΓ, is a (normalised) In particular, the measures µ x are in the same measure class for all x P X, and, by Proposition 3.10, they depend continuously on x for the weak-star convergence of measures. Note that a Patterson density for pΓ, F ˘q is also a Patterson density for pΓ, F ˘`sq for every s P R, since the definition involves only the normalised potential r F ˘´δ. If F " 0, we get the usual notion of a Patterson-Sullivan density (of dimension δ Γ ) for the group Γ, see for instance [START_REF] Patterson | The limit set of a Fuchsian group[END_REF][START_REF] Sullivan | The density at infinity of a discrete group of hyperbolic motions[END_REF][START_REF] Nicholls | The ergodic theory of discrete groups[END_REF][START_REF] Coornaert | Mesures de Patterson-Sullivan sur le bord d'un espace hyperbolique au sens de Gromov[END_REF][START_REF] Bourdon | Structure conforme au bord et flot géodésique d'un CATp´1q espace[END_REF][START_REF] Roblin | Ergodicité et équidistribution en courbure négative[END_REF]. Proof. The Patterson construction (see [START_REF] Patterson | A lattice-point problem in hyperbolic space[END_REF], [Coo]) modified as in [START_REF] Paulin | Equilibrium states in negative curvature[END_REF]Section 3.6] gives the result. l

We refer to Theorem 4.5 for the uniqueness up to scalar multiple of the Patterson density when pΓ, F ˘q is of divergence type and to [START_REF] Das | Geometry and dynamics in Gromov hyperbolic metric spaces, with an emphasis on non-proper settings[END_REF]Coro. 17.1.8] for a characterisation of the uniqueness.

The Patterson densities satisfy the following extension of the classical Sullivan shadow lemma (which gives the claim when r F is constant, see [START_REF] Roblin | Ergodicité et équidistribution en courbure négative[END_REF]), and its corollaries. If µ is a Borel positive measure on a metric space pX, dq, the triple pX, d, µq is called a metric measure space. A metric measure space pX, d, µq is doubling if there exists c ě 1 such that for all x P X and r ą 0 µpBpx, 2 rqq ď c µpBpx, rqq .

Note that, up to changing c, the number 2 may be replaced by any constant larger than 1. See for instance [Hei]) for more details on doubling metric measure spaces. We refer for instance to [START_REF] Das | Geometry and dynamics in Gromov hyperbolic metric spaces, with an emphasis on non-proper settings[END_REF]Ex. 17.4.12] for examples of non-doubling Patterson(-Sullivan) measures, and to [START_REF] Das | Geometry and dynamics in Gromov hyperbolic metric spaces, with an emphasis on non-proper settings[END_REF]Prop. 17.4.4] for a characterisation of the doubling property of the Patterson measures when Γ is geometrically finite and F " 0.

A family pX, µ i , d i q iPI of Borel positive measures µ i and distances d i on a common set X is called uniformly doubling if there exists c ě 1 such that for all i P I, x P X and r ą 0

µ i pB d i px, 2 rqq ď c µ i pB d i px, rqq .
Lemma 4.2. Let pµ x q xPX be a Patterson density for the pair pΓ, F ˘q, and let K be a compact subset of X.

(1) [Mohsen's shadow lemma] If R is large enough, there exists C ą 0 such that for all γ P Γ and x, y P K,

1 C e ş γy x p r F ˘´δq ď µ x `Ox Bpγy, Rq ˘ď C e ş γy x p r F ˘´δq .
(2) For all x, y P X, there exists c ą 0 such that for every n P N (3) For every R ą 0 large enough, there exists C " CpRq ą 0 such that for all γ P Γ and all x, y P K µ x pO x Bpγy, 5Rqq ď C µ x pO x Bpγy, Rqq .

(4) If Γ is convex-cocompact, then the metric measure space pΛΓ, d x , µ x is doubling for every x inX, and the family of metric measure spaces pΛΓ, d x , µ x q xPCΛΓ is uniformly doubling.

19/12/2016

Proof. For the first assertion, the proof of [START_REF] Paulin | Equilibrium states in negative curvature[END_REF]Lem. 3.10] (see also [START_REF] Coudène | Gibbs measures on negatively curved manifolds[END_REF]Lem. 4] with the multiplicative rather than additive convention, as well as [Moh]) extends, using Proposition 3.10 (2), (4) instead of [START_REF] Paulin | Equilibrium states in negative curvature[END_REF]Lem. 3.4 (i),(ii)]. The second assertion is similar to the one of [START_REF] Paulin | Equilibrium states in negative curvature[END_REF]Lem. 3.11 (i)], and the proof of the last two assertions is similar to the one of [START_REF] Paulin | Equilibrium states in negative curvature[END_REF]Prop. 3.12], using Lemma 2.2 instead of [START_REF] Hersonsky | On the almost sure spiraling of geodesics in negatively curved manifolds[END_REF]Lem. 2.1]. The uniformity in the last assertion follows from the compactness of ΓzCΛΓ and the invariance and continuity properties of the Patterson densities. l

Gibbs measures

We fix from now on two Patterson densities pµ x q xPX for the pairs pΓ, F ˘q.

The Gibbs measure r m F on G X (associated with this ordered pair of Patterson densities) is the measure r m F on G X given by the density d r m F p q " e C ´ ´px 0 , p0qq `C`

`px 0 , p0qq dµ x0 p ´q dµ x0 p `q dt (4.3) in Hopf's parametrisation with respect to the basepoint x 0 . The Gibbs measure r m F is independent of x 0 , and it is invariant under the actions of the group Γ and of the geodesic flow. Thus, it defines a measure m F on ΓzG X which is invariant under the quotient geodesic flow, called the Gibbs measure on ΓzG X. The proofs of these claims are analogous to the ones of [START_REF] Paulin | Equilibrium states in negative curvature[END_REF]§3.7]. If F " 0 and the Patterson densities pµ x q xPX and pµ x q xPX coincide, the Gibbs measure m F coincides with the Bowen-Margulis measure m BM (associated with this Patterson density), see for instance [START_REF] Roblin | Ergodicité et équidistribution en courbure négative[END_REF].

Remark 4.3. The (Borel positive) measure given by the density dλpξ, ηq " e C ξ px 0 , pq `Cὴ px 0 , pq dµ x0 pξq dµ x0 pηq (4.4) on B 2 8 X is independent of p P sξ, ηr , locally finite and invariant under the diagonal action of Γ on B 2 8 X. It is a geodesic current for the action of Γ on the Gromov-hyperbolic proper metric space X in the sense of Ruelle-Sullivan-Bonahon, see for instance [Bon] and references therein.

Let us now indicate why the terminology of Gibbs measures is indeed appropriate. This explanation will be the point of Proposition 4.4, but we need to give some definitions first.

For all P G X and r ą 0, T, T 1 ě 0, the dynamical (or Bowen) ball around is

Bp ; T, T 1 , rq " 1 P G X : sup tPr´T 1 ,T s dp ptq, 1 ptqq ă r ( .
Bowen balls have the following invariance properties: for all s P r´T 1 , T s and γ P Γ, g s Bp ; T, T 1 , rq " Bpg s ; T ´s, T 1 `s, rq and γBp ; T, T 1 , rq " Bpγ ; T, T 1 , rq .

The following inclusion properties of the dynamical balls are immediate: If r ď s, T ě S, T 1 ě S 1 , then Bp ; T, T 1 , rq is contained in Bp ; S, S 1 , sq. The dynamical balls are almost independent on r (see [START_REF] Paulin | Equilibrium states in negative curvature[END_REF]Lem. 3.14]): For all r 1 ě r ą 0, there exists T r, r 1 ě 0 such that 55 19/12/2016 for all P G X and T, T 1 ě 0, the dynamical ball Bp ; T `Tr, r 1 , T 1 `Tr, r 1 , r 1 q is contained in Bp ; T, T 1 , rq.

For every P ΓzG X, let us define Bp ; T, T 1 , r 1 q as the image by the canonical projection G X Ñ ΓzG X of Bp r ; T, T 1 , r 1 q, for any preimage r of in G X.

A pg t q tPR -invariant measure m 1 on ΓzG X satisfies the Gibbs property for the potential F with constant cpF q P R if for every compact subset K of ΓzG X, there exist r ą 0 and c K, r ě 1 such that for all large enough T, T 1 ě 0, for every P ΓzG X with g ´T 1 , g T P K, we have

1 c K, r ď m 1 pBp ; T, T 1 , rqq e ş T ´T 1 pF pv g t q´cpF qq dt ď c K, r .
We refer to [START_REF] Paulin | Equilibrium states in negative curvature[END_REF]Sect. 3.8] for equivalent variations on the definition of the Gibbs property. The following result shows that the Gibbs measures indeed satisfy the Gibbs property on the dynamical balls of the geodesic flow, thereby justifying their names. We refer for instance to [START_REF] Paulin | Equilibrium states in negative curvature[END_REF]Sect. 3.8] for the explanations of the connection with symbolic dynamics mentioned in the introduction. See also Proposition 4.12 for a discussion of the case when X is a simplicial tree -here the correspondence with symbolic dynamics is particularly clear.

Proposition 4.4. Let m F be the Gibbs measure on ΓzG X associated with a pair of Patterson densities pµ x q xP Ă M for pΓ, r F ˘q. Then m F satisfies the Gibbs property for the potential F , with constant cpF q " δ.

Proof. The proof is similar to the one of [START_REF] Paulin | Equilibrium states in negative curvature[END_REF]Prop. 3.16] (in which the key Lemma 3.17 uses only CATp´1q arguments), up to replacing the use of [START_REF] Paulin | Equilibrium states in negative curvature[END_REF]Lem. 3.4 (1)] by finitely many applications of Proposition 3.10 (2). l

The basic ergodic properties of the Gibbs measures are summarised in the following result. The case when r F is constant is due to [START_REF] Roblin | Ergodicité et équidistribution en courbure négative[END_REF], see also [START_REF] Burger | CAT(´1) spaces, divergence groups and their commensurators[END_REF]§6].

Theorem 4.5 (Hopf-Tsuji-Sullivan-Roblin). The following conditions are equivalent (i) The pair pΓ, F q is of divergence type.

(ii) ´The conical limit set of Γ has positive measure with respect to µ x for some (equivalently every) x P X.

(ii) `The conical limit set of Γ has positive measure with respect to µ x for some (equivalently every) x P X.

(iii) The dynamical system pB 2 8 X, Γ, pµ x b µ x q |B 2 8 X q is ergodic and conservative for some (equivalently every) x P X.

(iv) The dynamical system pΓzG X, pg t q tPR , m F q is ergodic and conservative.

If one of the above conditions is satisfied, then

(1) the measures µ x have no atoms for any x P X,

(2) the diagonal of B 8 X ˆB8 X has measure 0 for µ x b µ x ,

(3) the Patterson densities pµ x q xPX are unique up to a scalar multiple, and ˘" ope δ n q .

56
Proof. The proof of the equivalence claim is similar to the one of [START_REF] Paulin | Equilibrium states in negative curvature[END_REF]Theo. 5.4], using Proposition 3.10 (2), (4) instead of [START_REF] Paulin | Equilibrium states in negative curvature[END_REF]Lem. 3.4], and Lemma 4.2 instead of [START_REF] Paulin | Equilibrium states in negative curvature[END_REF]Lem. 3.10]. The claims (1) to (4) are proved as in [START_REF] Paulin | Equilibrium states in negative curvature[END_REF]Sect. 5.3]. l

The following corollary follows immediately from Poincaré's recurrence theorem and the Hopf-Tsuji-Sullivan-Roblin theorem, see [START_REF] Paulin | Equilibrium states in negative curvature[END_REF]Thm. 5.15] for the argument written for the manifold case.

Corollary 4.6. If m F is finite, then

(1) the pair pΓ, F ˘q is of divergence type,

(2) the Patterson densities pµ x q xPX are unique up to a multiplicative constant and the Gibbs measure m F is uniquely defined up to a multiplicative constant.

(3) the support of m F is the image ΩΓ in ΓzG X of r ΩΓ " t P G X : ˘P Λ c Γu, and

(4) the geodesic flow is ergodic for m F . l

As the finiteness of the Gibbs measures will be a standing hypothesis in many of the following results, we now give criteria for Gibbs measures to be finite. Recall (see Section 2.2) that the discrete nonelementary group of isometries Γ of X is geometrically finite if every element of ΛΓ is either a conical limit point or a bounded parabolic limit point of Γ.

Theorem 4.7. Assume that Γ is a geometrically finite discrete group of isometries of X.

(1) If pΓ, F ˘q is of divergence type, then the Gibbs measure m F is finite if and only if for every bounded parabolic limit point p of Γ, the series ÿ αPΓp dpx, αyq e ş αy x p r F ˘´δq converges, where Γ p is the stabiliser of p in Γ.

(2) If we have δ Γp, F p ă δ, for every bounded parabolic limit point p of Γ with stabiliser Γ p in Γ and with F p : Γ p zX Ñ R the map induced by r F ˘, then pΓ, F q is of divergence type. In particular, m F is finite.

When X is a manifold, this result is due to [DaOP, Theo. B] for the case F " 0, and to [Cou] and [START_REF] Paulin | Equilibrium states in negative curvature[END_REF]Theo. 8.3,8.4] for the general case of Hölder-continuous potentials. When F " 0 but on much more general assumptions on X with optimal generality, this result is due to [START_REF] Das | Geometry and dynamics in Gromov hyperbolic metric spaces, with an emphasis on non-proper settings[END_REF]Theo. 17.1.2].

Proof. The proof is similar to the manifold case in [PauPS], which follows closely the proof of [DaOP]. Note that the convergence or divergence of the above series does not depend on the choice of the sign ˘.

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Let Par Γ be the set of bounded parabolic limit points of Γ. By [Rob2, Lem. 1.9] 3 , there exists a Γ-equivariant family pH p q pPPar Γ of pairwise disjoint closed horoballs, with H p centred at p, such that the quotient M 0 " Γz `C ΛΓ ´ď pPPar Γ H p ȋs compact. Using Theorem 4.5, the HC-property and Proposition 3.10 instead of [START_REF] Paulin | Equilibrium states in negative curvature[END_REF]Lem. 3.4], the proofs of [START_REF] Paulin | Equilibrium states in negative curvature[END_REF]Theo. 8.3,8.4] then extend to our situation. l

Recall that the length spectrum of Γ on X is the subgroup of R generated by the translation lengths in X of the elements of Γ.

Recall that a continuous-time 1-parameter group ph t q tPR of homeomorphisms of a topological space Z is topologically mixing if for all nonempty open subsets U, V of Z, there exists t 0 P R such that for all t ě t 0 , we have U X h t V ‰ H.

We have the following result, due to [START_REF] Babillot | On the mixing property for hyperbolic systems[END_REF]Thm. 1] in the manifold case, with developments by [START_REF] Roblin | Ergodicité et équidistribution en courbure négative[END_REF] when r F " 0, and by [START_REF] Paulin | Equilibrium states in negative curvature[END_REF]Sect. 8] for manifolds with pinched negative curvature.

Theorem 4.8. If the Gibbs measure is finite, then the following assertions are equivalent :

(1) the geodesic flow of ΓzX is mixing for the Gibbs measure,

(2) the geodesic flow of ΓzX is topologically mixing on its nonwandering set, which is the quotient under Γ of the space of geodesic lines in X both of whose endpoints belong to ΛΓ.

(3) the length spectrum of Γ on X is not contained in a discrete subgroup of R. l

In the manifold case, the third assertion of Theorem 4.8 is satisfied, for example, if Γ has a parabolic element, if ΛΓ is not totally disconnected (hence if ΓzX is compact), or if X is a surface or a (rank-one) symmetric space, see for instance [START_REF] Dal | Remarques sur le spectre des longueurs d'une surface et comptage[END_REF][START_REF] Dal | Topologie du feuilletage fortement stable[END_REF].

Error terms for the mixing property will be described in Chapter 9. The above result holds for the continuous time geodesic flow when X is a metric tree. See Proposition 4.15 for a version of this theorem for the discrete time geodesic flow on simplicial trees. At least when X is an R-tree and Γ is a uniform lattice (so that ΓzX is a finite metric graph), we have a stronger result under additional regularity assumptions, see Section 9.2.

We end this Section by an elementary remark on the independence of Gibbs measures upon replacement of the potential F by a cohomologous one.

Remark 4.9. Let r F ˚: T 1 X Ñ R be a potential for Γ cohomologous to r F and satisfying the (HC)-property. As usual, let r F ˚`" r F ˚and r F ˚´" r F ˚˝ι, and let F ˚: ΓzT 1 X Ñ R be the induced map. Let r G : T 1 X Ñ R be a continuous Γ-invariant function such that, for every P G X, the map t Þ Ñ r Gpv g t q is differentiable and r F ˚pv q ´r F pv q " d dt |t"0 r Gpv g t q. For all x P X and ξ P B 8 X, let x, ξ be any geodesic line with footpoint x, ξ p0q " x and endpoint p x, ξ q `" ξ, and let ξ, x be any geodesic line with ξ, x p0q " x and origin p ξ, x q ´" ξ. Note that the value r Gpv x, ξ q is independent of the choice of x, ξ , by the continuity of r G, and similarly for r Gpv ξ, x q. In particular, for all γ P Γ, by the Γ-invariance of r G, we have r Gpv x, γ ´1ξ q " r Gpv γx, ξ q and r G ˝ι pv x, ξ q " r Gpv ξ, x q . (4.5)

3 See also [START_REF] Paulin | Groupes géométriquement finis d'automorphismes d'arbres et approximation diophantienne dans les arbres[END_REF] for the case of simplicial trees and [START_REF] Das | Geometry and dynamics in Gromov hyperbolic metric spaces, with an emphasis on non-proper settings[END_REF]Thm. 12.4.5] for a greater generality on X.

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Note that r F ˚´" r F ˚˝ι and r F ´" r F ˝ι are cohomologous, since if r G ˚" ´r G ˝ι, for every P G X, we have r F ˚˝ιpv q ´r F ˝ιpv q " r F ˚pv ι q ´r F pv ι q "

d dt |t"0 r Gpv g t ι q " d dt |t"0 r Gpιv g ´t q " d dt |t"0 r G ˚pv g t q .
As already seen in Lemma 3.7, the critical exponent δ Γ, F ˚˘is equal to the critical exponent δ Γ, F ˘, and independent of the choice of ˘, and we denote by δ the common value in the definition of the Gibbs cocycle.

Note that if C ˚˘" C Γ, F ˚˘is the Gibbs cocycle associated with pΓ, F ˚˘q, then C ˚˘and C ˘are cohomologous:

C ˚ξ px, yq ´Cξ px, yq " r Gpv y, ξ q ´r Gpv x, ξ q , (4.6)

and similarly C ˚ξ px, yq ´Cξ px, yq " r G ˚pv y, ξ q ´r G ˚pv x, ξ q . (4.7)

Let pµ x q xP Ă M be a Patterson density for pΓ, F ˘q. In order to simplify the notation, let r G `" r G and r G ´" r G ˚. The family of measures pµ ˚x q xPX defined by setting, for all x P X and ξ P B 8 X,

dµ ˚x pξq " e r G ˘pv x, ξ q dµ x pξq , (4.8)
is also a Patterson density for pΓ, F ˚˘q. Indeed, the equivariance property (4.1) for pµ ˚x q xPX follows from the one for pµ x q xPX and from Equation (4.5). The absolutely continuous property (4.2) for pµ ˚x q xPX follows from the one for pµ x q xPX and Equations (4.6) and (4.7). Assume that the Patterson density for pΓ, F ˚˘q defined by Equation (4.8) is chosen in order to construct the Gibbs measure r m F ˚for pΓ, F ˚q on G X. Then using ' Hopf's parametrisation with respect to the base point x 0 and Equation (4.3) with F replaced by F ˚for the first equality, ' Equations (4.6), (4.7), (4.8) and cancellations for the second equality, ' the definition of r G ˚" ´r G ˝ι and again Equation (4.3) for the third equality, ' Equation (4.5) so that ι v p0q, ´" v ´, p0q , and the fact that we may choose ´, p0q " and p0q, `" for the last equality, we have d r m F ˚p q " e C ˚´ ´px 0 , p0qq `C˚`

`px 0 , p0qq dµ ˚x 0 p ´q dµ ˚x 0 p `q dt " e

C ´ ´px 0 , p0qq `r G ˚pv p0q, ´q `C`

`px 0 , p0qq `r Gpv p0q, ´q dµ x0 p ´q dµ x0 p `q dt " e ´r G˝ιpv p0q, ´q `r Gpv p0q, ´q d r m F p q " e ´r Gpv q`r Gpv q d r m F p q , hence r m F ˚" r m F . In particular, since the Gibbs measure, when finite, is independent up to a multiplicative constant on the choice of the

Patterson densities for simplicial trees

In this Section and the following one, we specialise and modify the general framework of the previous sections to treat simplicial trees. Recall 4 that a simplicial tree X is a metric tree whose edge length map is constant equal to 1. The time 1 map of the geodesic flow pg t q tPR on the space p G X of all generalised geodesic lines of the geometric realisation X " |X| 1 of X preserves for instance its subset of generalised geodesic lines whose footpoints are at distance at most 1{4 from vertices. Since both this subset and its complement have nonempty interior in p G X, the geodesic flow on p G X has no mixing or ergodic measure with full support. This is why we considered the discrete time geodesic flow pg t q tPZ on p G X in Section 2.7.

Let X be a locally finite simplicial tree without terminal vertices, and let X " |X| 1 be its geometric realisation. Let Γ be a nonelementary discrete subgroup of AutpXq. Let r F : T 1 X Ñ R be a potential for Γ, and let r F `" r F , r F ´" r F ˝ι. Let C ˘: B 8 X ˆX ˆX Ñ R be the associated Gibbs cocycles. Let pµ x q xPX be two Patterson densities on B 8 X for the pairs pΓ, F ˘q Note that only the restrictions of the cocycles C ˘to B 8 X ˆV X ˆV X are useful and that it is often convenient and always sufficient to replace the cocycles by finite sums involving a system of conductances (as defined in Section 3.5), see below. Furthermore, only the restriction pµ x q xPV X of the family of Patterson densities to the set of vertices of X is useful.

Example 4.10. Let X be a simplicial tree and let r c : EX Ñ R be a system of conductances on X. For all x, y in V X and ξ P B 8 X, with the usual convention on the empty sums, let

c ξ px, yq " m ÿ i"1 r cpe i q ´n ÿ j"1
r cpf j q and c ξ px, yq "

m ÿ i"1 r cpe i q ´n ÿ j"1 r cpf j q ,
where, if p P V X is such that rp, ξr " rx, ξr X ry, ξr , then pe 1 , e 2 , . . . , e m q is the geodesic edge path in X from x " ope 1 q to p " tpe m q and pf 1 , f 2 , . . . , f n q is the geodesic edge path in X from v " opf 1 q to p " tpf n q.

f 1 f n ξ p x e 1 e m y f 2
With δ c defined in the end of Section 3.5 and with C ˘the Gibbs cocycles for pΓ, r F c q, by Proposition 3.11, we have, for all ξ P B 8 X and x, y P V X, C ξ px, yq " ´cξ px, yq `δc dpx, yq , and Equation (4.2) gives dµ x pξq " e c ξ px,yq´δc dpx,yq dµ y pξq .

4 See Section 2.7.

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Using the particular structure of trees, we can prove a version of the Shadow Lemma 4.2 where one can take the radius R to be 0. When F " 0, this result is due to Coornaert [Coo].

Lemma 4.11 (Mohsen's shadow lemma for trees). Let K be a finite subset of V X. There exists C ą 0 such that for all γ P Γ and x, y P K with y P C ΛΓ, we have

1 C e ş γy x p r F ˘´δq ď µ x `Ox tγyu ˘ď C e ş γy x p r F ˘´δq .
Proof. The structure of the proof is the same as in Lemma 4.2 with some differences in the details towards the end of the argument. Note that C ξ px, γyq `şγy x p r F ˘´δq " 0 if ξ P O x tγyu (that is, if γy P rx, ξr ), by Equation (3.8).

First, one argues as in the proof of [START_REF] Paulin | Equilibrium states in negative curvature[END_REF]Lem. 3.10], that it suffices to prove that there exists C ą 0 such that for all γ P Γ and x, y P K with γy P C ΛΓ, we have

1 C ď µ γy `Ox tγyu ˘ď C .
Now, the argument for proving the upper bound is the same as in loc. cit., using Proposition 3.10 (2) instead of [START_REF] Paulin | Equilibrium states in negative curvature[END_REF]Lem. 3.4 (i)].

In order to prove the lower bound, we assume by contradiction that there exist sequences px i q iPN , py i q iPN in K with y i P C ΛΓ and pγ i q iPN in Γ such that µ γi y i pO x i tγ i y i uq converges to 0 as i Ñ `8. Up to extracting a subsequence, since K is finite, we may assume that the sequences px i q iPN and py i q iPN are constant, say with value x and y respectively. Since y P C ΛΓ, as every point in C ΛΓ belongs to at least one geodesic line between two limit points of Γ, the geodesic segment from x to γ i y may be extended to a geodesic ray from x to a limit point. Since the support of µ z is equal to ΛΓ for any z P X, we have µ γi y pO x tγ i yuq ą 0 for all i P N. Thus, up to taking a subsequence, we can assume that γ ´1 i x converges to ξ P ΛΓ (otherwise by discreteness, we may extract a subsequence so that pγ i q iPN is constant, and µ γi y pO x tγ i yuq cannot converge to 0).

Since X is a tree, there exists a positive integer N such that O γ ´1 i x tyu " O γ ´1 N x tyu " O ξ tyu for all i ě N . As above, O ξ tyu meets ΛΓ since y P C ΛΓ, thus µ y pO ξ tyuq ą 0. But for every i ě N , µ y `Oξ tyu ˘" pγ ´1 i q ˚µγ i y `Oγ ´1 i x tyu ˘" µ γi y `Ox tγ i yu tends to 0 as i Ñ `8, a contradiction. l

Let r φ μ : V X Ñ r0, `8r be the total mass functions of the Patterson densities :

r φ μ pxq " }µ x }
for every x P V X. These maps are Γ-invariant by Equation (4.1), hence they induce maps φ μ : ΓzV X Ñ r0, `8r. In the case of real hyperbolic manifolds and vanishing potentials, the total mass functions have important links to the spectrum of the hyperbolic Laplacian (see [START_REF] Sullivan | Related aspects of positivity in Riemannian manifolds[END_REF]). See also [START_REF] Coornaert | Positive eigenfunctions of the Laplacian and conformal densities on homogeneous trees[END_REF][START_REF] Coornaert | Spherical functions and conformal densities on spherically symmetric CATp´1q-spaces[END_REF] for the case of simplicial trees and the discrete Laplacian, Section 6.1 for a generalisation of the result of Coornaert and Papadopoulos, and for instance [BerK] for developments in the field of quantum graphs. 61 19/12/2016

Gibbs measures for metric and simplicial trees

Let pX, λq be a locally finite metric tree without terminal vertices, and let X " |X| λ be its geometric realisation. Let Γ be a nonelementary discrete subgroup of AutpX, λq. Let r F : T 1 X Ñ R be a potential for Γ. Let pµ x q xPV X be two Patterson densities on B 8 X for the pairs pΓ, F ˘q.

The Gibbs measure r m F on the space of discrete geodesic lines G X of X, invariant under Γ and under the discrete time geodesic flow pg t q tPZ of p G X, is defined analogously with the continuous time case, using the discrete Hopf parametrisation for any basepoint x 0 P V X, by d r m F p q " e C ´ ´px 0 , p0qq `C`

`px 0 , p0qq dµ x0 p ´q dµ x0 p `q dt , (4.10)

where now dt is the counting measure on Z. We again denote by m F the measure it induces on ΓzG X.

In this Section, we prove that the Gibbs measures in the case of trees satisfy a Gibbs property even closer to the one in symbolic dynamics, we give an analytic finiteness criterion of the Gibbs measure for metric trees, and recall the ergodic properties of tree lattices.

As recalled in the introduction, Gibbs measures were first introduced in statistical mechanics and consequently in symbolic dynamics, see for example [START_REF] Bowen | Equilibrium states and the ergodic theory of Anosov diffeomorphisms[END_REF], [ParP], [PauPS]. In order to motivate the terminology used here, we recall the definition of a Gibbs measure for the full two-sided shift on a finite alphabet: 5 Let Σ n " t1, 2, . . . , nu Z be the product space of sequences x " px n q nPZ indexed by Z in the finite set t1, 2, . . . , nu, and let σ : Σ n Ñ Σ n be the shift map defined by σppx n q nPZ q " px n`1 q nPZ . A shift-invariant probability measure µ on Σ n satisfies the Gibbs property for an energy function φ :

Σ n Ñ R if 1 C ď µpra ´m´, a ´m´`1 , . . . , a m `´1 , a m `sq e ´P pm ´`m ``1q`ř m k"´m ´φpσ k xq ď C
for some constants C ě 1 and P P R (called the pressure) and for all m ˘in N and x in the cylinder ra ´m´, a ´m´`1 , . . . , a m `´1 , a m `s that consists of those x P Σ n for which x k " a k for all k P r´m ´, m `s. Let x ´, x `P V X and let x 0 P V X X rx ´, x `s. Let us define the tree cylinder of the triple px ´, x 0 , x `q by rx ´, x 0 , x `s " t P G X : ˘P O x 0 tx ˘u, p0q " x 0 u .

These cylinders are close to dynamical balls that have been introduced in Section 4.2, and the parallel with the symbolic case is obvious, as this cylinder is the set of geodesic lines which coincides on r´m ´, m `s, where m ˘" dpx 0 , x ˘q, with a given geodesic line passing through x ˘and through x 0 at time t " 0. The Gibbs measure r m F on the space of discrete geodesic lines G X satisfies a variant of the Gibbs property which is even closer to the one in symbolic dynamics than the general case described in Proposition 4.4. Proposition 4.12 (Gibbs property). Let K be a finite subset of V X X C ΛΓ. There exists C 1 ą 1 such that for all γ P Γ and x ˘P ΓK and all x 0 P V X X rx ´, x `s,

1 C 1 ď r m F prx ´, x 0 , x `sq e ´δ dpx ´, x `q`ş x x ´r F ď C 1
5 See Section 5.1 for the appropriate definition when the alphabet is countable.

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Proof. The result is immediate if dpx ´, x `q is bounded, since the above denominator and numerator take only finitely many values, and the numerator is nonzero since x ˘P C ΛΓ, hence rx ´, x 0 , x `s meets the support of r m F . We may hence assume that dpx ´, x `q ě 2. Using the invariance of r m F under the discrete time geodesic flow, we may thus assume that

x 0 ‰ x ´, x `.
Using the discrete Hopf parametrisation with respect to the vertex x 0 , we have, by Lemma 4.11,

r m F prx ´, x 0 , x `sq " µ x0 pO x 0 tx ´uq µ x0 pO x 0 tx `uq ď C 2 e ş x x 0 p r F ˝ι´δq e ş x x 0 p r F ´δq " C 2 e ş x x ´p r F ´δq .
This proves the upper bound in Proposition 4.12 with C 1 " C 2 and the lower bound follows similarly. l

Next, we give a finiteness criterion of the Gibbs measure for metric trees in terms of the total mass functions of the Patterson densities, extending the case when Γ is torsion free and r F " 0, due to [START_REF] Coornaert | Upper and lower bounds for the mass of the geodesic flow on graphs[END_REF]Theo. 1.1].

Proposition 4.13. Let pX, λ, Γ, r F q be as in the beginning of this Section.

(1) If pX, λq is simplicial and } ¨}2 is the Hilbert norm of L 2 pΓzV X, vol Γz zX q, we have

}m F } ď }φ μ } 2 }φ μ } 2 .
(2) In general, with } ¨}2 the Hilbert norm of L 2 pΓzEX, Tvol Γz zX,λ q, we have 6 }m F } ď }φ μ ˝o} 2 }φ μ ˝o} 2 .

Proof.

(1) The simplicial assumption on pX, λq means that all edges have length 1. The space ΓzG X is the disjoint union of the subsets t P ΓzG X : πp q " p0q " rxsu as the class rxs " Γx of x P V X ranges over ΓzV X. By Equation (4.10), using Hopf's decomposition with respect to the basepoint x, we have dp r m F q |t PG X : p0q"xu p q " dµ x p ´q dµ x p `q .

Let ∆ rxs be the unit Dirac mass at rxs. By ramified covering arguments, we hence have the following equality of measures on the discrete set ΓzV X:

π ˚mF " ÿ rxsPΓzV X 1 |Γ x | µ x ˆµx `tp ´, `q P B 2 8 X : x P s ´, `ru ˘∆rxs . (4.11)
Thus, using the Cauchy-Schwarz inequality,

}m F } " }π ˚mF } ď ÿ rxsPΓzV X 1 |Γ x | }µ x ˆµx } " xφ μ , φ μ y 2 ď }φ μ } 2 }φ μ } 2 .
This proves Assertion (1) of Proposition 4.13.

6 Recall that o : ΓzEX Ñ ΓzV X is the initial vertex map, see Section 2.7.

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(2) The argument is similar to the proof of the simplicial case. Since the singletons in R have zero Lebesgue measure, the space ΓzG X is, up to a measure zero subset for m F , the disjoint union for res P ΓzEX of the sets A res consisting of the elements P ΓzG X such that p0q belongs to the interior of the edge res. We fix a representative e of each res P ΓzEX. For every t P r0, λpeqs, let e t be the point of e at distance t from opeq. By Equation (4.3), using Hopf's decomposition with respect to the basepoint opeq in A res , we have as above Let us give some corollaries of this proposition in the case of simplicial trees. It follows from Assertion (1) of Proposition 4.13 that if the L 2 -norms of the total mass of the Patterson densities are finite, then the Gibbs measure m F is finite. Taking r F " 0 and pµ x q xPV X " pµ x q xPV X , so that the Gibbs measure m F is the Bowen-Margulis measure m BM , it follows from this proposition that

}m F } " ÿ resPΓzEX 1 |Γ e | ż ´PB
}m BM } ď }φ μ } 2 2 ď VolpΓz zXq sup xPV X }µ x } 2 . (4.12)
In particular, if Γ is a lattice in X and if the total mass of the Patterson density is bounded, then the Bowen-Margulis measure m BM is finite.

The following statement summarises the basic ergodic properties of the lattices of X when F " 0.

Proposition 4.14. Let pX, λq be a metric or simplicial tree, with geometric realisation X. Assume that pX, λq is uniform and that Γ is a lattice in AutpX, λq. Then

(1) Γ is of divergence type, and its critical exponent δ Γ is the Hausdorff dimension of any visual distance d x on B 8 X " ΛΓ;

(2) the Patterson density pµ x q xPX coincides, up to a scalar multiple, with the family of Hausdorff measures pµ Haus x q xPX of dimension δ Γ of the visual distances pB 8 X, d x q; in particular, it is AutpX, λq-equivariant: for all γ P AutpX, λq, we have γ ˚µx " µ γx ;

(3) the Bowen-Margulis measure r m BM of Γ on G X is AutpX, λq-invariant, and the Bowen-Margulis measure m BM of Γ on ΓzG X is finite.

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Proof. Let Γ 1 be any uniform lattice of pX, λq, which exists since the metric tree pX, λq is uniform. It is well-known (see for instance [Bou]) that the critical exponent δ Γ 1 of Γ 1 is finite and equal to the Hausdorff dimension of any visual distance pB 8 X, d x q, and that the family pµ Haus

x q xPV X of Hausdorff measures of the visual distances pB 8 X, d x q is a Patterson density for any discrete nonelementary subgroup of AutpX, λq with critical exponent equal to δ Γ 1 .

By [START_REF] Burger | CAT(´1) spaces, divergence groups and their commensurators[END_REF]Coro. 6.5(2)], the lattice Γ in AutpX, λq is of divergence type and δ Γ " δ Γ 1 . By the uniqueness property of the Patterson densities when Γ is of divergence type (see Theorem 4.5), the family pµ x q xPV X coincides, up to a scalar multiple, with pµ Haus x q xPV X . As the graph Γ 1 zX is compact, the total mass function of the Hausdorff measures of the visual distances is bounded, hence so is p}µ x }q xPV X . By Proposition 4.13, since Γ is a lattice of X, this implies that the Bowen-Margulis measure m BM of Γ is finite. l

Note that as in [DaOP], when pX, λq (or its minimal nonempty Γ-invariant subtree) is not assumed to be uniform, there are examples of Γ that are lattices (or are geometrically finite) whose Bowen-Margulis measure m BM is infinite, see Section 15.5 for more details.

Let us now discuss the mixing properties of the discrete time geodesic flow on ΓzG X for the Gibbs measure m F .

Let L Γ be the length spectrum of Γ, which is, in this simplicial case, the subgroup of Z generated by the translation lengths of the elements of Γ.

Since dpx, γxq " 2 dpx, Ax γ q ` pγq if an isometry γ of X is loxodromic and dpx, γxq " 2 dpx, Fix γ q if γ is elliptic with fixed point set Fix γ , the following assertions are equivalent : p1q L Γ Ă 2Z p2q @ x P X, @ γ P Γ, dpx, γxq P 2Z .

(4.13)

Let G even X (respectively p G even X) be the subset of G X (respectively p G X) that consists of the geodesic lines (respectively generalised geodesic lines) in X whose origin is at an even distance from the basepoint x 0 .

Recall that a discrete-time 1-parameter group ph n q nPZ of homeomorphisms of a topological space Z is topologically mixing if for all nonempty open subsets U, V of Z, there exists n 0 P N such that for all n ě n 0 , we have U X h n V ‰ H.

The following result is well-known. When r F " 0, see for instance [START_REF] Roblin | Ergodicité et équidistribution en courbure négative[END_REF] for the equivalence of the first, second and fourth claims, and the arguments of [START_REF] Broise-Alamichel | Sur le codage du flot géodésique dans un arbre[END_REF]Prop. 3.3] for what remains to be proved. Proposition 4.15. Assume that the smallest nonempty Γ-invariant simplicial subtree of X is uniform, without vertices of degree 2, and that m F is finite. Then the following assertions are equivalent:

• the length spectrum of Γ is nonarithmetic, that is L Γ " Z;

• the discrete time geodesic flow on ΓzG X is topologically mixing on its nonwandering set;

• the quotient graph ΓzX is not bipartite;

• the Gibbs measure m F is mixing under the discrete time geodesic flow pg t q tPZ on ΓzG X. Otherwise L Γ " 2Z, and the square of the discrete time geodesic flow pg 2t q tPZ is topologically mixing on the nonwandering subset of ΓzG even X and mixing for the restriction of the Gibbs measure m F to ΓzG even X. l 65 19/12/2016 By Proposition 4.14, the general assumptions of Proposition 4.15 are satisfied if X is uniform, without vertices of degree 2, Γ is a lattice of X and r F " 0. Thus, if we assume furthermore that ΓzX is not bipartite, then the Bowen-Margulis measure m BM of Γ is mixing under the discrete time geodesic flow on ΓzG X.

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Chapter 5

Symbolic dynamics of geodesic flows on trees

Two-sided topological Markov shifts

In this short and independent Section, that will be used in Sections 5.2, 5.3, 5.4, 9.2 and 9.3, we recall some definitions concerning symbolic dynamics on countable alphabets.1

A (two-sided, topological) Markov shift2 is a topological dynamical system pΣ, σq constructed from a countable discrete alphabet A and a transition matrix A " pA i, j q i, jPA P t0, 1u A ˆA , where Σ is the closed subset of the topological product space A Z defined by Σ " x " px n q nPZ P A Z : @ n P Z, A xn,x n`1 " 1u , and σ : Σ Ñ Σ is the (two-sided) shift defined by pσpxqq n " x n`1 for all x P Σ and n P Z. Note that to be given pA , Aq is equivalent to be given an oriented graph with countable set of vertices A (and set of oriented edges a subset of A ˆA ) and with incidence matrix A such that A i, j " 1 if there is an oriented edge from the vertex i to the vertex j and A i, j " 0 otherwise. For all p ď q P Z, a finite sequence pa n q pďnďq P A tp,...,qu is admissible (or A-admissible when we need to make A precise) if A an, a n`1 " 1 for all n P tp, . . . , q ´1u. A topological Markov shift is transitive if for all x, y P A , there exists an admissible finite sequence pa n q pďnďq with a p " x and a q " y. This is equivalent to require the dynamical system pΣ, σq to be topologically transitive: for all nonempty open subsets U, V in Σ, there exists n P Z such that A X σ n pBq ‰ H.

Note that the product space A Z is not locally compact when A is infinite. When the matrix A has only finitely many nonzero entries on each line and each colum, then pΣ, σq is also called a subshift of finite type (on a countable alphabet). The topological space Σ is then 19/12/2016 locally compact: By diagonal extraction, for all p ď q in Z and a p , a p`1 , . . . , a q´1 , a q in A , every cylinder ra p , a p`1 , . . . , a q´1 , a q s " px n q nPZ P Σ : @ n P tp, . . . , qu, x n " a n (

is a compact open subset of Σ.

Given a continuous map F symb : Σ Ñ R and a constant c F symb P R, we say that a measure P on Σ, invariant under the shift σ, satisfies the Gibbs property3 with Gibbs constant c F symb for the potential F symb if for every finite subset E of the alphabet A , there exists C E ě 1 such that for all p ď q in Z and for every x " px n q nPZ P Σ such that x p , x q P E, we have

1 C E ď Pprx p , x p`1 , .
. . , x q´1 , x q sq e ´cF symb pq´p`1q`ř q n"p F symb pσ n xq ď C E .

(5.1)

Coding discrete time geodesic flows on simplicial trees

Let X be a locally finite simplicial tree without terminal vertices, with X " |X| 1 its geometric realisation. Let Γ be a nonelementary discrete subgroup of Aut X, and let r F : T 1 X Ñ R be a potential for Γ.

In this Section, we give a coding of the discrete time geodesic flow pg t q tPZ on the nonwandering subset of ΓzG X by a locally compact transitive (two-sided) topological Markov shift. This explicit construction will be useful later on to study the variational principle (see Section 5.4) and rates of mixing (see Section 9.2).

The main technical aspect of this construction, building on [START_REF] Broise-Alamichel | Sur le codage du flot géodésique dans un arbre[END_REF]§6], is to allow the case when Γ has torsion. When Γ is torsion free and Γ{X is finite, the construction is well-known, we refer for instance to [START_REF] Coornaert | Symbolic coding for the geodesic flow associated to a word hyperbolic group[END_REF] for a more general setting when the potential is 0. In order to consider for instance non-uniform tree lattices, it is important to allow torsion in Γ. Our direct approach also avoids the assumption that the discrete subgroup Γ is full, that is, equal to the subgroup consisting of the elements g P AutpXq such that p ˝g " p where p : X Ñ ΓzX is the canonical projection, as in [Kwo] (building on [START_REF] Burger | CAT(´1) spaces, divergence groups and their commensurators[END_REF]7.3]).

Let X 1 be the minimal nonempty Γ-invariant simplicial subtree of X, whose geometric realisation is C ΛΓ. Since we are only interested in the support of the Gibbs measures, we will only code the geodesic flow on the non-wandering subset ΓzX 1 of ΓzG X. The same construction works with the full space ΓzG X, but the resulting Markov shift is then not necessarily transitive.

Let pY, G ˚q " Γz zX 1 be the quotient graph of groups of X 1 by Γ (see for instance Example 2.10), and let p : X 1 Ñ Y " ΓzX 1 be the canonical projection. We denote by r1s " H the trivial double coset in any double coset set HzG{H of a group G by a subgroup H.

We consider the alphabet A consisting of the triples pe ´, h, e `q where ' e ˘P EY satisfy tpe ´q " ope `q and ' h P ρ e ´pG e ´qzG ope `q{ρ e `pG e `q satisfy h ‰ r1s if e `" e

´.

This set is countable (and finite if and only if the quotient graph ΓzX 1 is finite), we endow it with the discrete topology. We consider the (two-sided) topological Markov shift with alphabet A and transition matrix A pe ´, h, e `q, pe 1´, h 1 , e 1`q " 1 if e `" e 1 ´and 0 otherwise. Since for every edge e of Y the structural monomorphism

ρ e : G e " Γ r e ÝÑ G tpeq " Γ Ą tpeq is the map g Þ Ñ g ´1
e gg e , the double coset of h i p q in ρ e í pG e í qzG ope ì q {ρ e ì pG e `q does not depend on the choice of the γ i 's, and we again denote it by h i p q.

The next result shows that, under the only assumptions on Γ that it is discrete and nonelementary, the time-one discrete geodesic flow g 1 on its nonwandering subset of ΓzG X is topologically conjugate to a locally compact transitive (two-sided) topological Markov shift.

Theorem 5.1. If X 1 " C ΛΓ, the map Θ : ΓzG X 1 Ñ Σ defined by Γ Þ Ñ pe í p q, h i p q, e ì p qq iPZ is a homeomorphism which conjugates the time-one discrete geodesic flow g 1 and the shift σ, and the topological Markov shift pΣ, σq is a locally compact and transitive.

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Furthermore, if we endow ΓzG X 1 with the quotient distance of4 dp , 1 q " e ´max nPN : |r´n,ns " 1 |r´n,ns ( on G X 1 and Σ with the distance5 dpx, x 1 q " e ´max nPN : @ i P t´n,...,nu,

x i " x 1 i ( ,
then Θ is a bilipschitz homeomorphism. Finally, if X 1 is a uniform tree without vertices of degree at most 2, if the Gibbs measure m F of Γ is finite, and if the length spectrum L Γ of Γ is equal to Z, then the topological Markov shift pΣ, σq is topologically mixing.

The following diagram hence commutes

ΓzG X 1 g 1 ÝÑ ΓzG X 1 Θ Ó Ó Θ Σ σ ÝÑ Σ .
Note that when Y is finite (or equivalently when Γ is cocompact), the alphabet A is finite (hence pΣ, σq is a standard subshift of finite type). When furthermore the vertex groups of pY, G ˚q are trivial (or equivalently when Γ acts freely, and in particular is a finitely generated free group), this result is well-known, but it is new if the vertex groups are not trivial. Compare with the construction of [START_REF] Coornaert | Symbolic coding for the geodesic flow associated to a word hyperbolic group[END_REF], whose techniques might be applied since Γ is wordhyperbolic if Y is finite, up to replacing Gromov's (continuous time) geodesic flow of Γ by the (discrete time) geodesic flow on G X 1 , thus avoiding the suspension part (see also the end of loc. cit. when Γ is a free group).

Proof. For all P G X 1 and γ P Γ, we can take γ i pγ q " γ i p qγ ´1, and since ppγf i q " ppf i q, we have e ȋ pγ q " e ȋ p q and h i pγ q " h i p q, hence the map Θ is well defined.

By construction, the map Θ is equivariant for the actions of g 1 on ΓzG X 1 and σ.

For every N P N, if , 1 P G X 1 are close enough, then for ´N ´1 ď i ď N `1, we have f i p q " f i p 1 q and we may take γ i p q " γ i p 1 q, so that e ȋ p q " e ȋ p 1 q and h i p q " h i p 1 q for ´N ď i ď N . Therefore Θ is continuous.

Furthermore, with the distances indicated in the statement of Theorem 5.1, if , 1 P G X 1 satisfy |r´n, ns " 1 |r´n, ns for some n P N, then we have e ȋ p q " e ȋ p 1 q for ´n ď i ď n ´1, and we may take γ i p q " γ i p 1 q for ´n ď i ď n ´1, so that h i p q " h i p 1 q for ´n ď i ď n ´1. Therefore, we have dp ΘpΓ q, ΘpΓ 1 q q ď e dpΓ , Γ 1 q , and Θ is Lipschitz.

Let us construct an inverse Ψ : Σ Ñ ΓzG X 1 of θ, by a more general construction that will be useful later on. Let I be a nonempty interval of consecutive integers in Z, either finite or equal to Z (the definition of the inverse of θ only requires the second case I " Z). For all e ´, e `P EY such that tpe ´q " ope `q, we fix once and for all a representative of every double coset in ρ e ´pG e ´qzG ope `q{ρ e `pG e `q, and we will denote this double coset by its representative. e í " e ì´1 e ì`1 e ì " e í`1

g ´1 e í Ă e í h i h i`1 Č ope ì q Y X f i´1 f i α i´1 α i f i`1 p g ´1 e ì Ă e ì g ´1 e ì Ă e ì g ´1 e ì g e ì
Let us construct by induction a geodesic segment r w in X 1 (which will be a discrete geodesic line if I " Z), well defined up to the action of Γ, as follows.

We fix i 0 P I (for instance i 0 " 0 if I " Z or i 0 " min I if I is finite), and

α i 0 P Γ. Let us define f i 0 " f i 0 pwq " α i 0 g ´1 e ì0 Ă e ì0 .
Let us then define

α i 0 ´1 " α i 0 ´1pwq " α i 0 g ´1 e ì0 g e ì0 h ´1 i 0 and f i 0 ´1 " f i 0 ´1pwqα i 0 ´1 g ´1 e í0 Ă e í0 .
We have α i 0 ´1 h i 0 g ´1 e ì0 Ă e ì0 " f i 0 and pf i 0 ´1, f i 0 q is a geodesic edge path of length 2 (as the image by α i 0 ´1 of such a path).

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19/12/2016 Let i ´1, i 1 P I be such that i 1 ď i 0 ď i ´1. Assume by increasing induction on i and decreasing induction on i 1 that a geodesic edge path pf i 1 ´1 " f i 1 ´1pwq, . . . , f i´1 " f i´1 pwqq in X 1 and a sequence pα i 1 " α i 1 ´1pwq, . . . , α i´1 " α i´1 pwqq in Γ have been constructed such that f j " α j g ´1 e j Ă e j and α j " α j´1 h j g ´1 e j g e j for every j P N such that i 1 ´1 ď j ď i ´1, with besides j ě i 1 for the equality on the right.

If i does not belong to I, we stop the construction on the right hand side at i ´1. If on the contrary i P I, let us define (see the above picture)

α i " α i´1 h i g ´1 e ì g e ì and f i " f i pwq " α i g ´1 e ì Ă e ì . Then pf i´1 , f i q " `αi´1 g ´1 e í Ă e í , α i´1 h i g ´1 e ì Ă e ì ˘,
is a geodesic edge path of length 2 (as the image by α i´1 of such a path). As an edge path is geodesic if and only if it has no back-and-forth, pf i 1 , . . . , f i q is a geodesic edge path in X 1 . Thus the construction holds at rank i on the right. If i 1 ´1 does not belong to I, we stop the construction on the left side at i 1 . Otherwise we proceed as for the construction of α i 0 ´1 and f i 0 ´1 in order to construct α i´1 and f i´1 with the required properties.

If I " rp, qs X Z with p ď q P Z, let I 1 " rp ´1, qs X Z. If I " Z, let I 1 " Z. We have thus constructed a geodesic edge path pf i q iPI 1 " pf i pwqq iPI 1

(5.3) in X 1 . We denote by r w its parametrisation by R if I " Z and by rp ´1, q `1s if I " rp, qs X Z, in such a way that r wpiq " opf i q for all i P I. In particular, f i " r wpri, i `1sq for all i P I 1 . When I " rp, qs X Z, we consider r w as a generalised discrete geodesic line, by extending it to a constant on s ´8, p ´1s and on rq, `8r .

The orbit Γ r w of r w does not depend on the choice of α i 0 , since replacing α i 0 by α 1 i 0 replaces f i by α 1 i 0 α ´1 i 0 f i for all i P I 1 , hence replaces r w by α 1 i 0 α ´1 i 0 r w. This also implies that Γ r w does not depend on the choice of i 0 P I.

Assume from now on that I " Z, and define Ψ : Σ Ñ ΓzG X 1 by Ψpwq " Γ r w .

With the distances indicated in the statement of Theorem 5.1, let w " pe í , h i , e ì q iPZ and w 1 " pe 1 i ´, h 1 i , e 1 i `qiPI in Σ satisfy e ȋ " e 1 i ˘and h i " h 1 i for all i P t´n, . . . , nu for some n P N. Then we may take the same i 0 " 0 and α i 0 in the construction of r w and Ă w 1 . We thus have α i pwq " α i pw 1 q and f i pwq " f i pw 1 q for ´n ď i ď n. Therefore, we have dp Ψpwq, Ψpw 1 q q ď dpw, w 1 q , and Ψ is Lipschitz.

Let us prove that Ψ is indeed the inverse of Θ. As in the construction of Θ, for all P G X 1 and i P Z, we define f i " pri, i `1sq, e ì " ppf i q and e í " e ì´1 . We denote by γ 1 i P Γ an element sending f i to g ´1 e í Ă e í for all i P Z (see the picture below): with the notation above the statement of Theorem 5.1, we have

γ 1 i " g ´1 e í γ i p q. 72 19/12/2016 e í " e ì´1 e ì`1 e ì " e í`1 Y f i´1 f i f i`1 g ´1 e ì Ă e ì g ´1 e í Ă e í h 1 i h 1 i`1 γ 1 i X γ 1 i´1 Č ope ì q γ 1 i´1 f i p g ´1 e ì Ă e ì g ´1 e ì g e ì Then γ 1
i is well defined up to multiplication on the left by an element of

Γ g ´1 e í
Ă e

í " ρ e í pG e í q.

Let h 1 i be an element in G ope ì q sending g ´1 e ì Ă e ì to γ 1 i´1 f i . It exists since these two edges have the same origin Č ope ì q, and same image by p:

ppγ 1 i´1 f i q " ppf i q " e ì " pp Ă e ì q " pp g ´1 e ì Ă e ì q .
Furthermore, it is well defined up to multiplication on the right by an element of

Γ g ´1 e ì Ă e
ì " ρ e ì pG e ì q, and we have (see the above picture)

γ 1 i´1 γ 1 i ´1 g ´1 e í g e ì P h 1 i ρ e ì pG e ì q
By the construction of Θ (see Equation (5.2) with γ 1 i " g ´1 e í γ i p q for all i P Z), we have Θp q " pe í , ρ e í pG e í q h 1 i ρ e ì pG e ì q, e ì p qq iPZ .

Let h i be the chosen representative of the double coset ρ e í pG e í q h 1 i ρ e ì pG e ì q : there exists α P ρ e í pG e í q and β P ρ e ì pG e ì q such that h i " αh 1 i β. Up to replacing γ 1 i by α ´1γ 1 i and h 1 i by h 1 i β, we then may have

h 1 i " h i . By taking α i 0 " γ 1 i 0 ´1, we have α i " γ 1 i ´1
for all i P Z, and an inspection of the above two constructions gives that Θ ˝Ψ " id and Ψ ˝Θ " id.

Since the discrete time geodesic flow is topologically transitive on its nonwandering subset and by conjugation, the topological Markov shift pΣ, σq is topologically transitive.

If X 1 is a uniform tree without vertices of degree at most 2, and if the length spectrum of Γ is nonarithmetic and if the Gibbs measure m F for Γ is finite, then by Proposition 4.15, the discrete time geodesic flow on ΓzG X 1 is topologically mixing, hence by conjugation by Θ, the topological Markov shift pΣ, σq is topologically mixing. This concludes the proof of Theorem 5.1. l

When the length spectrum L Γ of Γ is different from Z, the topological Markov shift pΣ, σq constructed above is not always topologically mixing. We now modify the above construction in order to take care of this problem.

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Recall that X 1 " C ΛΓ and that G even X 1 is the space of geodesic lines P G X 1 whose origin p0q is at even distance from the basepoint x 0 (we assume that x 0 P X 1 ), which is invariant under the time-two discrete geodesic flow g 2 and, when L Γ " 2Z, under Γ.

Consider A even the alphabet consisting of the quintuples pf ´, h ´, f 0 , h `, f `q where the triples pf ´, h ´, f 0 q and pf 0 , h `, f `q belong to A and opf 0 q is at even distance from the image in Y " ΓzX 1 of the basepoint x 0 . Let A even " pA even, i,j q i,jPAeven be the transition matrix with line and column indices in A even such that for all i " pf ´, h ´, f 0 , h `, f `q and j " pf ´, h ´, f 0 ˚, h `, f `q, we have A even, i,j " 1 if and only if f `" f ´. We denote by pΣ even , σ even q the associated topological Markov shift. We endow Σ even with the slightly modified distance d even px, x 1 q " e ´1 2 max nPN : @ k P t´n, ..., nu, x k " x 1 k ( , where x " px k q kPZ and x 1 " px 1 k q kPZ are in Σ even . We have a canonical injection inj : Σ even Ñ Σ sending the sequence pf ń , h ń , f 0 n , h ǹ , f ǹ q nPZ to pe ń , h n , e ǹ q nPZ with, for every n P Z,

e 2n " f ń , h 2n " h ń , e 2n " f 0 n , e 2n`1 " f 0 n , h 2n`1 " h ǹ , e 2n`1 " f ǹ ,
By construction, inj is clearly a homeomorphism onto its image, and ΘpΓzG even X 1 q " injpΣ even q .

If two sequences in Σ even coincide between ´n and n, then their images by inj coincide between ´2n and 2n. Conversely, if the images by inj of two sequences in Σ even coincide between ´2n ´1 and 2n `1, then these sequences coincide between ´n and n. Hence inj is bilipschitz, for the above distances. Let us define Θ even " inj ´1 ˝Θ|ΓzGevenX 1 : ΓzG even X 1 Ñ Σ even . The following diagram hence commutes

ΓzG even X 1 Θeven Ý ÝÝÝ Ñ Σ even § § đ § § đ inj ΓzG X 1 Θ Ý ÝÝÝ Ñ Σ ,
where the vertical map on the left hand side is the inclusion map.

Theorem 5.2. Assume that X 1 " C ΛΓ is a uniform tree without vertices of degree at most 2, that the Gibbs measure m F of Γ is finite, and that the length spectrum L Γ of Γ is equal to 2Z. Then the map Θ even : ΓzG even X 1 Ñ Σ even is a bilipschitz homeomorphism which conjugates the time-two discrete geodesic flow g 2 and the shift σ even , and the topological Markov shift pΣ even , σ even q is locally compact and topologically mixing.

Proof. The only claims that remains to be proven is the last one, which follows from Proposition 4.15, by conjugation. l

Let us now study the properties of the image by the coding map Θ of finite Gibbs measures on ΓzG X.

Let pµ x q xPV X be two Patterson densities on B 8 X for the pairs pΓ, F ˘q, where as previously r F `" r F , r F ´" r F ˝ι. Assume that the associated Gibbs measure m F on ΓzG X (using the convention for discrete time of Section 4.3) is finite.

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Let us define

P " 1 }m F } Θ ˚mF (5.4)
as the image of the Gibbs measure m F (whose support is ΓzG X 1 ) by the homeomorphism Θ, normalised to be a probability measure. It is a probability measure on Σ, invariant under the shift σ.

Let pZ n q nPZ be the random process classically associated with the full shift σ on Σ: it is the random process on the Borel space Σ indexed by Z with values in the discrete alphabet A , where Z n : Σ Ñ A is the (continuous hence measurable) n-th projection px k q kPN Þ Ñ x n for all n P Z.

The following result summarises the properties of the probability measure P. We start by recalling and giving some notation used in this proposition.

For every admissible finite sequence w " pa p , . . . , a q q in A , where p ď q P Z, we denote ' by rws " ra p , . . . , a q s " tpx n q nPZ P Σ : @ n P tp, . . . , qu, x n " a n ( the associated cylinder in Σ, ' by p w the associated geodesic edge path in X with length q ´p `2 constructed in the proof of Theorem 5.1 (see Equation (5.3)), with origin p w ´and endpoint p w `.

For every geodesic edge path α " pf p´1 , . . . , f q q in X 1 , we define (See Section 2.7 for the notation, and the picture below)

B ὰ X 1 " B fq X 1 and B ά X 1 " B f p´1 X 1 ,

and

G α X " t P G X : pp ´1q " opf p´1 q and pq `1q " tpf q qu .

f q f p´1 B ὰ X 1 B ά X 1 α We define a map F symb : Σ Ñ R by F symb pxq " ż tpe 0 q ope 0 q

F

(5.5) if x " px i q iPZ with x 0 " pe 0 , h 0 , e 0 q. Note that F symb is locally constant (constant on each cylinder of length 1 at time 0), hence continuous: for all px n q nPZ , py n q nPZ P Σ, if x 0 " y 0 , then F symb pxq " F symb pyq. For instance, if F " F c is the potential associated with a system of conductances c : ΓzEX 1 Ñ R (see Section 3.5), then

F symb pxq " cpe 0 q . Note that if c, c 1 : ΓzEX 1 Ñ R are cohomologous systems of conductances on ΓzEX 1 , then the corresponding maps F symb , F 1 symb : Σ Ñ R are cohomologous. Indeed if f : ΓzV X Ñ R
is a map such that c 1 peq ´cpeq " f ptpeqq ´f popeqq for all e P ΓzEX, with G : Σ Ñ R the map defined by Gpxq " f pope 0 qq if x " px i q iPZ with x 0 " pe 0 , h 0 , e 0 q, then G is locally constant, hence continuous, and since tpe 0 q " ope 1 q, we have, for every x P Σ, F 1 symb pxq ´Fsymb pxq " Gpσxq ´Gpxq .

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Definition 5.3. Let X be a locally finite simplicial tree. A nonelementary discrete subgroup Γ 1 of AutpXq is Markov-good if for every n P N´t0u and every geodesic edge path pe 0 , . . . , e n`1 q in C ΛΓ 1 , we have

|Γ 1 e 0 X ¨¨¨X Γ 1 en | |Γ 1 e n´1 X Γ 1 en X Γ 1 e n`1 | " |Γ 1 e 0 X ¨¨¨X Γ 1 e n`1 | |Γ 1 e n´1 X Γ 1 en | .
(5.6)

Remark 5.4.

(1) Note that Equation (5.6) is automatically satisfied if n " 1 and that Γ 1 is Markov-good if Γ 1 acts freely on X.

(2) A group action on a tree is 2-acylindrical6 if the stabiliser of any geodesic edge path of length 2 is trivial. If Γ 1 is 2-acylindrical on X, then Γ 1 is Markov-good, since all groups appearing in Equation 5.6 are trivial.

(3) If X has degrees at least 3 and if Γ 1 is a noncocompact geometrically finite lattice of X, then Γ 1 is not Markov-good.

Proof.

(3) Since the quotient graph Γ 1 zX is infinite, the graph of groups Γ 1 z zX contains at least one cuspidal ray. Consider a geodesic ray in X with consecutive edges pf n q nPN mapping injectively onto this cuspiday ray, pointing towards its end. Their stabilisers in Γ 1 are hence nondecreasing: we have Γ 1 fn Ă Γ 1 f n`1 for all n P N. By the finiteness of the volume, there exists n ě 3 such that Γ 1 f n´2 is strictly contained in Γ 1 f n´1 . Since X has degrees at least 3, there exists γ P Γ 1 fixing tpf n´1 q but not fixing f n´1 . Let e 0 " f 0 , . . . , e n´1 " f n´1 , e n " γ f n´1 and e n`1 " γ f n´2 . Then pe 0 , . . . , e n`1 q is a geodesic edge path in X (equal to

C ΛΓ 1 since Γ 1 is a lattice). Since Γ 1 e 0 X ¨¨¨X Γ 1 en " Γ 1 f 0 , Γ 1 e n´1 X Γ 1 en X Γ 1 e n`1 " Γ 1 f n´2 , Γ 1 e 0 X ¨¨¨X Γ 1 e n`1 " Γ 1 f 0 , Γ 1 e n´1 X Γ 1 en " Γ 1 f n´1 and |Γ 1 f n´2 | ‰ |Γ 1 f n´1 |, the subgroup Γ 1 is not Markov-good. l
Recall that a random process pZ n q nPZ on pΣ, Pq is a Markov chain if and only if for all p ď q in Z and a p , . . . , a q , a q`1 in A , we have PpZ q`1 " a q`1 | Z q " a q , . . . , Z p " a p q " PpZ q`1 " a q`1 | Z q " a q q .

(5.7)

Proposition 5.5.

(1) For every admissible finite sequence w in A , we have

Pprwsq "

µ ṕ w ´pB ṕ w X 1 q µ p w `pB p w X 1 q e ş p w p w ´p r F ´δq |Γ p w | }m F } .
(2) The random process pZ n q nPZ on pΣ, Pq is a Markov chain if and only if Γ is Markov-good.

(3) The measure P on the topological Markov shift Σ satisfies the Gibbs property with Gibbs constant δ for the potential F symb .

It follows from the above Assertion (2) and from Remark 5.4 that when X has degrees at least 3 and Γ is a noncocompact geometrically finite lattice of X, then pZ n q nPZ is not a Markov chain. The fact that codings of discrete time geodesic flows on trees might not satisfy the Markov chain property had been noticed by Burger and Mozes around the time the paper [BuM] was published.7 When proving the variational principle in Section 5.4 and the exponential decay of correlations in Section 9.2, we will hence have to use tools that are not using the Markov chain property.

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Proof.

(1) Let w " pa p , . . . , a q q where p ď q P Z, be an admissible finite sequence in A . By the construction of Θ, the preimage Θ ´1prwsq is equal to the image

ΓG p w X 1 of G p w X 1 in ΓzG X 1 . Hence, since Γ p w is the stabiliser of G p w X 1 in Γ, Pprwsq " 1 }m F } m F pΓG p w X 1 q " 1 |Γ p w | }m F } r m F pG p w X 1 q .
In the expression of r m F given by Equation (4.10), let us use as basepoint x 0 the origin p w ´of the edge path p w, and note that all elements of G p w X 1 pass through p w ´at time t " p ´1, so that by invariance of r m F under the geodesic flow , we have

r m F pG p w X 1 q " ż PG p w X 1 d r m F pg 1´p q " ż ´PB ṕ w `X1 ż `PB p w `X1 dµ ṕ w ´p ´qdµ p w ´p `q " µ ṕ w ´pB ṕ w `X1 q µ p w ´pB p w X 1 q " µ ṕ w ´pB ṕ w X 1 q µ p w `pB p w X 1 q e ş p w p w ´pF ´δq ,
where this last equality follows by Equations (4.2) and (3.8) with x " p w ´and y " p w `, since for every `P B p w `X1 , we have p w `P r p w `, `r .

(2) Let us fix p ď q in Z and a p , . . . , a q , a q`1 in A , and let us verify Equation (5.7). We may assume that α " pa p , . . . , a q , a q`1 q is an admissible sequence. Let α ˚" pa p , . . . , a q q, which is also an admissible sequence. Let us consider Q α " PpZ q`1 " a q`1 | Z q " a q , . . . , Z p " a p q PpZ q`1 " a q`1 | Z q " a q q " Ppra p , . . . , a q`1 sq Ppra q sq Ppra p , . . . , a q sq Ppra q , a q`1 sq .

Let us replace each one of the four terms in this ratio by its value given by Assertion (1).

Since B ṕ α X 1 " B x α˚X 1 , B p α X 1 " B `{ aq,a q`1 X 1 , B x α˚X 1 " B x
aq X 1 and B ṕ aq X 1 " B ´{ aq,a q`1 X 1 , all Patterson measure terms cancel. Denoting by y 1 the common origin of p α and x α ˚, by y 2 the common origin of p a q and { a q , a q`1 , by y 3 the common endpoint of p a q and x α ˚, and by y 4 the common endpoint of { a q , a q`1 and p α, we thus have by Assertion (1)

Q α " |Γ x α˚| |Γ { aq,a q`1 | |Γ p α | |Γ x aq | e ş y 4 y 1 p r F ´δq e ş y 3 y 2 p r F ´δq e ş y 3 y 1 p r F ´δq e ş y 4 y 2 p r F ´δq .
Since y 1 , y 2 , y 3 , y 4 are in this order on ry 1 , y 4 s, we have

Q α " |Γ x α˚| |Γ { aq,a q`1 | |Γ p α | |Γ x aq | .
Since every geodesic edge path of length n `1 at least 3 in X 1 defines an admissible sequence of length n at least 2 in A , by Equation (5.6), we have Q α " 1 for every admissible sequence α in A if and only if Γ is Markov-good.

(3) Let E be a finite subset of the alphabet A , and let w " pa p , . . . , a q q with p ď q in Z and admissible sequence in A such that a p , a q P E. By Assertion (1), we have

Pprwsq " µ ṕ w ´pB ṕ w X 1 q µ p w `pB p w X 1 q e ş p w p w ´p r F ´δq |Γ p w | }m F } . 77 19/12/2016
Since a p , a q are varying in the finite subset E of A , the first and last edges of p w vary amongst the images under elements of Γ of finitely many edges of X. Since w is admissible, the open sets B p w X 1 are nonempty subsets of ΛΓ, hence they have positive Patterson measures. Furthermore, the quantities µ p w ˘pB p w X 1 q are invariant under the action of Γ on the first/last edge of p w. Hence there exists c 1 ě 1 depending only on E such that 1 ď |Γ

p w | ď |Γ p w ´| ď c 1 and 1 c 1 ď µ p w ˘pB p w X 1 q ď c 1 . Note that the length of p w is equal to q ´p `2. Therefore e ´δ c 3 1 }m F } e ´δpq´p`1q`ş p w p w ´r F ď Pprwsq ď e ´δ c 2 1 }m F } e ´δpq´p`1q`ş p w p w ´r F .
If p w " pf p´1 , f p , . . . , f q q and x P rws, we have by the definition of

F symb ż p w p w ´r F " q ÿ i"p´1 ż tpf i q opf i q r F " ż tpf p´1 q opf p´1 q r F `q ÿ i"p F symb pσ i pxqq .
Since F is continuous and opf p´1 q remains in a finite subset of V X 1 , there exists c 2 ą 0 depending only on E such that | r F pvq| ď c 2 for every v P T 1 X with πpvq P ropf p´1 q, tpf p´1 qs. Hence | ş tpf p´1 q opf p´1 q r F | ď c 2 , and Assertion (3) of Proposition 5.5 follows (see Equation ( 5.1) for the definition of the Gibbs property). l

Again in order to consider the case when the length spectrum L Γ of Γ is 2Z, we define

P even " 1 }pm F q | ΓzGevenX 1 } pΘ even q ˚pm F q | ΓzGevenX 1 ,
and pZ even, n q nPZ the random process associated with the full shift σ even on Σ even , with Z even, n : Σ even Ñ A even the n-th projection for every n P Z. By a proof similar to the one of Proposition 5.5, we have the following result. We define a map F symb, even : Σ even Ñ R by F symb, even pxq "

ż tpf 0 q opf 0 0 q F (5.8) if x " px i q iPZ with x 0 " pf 0 , h 0 , f 0 0 , h 0 , f 0 q.
As previously, F symb, even is locally constant, hence continuous.

Proposition 5.6. The measure P even on the topological Markov shift Σ even satisfies the Gibbs property with Gibbs constant δ for the potential F symb, even . l

Again, if Γ is a noncocompact geometrically finite lattice of X and X 1 has degrees at least 3, then pZ even, n q nPZ is not a Markov chain.

Coding continuous time geodesic flows on metric trees

Let pX, λq be a locally finite metric tree without terminal vertices, with X " |X| λ its geometric realisation. Let Γ be a nonelementary discrete subgroup of AutpX, λq, and let r

F : T 1 X Ñ R be a potential for Γ. Let X 1 " C ΛΓ, which is the geometric realisation |X 1 | λ of a metric 78 19/12/2016
subtree pX 1 , λq. Let pµ x q xPV X be two Patterson densities on B 8 X for the pairs pΓ, F ˘q, and assume that the associated Gibbs measure m F is finite. We also assume in this Section that the lengths of the edges of pX 1 , λq have a finite upper bound (which is in particular the case if pX 1 , λq is uniform). They have a positive lower bound by definition (see Section 2.7).

In this Section, we prove that the continuous time geodesic flow on ΓzG X 1 is isomorphic to a suspension of a transitive (two-sided) topological Markov shift on a countable alphabet, by an explicit construction that will be useful later on to study the variational principle (see Section 5.4) and rates of mixing (see Section 9.3). Since we are only interested in the support of the Gibbs measures, we will only give such a description for the geodesic flow on the nonwandering subset ΓzG X 1 of ΓzG X. The same construction works with the full space ΓzG X, but the resulting Markov shift is then not necessarily transitive.

We start by recalling (see for instance [START_REF] Brin | Introduction to dynamical systems[END_REF]§1.11]) the definitions of the suspension of a (invertible) discrete time dynamical system and of the first return map on a cross-section of a continuous time dynamical system, which allow to pass from transformations to flows and back, respectively.

Let pZ, µ, T q be a metric space Z endowed with an homeomorphism T and a T -invariant (Borel, positive) measure µ. Let r : Z Ñ s0, `8r be a continuous map, such that for all z P Z, the subset trpT n zq : n P Nu Y t´rpT ´pn`1q zq : n P Nu is discrete in R. Then the suspension (or also special flow) over pZ, µ, T q with roof function r is the following continuous time dynamical system pZ r , µ r , pT t r q tPR q : ' The space Z r is the quotient topological space pZ ˆRq{ " where " is the equivalence relation on Z ˆR generated by pz, s `rpzqq " pT z, sq for all pz, sq P Z ˆR. We denote by rz, ss the equivalence class of pz, sq. Note that F " tpz, sq : z P Z, 0 ď s ă rpzqu is a measurable strict fundamental domain for this equivalence relation. We endow Z r with the Bowen-Walters distance, see [BowW] and particularly the appendix in [BarS].

' For every t P R, the map T t r : Z r Ñ Z r is the map rz, ss Þ Ñ rz, s `ts. Equivalently, when pz, sq P F and t ě 0, then T t r prz, ssq " rT n z, s 1 s where n P N and s 1 P R are such that t `s "

n´1 ÿ i"0
rpT i zq `s1 and 0 ď s 1 ă rpT n zq .

' With ds the Lebesgue measure on R, the measure µ r is the pushforward of the restriction to F of the product measure dµ ds by the restriction to F of the canonical projection pZ Rq Ñ Z r .

Note that pT t r q tPR is indeed a continuous 1-parameter group of homeomorphisms of Z r , preserving the measure µ r . The measure µ r is finite if and only if ş Z r dµ is finite, since

}µ r } " ż Z r dµ .
We will denote by pZ, µ, T q r the continuous time dynamical system pZ r , µ r , pT t r q tPR q thus constructed.

Conversely, let pZ, µ, pφ t q tPR q be a metric space Z endowed with a continuous 1-parameter group of homeomorphisms pφ t q tPR , preserving a (Borel, positive) measure µ. Let Y be a 79 19/12/2016 cross-section of pφ t q tPR , that is a closed subspace of Z such that for every z P Z, the set tt P R : φ t pzq P Y u is nonempty and discrete. Let τ : Y Ñ s0, `8r be the (continuous) first return time on the cross-section Y : for every y P Y , τ pyq " mintt ą 0 : φ t pyq P Y u .

Let φ Y : Y Ñ Y be the (homeomorphic) first return map to (or Poincaré map of) the crosssection Y , defined by

φ Y : y Þ Ñ φ τ pyq pyq .
By the invariance of µ under the flow pφ t q tPR , the restriction of µ to tφ t pyq : y P Y, 0 ď t ă τ pyqu disintegrates8 by the (well-defined) map φ t pyq Þ Ñ y over a measure µ Y on Y , which is invariant under the first return map φ Y :

dµpφ t pyqq " dt dµ Y pyq .
Note that if r has a positive lower bound and if µ is finite, then µ Y is finite, since

}µ} ě }µ Y } inf r .
Hence pY, µ Y , φ Y q is a discrete time dynamical system.

Recall that an isomorphism from a continuous time dynamical system pZ, µ, pφ t q tPR q to another one pZ 1 , µ 1 , pφ 1 t q tPR q is an homeomorphism between the underlying spaces preserving the underlying measures and commuting with the underlying flows.

Example 5.7. If pZ, µ, T q and pZ 1 , µ 1 , T 1 q are (invertible) discrete time dynamical systems, endowed with roof functions r : Z Ñ s0, `8r and r 1 : Z 1 Ñ s0, `8r respectively, if θ : Z Ñ Z 1 is a measure preserving homeomorphism commuting with the transformations T and T 1 (that is, θ ˚µ " µ 1 , θ ˝T " T 1 ˝θ) and such that r 1 ˝θ " r , then the map

p θ : Z r Ñ Z 1
r 1 defined by rz, ss Þ Ñ rθpzq, ss is an isomorphim between the suspensions pZ, µ, T q r and pZ 1 , µ 1 , T 1 q r 1 .

It is well known (see for instance [START_REF] Brin | Introduction to dynamical systems[END_REF]§1.11]) that the above two constructions are inverses one to another, up to isomorphism. In particular, we have the following result.

Proposition 5.8. The suspension pY, µ Y , φ Y q τ over pY, µ Y , φ Y q with roof function τ is isomorphic to pZ, µ, pφ t q tPR q by the map f Y : ry, ss Þ Ñ φ s y. l

In order to describe the continuous time dynamical system `ΓzG X 1 , m F }m F } , pg t q tPR ˘as a suspension over a topological Markov shift, we will start by describing it as a suspension of the discrete time geodesic flow on ΓzG X 1 . Note that the Patterson densities and Gibbs measures depend not only on the potential, but also on the lengths of the edges. 9 We hence need to relate precisely the continuous time and discrete time situations, and we will use in this Section the left exponent 7 to indicate a discrete time object whenever needed.

For instance, we set 7 X 1 " |X 1 | 1 and we denote by p 7 g t q tPZ the discrete time geodesic flow on ΓzG X 1 . Note that X 1 and 7 X 1 are equal as topological spaces (but not as metric spaces). The boundaries at infinity of X 1 and 7 X 1 , which coincide with their spaces of ends as topological spaces (by the assumption on the lengths of the edges), are hence equal and denoted by B 8 X.

We may assume by Section 3.5 that the potential r F : T 1 X Ñ R is the potential r F c associated with a system of conductances r c on the metric tree pX, λq for Γ. Let δ c " δ Fc . We denote by 7 r c : EX Ñ R the Γ-invariant system of conductances 7 r c : e Þ Ñ pcpeq ´δc qλpeq (5.9) on the simplicial tree X for Γ, by r F7 c : T 1 p 7 Xq Ñ R its associated potential, and by 7 c : ΓzEX Ñ R and F7 c : ΓzT 1 p 7 Xq Ñ R their quotient maps.

Note that the inclusion morphism AutpX, λq Ñ AutpXq is a homeomorphism onto its image (for the compact-open topologies), by the assumption on the lengths of the edges, hence that Γ is also a nonelementary discrete subgroup of AutpXq. Now, let pΣ, σ, Pq be the (two-sided) topological Markov shift conjugated to the discrete time geodesic flow `ΓzG X 1 , 7 g 1 ,

m F 7 c }m F 7 c
} ˘by the bilipschitz homeomorphism Θ : ΓzG X 1 Ñ Σ of Theorem 5.1 (where the potential F is replaced by F7 c ). Let r : Σ Ñ s0, `8r be the map

r : x Þ Ñ λpe 0 q
(5.10) if x " px n q nPZ P Σ and x 0 " pe 0 , h 0 , e 0 q P A . This map is locally constant, hence continuous on Σ, and has a positive lower bound, since the lengths of the edges of pX 1 , λq have a positive lower bound.

Theorem 5.9. Assume that the lengths of the edges of pX 1 , λq have a finite upper bound, and that the Gibbs measure m F is finite. Then there exists a ą 0 such that the continuous time dynamical system `ΓzG X 1 , m F }m F } , pg t q tPR ˘is isomorphic to the suspension pΣ, σ, a Pq r over pΣ, σ, a Pq with roof function r, by a bilipschitz homeomorphism Θ r : ΓzG X 1 Ñ Σ r .

Proof. Let

Y " t P ΓzG X 1 : p0q P ΓzV Xu .

Then the (closed) subset Y of ΓzG X 1 is a cross-section of the continuous time geodesic flow pg t q tPR , since every orbit meets Y and since the lengths of the edges of pX 1 , λq have a positive lower bound. Let τ : Y Ñ s0, `8r be the first return time, let µ Y be the measure on Y (obtained by disintegrating m F }m F } ), and let g Y : Y Ñ Y be the first return map associated with this cross-section Y .

9 The fact that the Patterson densities could be singular one with respect to another when the metric varies is a well known phenomenon, even when the potential vanishes. See for instance Kuusalo's theorem [Kuu] saying that the Patterson densities on the boundary at infinity of the real hyperbolic plane of two cocompact marked Fuchsian groups are absolutely continuous one with respect to the other if and only if the marked Fuchsian groups are conjugated, and the extension of this result in [START_REF] Hersonsky | On the rigidity of discrete isometry groups of negatively curved spaces[END_REF]. See also the result of [KaN] which parametrises the Culler-Vogtmann space using Patterson densities for cocompact and free actions of free groups on metric trees.

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We have a natural reparametrisation map R : Y Ñ ΓzG X 1 , defined by Þ Ñ 7 , where 7 pnq " pg n Y qp0q is the n-th passage of in V X, for every n P Z. Since there exists m, M ą 0 such that λpEXq Ă rm, M s, the map R is a bilipschitz homeomorphism. It commutes with the first return map g Y and the discrete time geodesic flow on ΓzG X 1 : R ˝gY " 7 g 1 ˝R .

The main point of this proof is the following result relating the measures µ Y and m F 7 c .

Lemma 5.10. (1) The family pµ x q xPV X is a Patterson density for pΓ, F7 c q on the boundary at infinity of the simplicial tree X 1 , and the critical exponent δ7 c of 7 c is equal to 0.

(

) We have R ˚µY }µ Y } " m F 7 c }m F 7 c } . 2
Proof.

(1) By the definition of the potential associated with a system of conductances (see Section 3.5), for every x, y P V X 1 , if pe 1 , . . . , e n q is the edge path in X with ope 1 q " x and tpe n q " y, then (noting that the integrals along paths depend on the lengths of the edges, the first one below being in X 1 , the second one in 7 X 1 )

ż y x p Ă F c ´δc q " n ÿ i"1
pr cpe i qλpe i q ´δc λpe i qq " the Poincaré series for the simplicial tree with potential r F7 c and for the metric tree with normalised potential r F c ´δc , respectively. We hence have 7 Qpsq ď Qp s M q ă `8 if s ą 0 and 7 Qpsq ě Qp s M q " `8 if s ă 0. Thus the critical exponent δ7 c of pΓ, F7 c q for the simplicial tree X 1 is equal to 0, hence F7 c is a normalised potential.

ż y x Ă F7 c . ( 5
By the definition (see Section 3.4) of the Gibbs cocycles (which uses the normalised potential), Equation (5.11) also implies that the Gibbs cocycles C ˘and 7 C ˘for pΓ, F c q and pΓ, F7 c q respectively coincide on B 8 X ˆV X ˆV X. Thus by Equations (4.1) and (4.2), the family pµ x q xPV X is indeed a Patterson density for pΓ, F7 c q: for all γ P Γ and x, y P V X, and for (almost) all ξ P B 8 X, γ ˚µx " µ γx and dµ

x dµ y pξq " e ´7C ξ px, yq .

(2) We may hence choose these families pµ x q xPV X in order to define the Gibbs measure m7 c associated with the potential F7 c on ΓzG X. Note that since we will prove that m7 c is finite, the normalised measure

m 7 c }m 7 c } is independent of this choice (see Corollary 4.6). Let r Y " t P G X 1 : p0q P V X 1 u be the (Γ-invariant) lift of the cross-section Y to G X 1 , let r R : r Y Ñ G X 1 be the lift of R,
Y " Γz r Y is µ Y . We have a partition of r Y into the closed-open subsets r Y x " t P G X 1 : p0q " xu as x varies in V X 1 .
Let us fix x P V X. By the definition of µ Y as a disintegration of m Fc }m Fc } with respect to the continuous time, by lifting to G X 1 , by using Hopf's parametrisation with respect to x and Equation (4.3) with x 0 " x , we have for every

P r Y x , d Ă µ Y p q " 1 }m Fc } dµ x p ´q dµ x p `q .
Note that p0q " 7 p0q, ´" 7

´, `" 7

`since the reparametrisation does not change the origin nor the two points at infinity. Hence by Assertion (1), we have

r R ˚p Ă µ Y q " 1 }m Fc } r m F 7 c .
As µ Y is a finite measure since τ has a positive lower bound, this implies that m F 7 c is finite. By renormalizing as probability measures, this proves Assertion (2). l

Let a " }µ Y } ą 0, so that by Lemma 5.10 (2) we have R ˚µY "

a m 7 c }m 7 c } . Let p r : ΓzG X 1 Ñ s0, `8s be the map p r : Γ Þ Ñ λ ` pr0, 1sq given
by the length for λ of the first edge followed by a discrete geodesic line P G X 1 . Note that p r is locally constant, hence continuous, and that p r is a roof function for the discrete time dynamical system pΓzG X 1 , 7 g 1 q. Also note that p r ˝R " τ and r ˝Θ " p r by the definitions of τ and r.

Let us finally define Θ r : ΓzG X 1 Ñ Σ r as the compositions of the following three maps

pΓzG X 1 , m F }m F } , pg t q tPR q f ´1 Y ÝÑ pY, µ Y , g Y q τ p R ÝÑ pΓzG X 1 , a m7 c }m7 c } , 7 g 1 q p r p Θ
ÝÑ pΣ, a P, σq r , (5.12)

where the first one is the inverse of the tautological isomorphism given by Proposition 5.8 and the last two ones, given by Example 5.7, are the isomorphisms p R and p Θ of continuous time dynamical systems obtained by suspensions of the isomorphisms R and Θ of discrete time dynamical systems. It is easy to check that Θ r is a bilipschitz homeomorphism, using the following description of the Bowen-Walters distance, see for instance [BarS, Appendix].

Proposition 5.11. Let pZ, µ, T q r be the suspension over an invertible dynamical system such that T is a bilipschitz homeomorphism, with roof function r having a positive lower bound and a finite upper bound. Let d BW : Z r ˆZr Ñ R be the map 10 defined (using the canonical representatives) by d BW prx, ss, rx 1 , s 1 sq " mintdpx, x 1 q `|s ´s1 |, dpT x, x 1 q `rpxq ´s `s1 , dpx, T x 1 q `rpyq `s ´s1 u 10 The map dBW is actually not a distance, but may replace the Bowen-Walters true distance when working up to multiplicative constants or bilipschitz homeomorphisms.

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Then there exists a constant C BW ą 0 such that the Bowen-Walters distance d on Σ r satisfies

1 C BW d BW ď d ď C BW d BW . l
This concludes the proof of Theorem 5.9. l

The variational principle for metric and simplicial trees

In this Section, we assume that X is the geometric realisation of a locally finite metric tree without terminal vertices pX, λq (respectively of a locally finite simplicial tree X without terminal vertices). Let Γ be a nonelementary discrete subgroup of AutpX, λq (respectively AutpXq).

We relate in this Section the Gibbs measures11 to the equilibrium states (see the definitions below) for the continuous time geodesic flow on ΓzG X (respectively for the discrete time geodesic flow on ΓzG X).

When X is a Riemannian manifold with pinched negative curvature such that the derivatives of the sectional curvature are uniformly bounded, and when the potential is Höldercontinuous, the analogs of the results of this Section are due to [START_REF] Paulin | Equilibrium states in negative curvature[END_REF]Thm. 6.1]. Their proofs generalise the proofs of Theorems 1 and 2 of [OP], with ideas and techniques going back to [LedS]. When Y is a compact locally CATp´1q-space, a complete statement about existence, uniqueness and Gibbs property of equilibrium states for any Hölder continuous potential is given in [ConLT].

The proof of the metric tree case will rely strongly (via the suspension process described in Section 5.3) on the proof of the simplicial tree case, hence we start by the latter.

The simplicial tree case.

Let X be a locally finite simplicial tree without terminal vertices, with geometric realisation X " |X| 1 . Let Γ be a nonelementary discrete subgroup of AutpXq and let r c : EX Ñ R be a system of conductances for Γ on X. Let r F c : T 1 X Ñ R be its associated potential, and let δ c be the critical exponent of c.

We define a map r

F c : G X Ñ R by r F c p q " r c pe 0 p qq " ż tpe 0 p qq ope 0 p qq r F c
for all P G X, where e 0 p q is the edge of X in which enters at time t " 0. This map is locally constant, hence continuous, and it is Γ-invariant, hence it induces a continuous map

F c : ΓzG X Ñ R. 12
The following result proves that the Gibbs measure of pΓ, F c q for the discrete time geodesic flow on ΓzG X is an equilibrium state for the potential F c . We start by recalling the definition of an equilibrium state,13 see also [START_REF] Bowen | Equilibrium states and the ergodic theory of Anosov diffeomorphisms[END_REF][START_REF] Ruelle | Thermodynamic formalism: The mathematical structure of equilibrium statistical mechanics[END_REF].

Let Z be a locally compact topological space, let T : Z Ñ Z be a homeomorphism, and let φ : Z Ñ R be a continuous map. Let M φ be the set of Borel probability measures m on Z, invariant under the transformation T , such that the negative part maxt0, ´φu of φ is m-integrable. Let h m pT q be the (metric) entropy of the transformation T with respect to m P M φ . The metric pressure for the potential φ of a measure m P M φ is

P φ pmq " h m pT q `żZ φ dm .
The pressure of the potential φ is

P φ " sup mPM φ P φ pmq .
A measure m 0 P M φ is an equilibrium state for the potential φ if P φ pm 0 q " P φ .

Theorem 5.12 (The variational principle for simplicial trees). Let X, Γ, r c be as above. Assume that δ c ă `8 and that there exists a finite Gibbs measure m c for F c such that the negative part of the potential F c is m c -integrable. Then mc }mc} is the unique equilibrium state for the potential F c under the discrete time geodesic flow on ΓzG X, and the pressure of F c coincides with the critical exponent δ c of c.

In order to prove this result, using the coding of the discrete time geodesic flow given in Section 5.2, the main tool is the following result of J. Buzzi in symbolic dynamics, building on works of Sarig and Buzzi-Sarig, whose proof is given in the Appendix.

Let σ : Σ Ñ Σ be a two-sided topological Markov shift 14 with (countable) alphabet A and transition matrix A, and let φ : Σ Ñ R be a continuous map.

For every n P N, we denote by var n φ " sup

x, y P Σ @ i P t´n, ..., nu, x i "y i |φpxq ´φpyq| the n-variation of φ. For instance, if φpxq depends only on x 0 , then var n φ " 0 (and hence ř nPN n var n φ " 0 converges).

A weak Gibbs measure for φ with Gibbs constant C " Cpmq P R is a σ-invariant Borel measure m on Σ such that for every a P A , there exists c a ě 1 such that for all n P N ´t0u and x P ras such that σ n pxq " x, we have

1 c a ď mprx 0 , . . . , x n´1 sq e ´Cpmq n`ř n´1 i"0 φpσ i xq ď c a .
(5.13) Theorem 5.13 (J. Buzzi, see Corollary A.5). Let pΣ, σq be a two-sided transitive topological Markov shift on a countable alphabet A and let φ : Σ Ñ R be a continuous map such that ř nPN n var n φ converges. Let m be a weak Gibbs measure for φ on Σ with Gibbs constant Cpmq, such that ş φ ´dm ă `8. Then the pressure of φ is finite, equal to Cpmq, and m is the unique equilibrium state. l

14 See Section 5.1 for definitions.

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Proof of Theorem 5.12. In Section 5.2, we constructed a transitive topological Markov shift pΣ, σq on a countable alphabet A and a homeomorphism Θ : ΓzG X 1 Ñ Σ which conjugates the time-one discrete geodesic flow g 1 on the nonwandering subset ΓzG X 1 of ΓzG X and the shift σ on Σ (see Theorem 5.1). Let us define a potential F c, symb : Σ Ñ R by F c, symb pxq " cpe 0 q (5.14) if x " px i q iPZ with x 0 " pe 0 , h 0 , e 0 q. Note that this potential is the one denoted by F symb in Equation (5.5), when the potential F on T 1 X is replaced by F c . By the construction of Θ, we have F c, symb ˝Θ " F c .

(5.15)

Note that all probability measures on ΓzG X invariant under the discrete time geodesic flow are supported on the nonwandering set ΓzG X 1 . The pushforward of measures Θ ˚hence gives a bijection from the space M Fc of g 1 -invariant probability measures on ΓzG X for which the negative part of F c is integrable to the space M F c, symb of σ-invariant probability measures on ΓzG X for which the negative part of F c, symb is integrable. This bijection induces a bijection between the subsets of equilibrium states. Since F c, symb pxq depends only on x 0 for every x P Σ, the series

ř nPN n var n F c, symb converges.
By definition (see Equation (5.4)), the measure P is the pushforward of m Fc }m Fc } by Θ, hence is a σ-invariant probability measure on ΓzG X for which the negative part of F c, symb is integrable, by the assumption of Theorem 5.12. By Proposition 5.5 (3), the measure P on Σ satisfies the Gibbs property with Gibbs constant δ c for the potential F c, symb , hence15 satisfies the weak Gibbs property with Gibbs constant δ c . Theorem 5.12 then follows from Theorem 5.13. l

Remark. It follows from Equation (5.15), from the remark above Proposition 5.5 and from the fact that Θ˝g 1 " σ˝Θ, that if c, c 1 : ΓzEX 1 Ñ R are cohomologous systems of conductances on ΓzEX 1 , then the corresponding maps F c , F c 1 : ΓzG X 1 Ñ R are cohomologous: there exists a continuous map G : ΓzG X 1 Ñ R such that for every P ΓzG X 1 , F c 1 p q ´Fc p q " Gpg 1 q ´Gp q .

The metric tree case.

Let pX, λq be a locally finite metric tree without terminal vertices with geometric realisation X " |X| λ , let Γ be a nonelementary discrete subgroup of AutpX, λq and let r c : EX Ñ R be a system of conductances for Γ on X. Let r F c : T 1 X Ñ R be its associated potential (see Section 3.5), and let δ c " δ Fc be the critical exponent of c.

Recall16 that we have a canonical projection G X Ñ T 1 X which associates to a geodesic line its germ v at its footpoint p0q. Let r F 6 c : G X Ñ R be the Γ-invariant map obtained by precomposing the potential r F c : T 1 X Ñ R with this canonical projection:

r F 6 c : Þ Ñ r F c pv q .
Let F 6 c : ΓzG X Ñ R be its quotient map, which is continuous, as a composition of continuous maps.

The following result proves that the Gibbs measure of pΓ, F c q for the continuous time geodesic flow on ΓzG X is an equilibrium state for the potential F 6 c . We start by recalling the definition of an equilibrium state for a possibly unbounded potential under a flow. 17

Given pZ, pφ t q tPR q a topological space endowed with a continuous one-parameter group of homeomorphisms and ψ : Z Ñ R a continuous map (called a potential), let M ψ be the set of Borel probability measures m on Z invariant under the flow pφ t q tPR , such that the negative part of ψ is m-integrable. Let h m pφ 1 q be the (metric) entropy of the geodesic flow with respect to m P M ψ . The metric pressure for ψ of a measure m P M ψ is

P ψ pmq " h m pφ 1 q `żZ ψ dm .
The pressure of the potential ψ is

P ψ " sup mPM ψ P ψ pmq .
An element m P M ψ is an equilibrium state for ψ if the least upper bound defining P ψ is attained on m. Note that if ψ 1 is another potential cohomologous to ψ, that is, if there exists a continuous map G : Z Ñ R, differentiable along every orbit of the flow, such that ψ 1 pxq ´ψpxq " d dt |t"0 Gpg t xq, then M ψ 1 " M ψ , for every m P M ψ , we have P ψ 1 pmq " P ψ pmq, P ψ 1 " P ψ and the equilibrium states for ψ 1 are exactly the equilibrium states for ψ.

Theorem 5.14 (The variational principle for metric trees). Let pX, λq, Γ, r c be as above. Assume that the lengths of the edges of pX, λq have a finite upper bound. 18 Assume that δ c ă `8 and that there exists a finite Gibbs measure m c for F c such that the negative part of the potential F 6 c is m c -integrable. Then mc }mc} is the unique equilibrium state for the potential F 6 c under the continuous time geodesic flow on ΓzG X, and the pressure of F 6 c coincides with the critical exponent δ c of c.

Using the description of the continuous time dynamical system `ΓzG X 1 , mc }mc} , pg t q tPR ˘as a suspension over a topological Markov shift (see Theorem 5.9), this statement reduces to well-known techniques in the thermodynamic formalism of suspension flows, see for instance [IJT], as well as [BarI, Kemp, IJ, JKL]. Our situation is greatly simplified by the fact that our roof function has a positive lower bound and a finite upper bound, and that our symbolic potential is constant on the 1-cylinders tx P Σ : x 0 " au for a in the alphabet.

Proof. Since finite measures invariant under the geodesic flow on ΓzG X are supported on its nonwandering set, up to replacing X by X 1 " C ΛΓ, we assume that X " X 1 .

Since equilibrium states are unchanged up to adding a constant to the potential, under the assumptions of Theorem 5.14, let us prove that mc }mc} is the unique equilibrium state for the potential F 6 c ´δc under the continuous time geodesic flow on ΓzG X, and that the pressure of F 6 c ´δc vanishes. The last claim of Theorem 5.14 follows, since

P F 6 c ´δc " P F 6 c ´δc .
17 This requires only minor modifications to the definition given in the Introduction for bounded potentials.

18 They have a positive lower bound by definition, see Section 2.7.

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We refer to the paragraphs before the statement of Theorem 5.9 for the definitions of ' the system of conductances 7 r c for Γ on the simplicial tree X, ' the (two-sided) topological Markov shift pΣ, σ, Pq on the alphabet A , conjugated to the discrete time geodesic flow `ΓzG X, 7 g 1 ,

m F 7 c }m F 7 c } ˘by the homeomorphism Θ : ΓzG X Ñ Σ, ' the roof function r : Σ Ñ s0, `8r
' and the suspension pΣ, σ, a Pq r " pΣ r , pσ t r q tPR , a P r q over pΣ, σ, a Pq with roof function r, conjugated to the continuous time geodesic flow `ΓzG X, mc }mc} , pg t q tPR ˘by the homeomorphism Θ r : ΓzG X Ñ Σ r defined at the end of the proof of Theorem 5.9. We will always (uniquely) represent the elements of Σ r as rx, ss with x P Σ and 0 ď s ă rpxq.

We denote by F 6 c, symb : Σ r Ñ R the potential defined by (5.16) which is continuous, as a composition of continuous maps. The key technical observation in this proof is the following one.

F 6 c, symb " F 6 c ˝Θ´1 r ,
Lemma 5.15. For every x P Σ, we have F7 c, symb pxq " ş rpxq 0 pF 6 c, symb ´δc qprx, ssq ds. For every x P Σ, the sign of F 6 c, symb prx, ssq is constant on s P r0, rpxqs.

Proof. Let x " px n q nPZ P Σ and x 0 " pe 0 , h 0 , e 0 q P A . By the definition of the first return time r in Equation (5.10), we have in particular rpxq " λpe 0 q .

By Equation (5.14) and by the definition of 7 r c in Equation (5.9), we have F7 c, symb pxq " 7 cpe 0 q " pcpe 0 q ´δc qλpe 0 q .

Using in the following sequence of equalities respectively ' the definitions of the potential F 6 c, symb in Equation (5.16) and of the suspension flow pσ t r q tPR , ' the fact that this suspension flow is conjugated to the continuous time geodesic flow by Θ r , ' the definition of Θ r using the reparametrisation map R of continuous time geodesic lines with origin on vertices to discrete geodesic lines,

' the definition of the potential F 6 c , ' the fact that the first edge followed by the discrete geodesic line Θ ´1x, hence by the geodesic line R ´1Θ ´1x, is e 0 and the relation between the potential F c associated with c and c (see Proposition 3.11), we have

ż rpxq 0 F 6 c, symb prx, ssq ds " ż rpxq 0 F 6 c `Θ´1 r σ s r rx, 0s ˘ds " ż rpxq 0 F 6 c `gs Θ ´1 r rx, 0s ˘ds " ż rpxq 0 F 6 c `gs R ´1Θ ´1x ˘ds " ż λpe 0 q 0 F c `vg s R ´1Θ ´1x ˘ds " cpe 0 qλpe 0 q . 88 19/12/2016
Since ş rpxq 0 δ c ds " δ c λpe 0 q, the first claim of Lemma 5.15 follows. The second claim follows by the definition of the potential F c associated with c, see Equation (3.11). l

By Equation (5.16), the pushforwards of measures by the homeomorphism Θ r , which conjugates the flows pg t q tPR and pσ t r q tPR , is a bijection from M

F 6 c to M F 6 c, symb
, such that

P F 6 c, symb
ppΘ r q ˚mq " P . In particular, we only have to prove that pΘ r q ˚mc }mc} " a P r is the unique equilibrium state for the potential F 6 c, symb ´δc under the suspension flow pσ t r q tPR , and that the pressure of F 6 c, symb ´δc vanishes. The uniqueness follows for instance from [START_REF] Iommi | Recurrence and transience for suspension flows[END_REF]Theo. 3.5], since the roof function r is locally constant and the potential g " F 6 c, symb is such that the map 19 from Σ to R defined by x Þ Ñ ş rptq 0 gprx, ssq ds is locally Hölder-continuous by Lemma 5.15 and since F7 c, symb is locally constant.

Let us now relate the σ-invariant measures on Σ with the pσ t r q tPR -invariant measures on Σ r . Recall that we denote the Lebesgue measure on R by ds and the points in Σ r by rx, ss with x P Σ and 0 ď s ď rpxq.

Lemma 5.16. The map S :

M F 7 c, symb Ñ M F 6 c, symb defined by d Spmqprx, ssq " 1 ş Σ r dm dµpxq ds
for every m P M F 7 c, symb is a bijection, such that

P F 6 c, symb ´δc pSpmqq " P F 7 c, symb pmq ş Σ r dm .
Proof. Note that ş Σ r dm is the total mass of the measure dµ r prx, ssq " dµpxq ds on Σ r . In particular, Spmq is indeed a probability measure.

Since r has a positive lower bound and a finite upper bound, it is well known since [AmK], see also [START_REF] Iommi | Recurrence and transience for suspension flows[END_REF]§2.4], that the map S defined above is a bijection from the set of σ-invariant probability measures m on Σ to the set of pσ t r q tPR -invariant probability measures on Σ r . Furthermore, for every σ-invariant probability measure m on Σ, we have the following Kac formula, by the definition of the probability measure Spmq and by Lemma 5.15,

ż Σr F 6 c, symb d Spmq ´δc " ż Σr pF 6 c, symb ´δc q d Spmq " 1 ş Σ r dm ż xPΣ ż rpxq 0 pF 6 c, symb ´δc qprx, ssq dmpxq ds " 1 ş Σ r dm ż Σ F7 c, symb dm .
(5.17)

19 denoted by ∆g in loc. cit.

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By the comment on the signs at the end of Lemma 5.15, this computation also proves that the negative part of F 6 c, symb is integrable for Spmq if and only if the negative part of F7 c, symb is integrable for m. Hence S is indeed a bijection from M F 7 c, symb to M F 6 c, symb . By Abramov's formula [Abr], see also [START_REF] Iommi | Recurrence and transience for suspension flows[END_REF]Prop. 2.14], we have

h Spmq pσ 1 r q " h m pσq ş Σ r dm .
(5.18)

The last claim of Lemma 5.16 follows by summation from Equations (5.17) and (5.18). l

By the proof of Theorem 5.12 (replacing the potential c by 7 c), the pressure of the potential F7 c, symb is equal to the critical exponent δ7 c of the potential 7 c, and by Lemma 5.10 (1), we have δ7 c " 0. Hence for every m P M F 7 c, symb , we have

P F 6 c, symb ´δc pSpmqq " P F 7 c, symb pmq ş Σ r dm ď P F 7 c, symb ş Σ r dm " δ7 c ş Σ r dm " 0 .
In particular, the pressure of the potential F 6 c, symb ´δc is at most 0, since S is a bijection. By the proof of Theorem 5.12 (replacing the potential c by 7 c), we know that P is an equilibrium state for the potential F7 c, symb . Hence

P F 6 c, symb ´δc pSpPqq " P F 7 c, symb pPq ş Σ r dP " 0 .
Therefore, SpPq is an equilibrium state of the potential F 6 c, symb ´δc , with pressure 0. But aP r , which is equal to Pr }Pr} since aP r is a probability measure, is by construction equal to SpPq.

The result follows. l

With slightly different notation, this result implies Theorem 1.1 in the Introduction.

Proof of Theorem 1.1. Any potential r F for Γ on T 1 X is cohomologous to a potential r F c associated with a system of conductances (see Proposition 3.12). If two potentials r F and r F 1 for Γ on T 1 X are cohomologous 20 then the potentials Þ Ñ r F pv q and Þ Ñ r F 1 pv q for Γ on G X are cohomologous for the definition given before the statement of Theorem 5.14. Since the existence and uniqueness of an equilibrium state depends only on the cohomology class of the potential on G X, the result follows. l 20 See the definition at the end of Section 3.2.

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Chapter 6

Random walks on weighted graphs of groups

Let X be a locally finite simplicial tree without terminal vertices, and let X " |X| 1 be its geometric realisation. Let Γ be a nonelementary discrete subgroup of AutpXq.

In Section 6.1, we define an operator ∆ c on the functions defined on the set of vertices of the quotient graph of groups Γz zX endowed with a system of conductances c : ΓzEX Ñ R. This operator is the infinitesimal generator of the random walk on Γz zX associated with the (normalised) exponential of this system of conductances. When Γ is torsion free and the system of conductances vanishes, the construction recovers the standard Laplace operator on the graph ΓzX.

Under appropriate anti-reversibility assumptions on the system of conductances, using techniques of Sullivan and Coornaert-Papadopoulos, we prove that the total mass of the Patterson densities is a positive eigenvector for the operator ∆ c associated with the system of conductances.

In Section 6.2, we study the nonsymmetric nearest neighbour random walks on V X associated with anti-reversible systems of conductivities, and we show that the Patterson densities are the harmonic measures of these random walks.

Laplacian operators on weighted graphs of groups

Let X be a locally finite simplicial tree without terminal vertices, and let X " |X| 1 be its geometric realisation. Let Γ be a nonelementary discrete subgroup of AutpXq. Let r c : EX Ñ R be a (Γ-invariant) system of conductances for Γ.

We define r c `" r c and r c ´: e Þ Ñ r cpeq, which is another system of conductances for Γ, and we denote by c ˘: ΓzEX Ñ R the quotient maps. Recall (see Section 3.5) that r c is reversible (respectively anti-reversible) if r c ´" r c `(respectively r c ´" ´r c `). For every x P V X, we define deg

r c ˘pxq " ÿ ePEX, opeq"x e r c ˘peq .
The quotient graph of groups Γz zX is endowed with the quotient maps c ˘:

ΓzEX Ñ R of r c ˘.
Also note that the quantity deg On the vector space C V X of maps from V X to C, we consider the operator ∆ c ˘, called the (weighted) Laplace operator of pX, c ˘q, 1 defined by setting, for all f P C V X and x P V X,

∆ c ˘f pxq " 1 deg r c ˘pxq ÿ ePEX, opeq"x e r c ˘peq `f pxq ´f ptpeqq ˘. (6.1)
This is the standard Laplace operator 2 of a weighted graph with the weight e Þ Ñ e r c ˘peq , except that usually one requires that r cpeq " r cpēq. Note that p ˘peq " e c ˘peq deg c ˘popeqq is a Markov transition kernel on the tree X, see Section 6.2.

The weighted Laplace operator ∆ c ˘is invariant under Γ: for all f P C V X and γ P Γ, we have ∆ c ˘pf ˝γq " p∆ c ˘f q ˝γ .

In particular, this operator induces an operator on functions defined on the quotient graph ΓzX, as follows.

Let pY, G ˚q be a graph of finite groups and let i : EY Ñ N ´t0u be the index map ipeq " rG opeq : G e s. For every function c : EY Ñ R, let deg c : V Y Ñ R be the positive function defined by deg c pxq "

ÿ ePEY, opeq"x ipeq e cpeq .
The Laplace operator 3 of pY, G ˚, cq is the operator

∆ c " ∆ Y,G˚,c on L 2 pV Y, vol Y,G˚q defined by ∆ c f : x Þ Ñ 1 deg c pxq ÿ ePEY, opeq"x ipeq e cpeq `f pxq ´f ptpeqq ˘.
Remark 6.1.

(1) Let pY, G ˚q " Γz zX be a graph of finite groups with p : V X Ñ V Y " ΓzV X the canonical projection. Let r c : EX Ñ R be a potential and let c : EY " ΓzEX Ñ R be the map induced by r c. An easy computation shows that for all f P C V Y and x P V Y, we have

∆ c f pxq " ∆ c r f pr xq if r f " f ˝p : V X Ñ C and r x P V X satisfies ppr xq " x.
(2) For every x P V Y, let ipxq "

ÿ ePEY, opeq"x ipeq .
Then ipxq is the degree of any vertex of any universal cover of pY, G ˚q above x. In particular, the map i : V Y Ñ R is bounded if and only if the universal cover of pY, G ˚q has uniformly bounded degrees. When c " 0, we denote the Laplace operator by ∆ " ∆ Y, G˚a nd for every x P V Y, we have

∆ 0 f pxq " 1 ipxq ÿ ePEY, opeq"x ipeq `f pxq ´f ptpeqq ˘.
We thus recover the Laplace operator of [Mor] on the edge-indexed graph pY, iq.

1 or on X associated with the system of conductances r c 2See for example [Car] with the opposite choice of the sign, or [ChGY]. 3 See for instance [Mor] when c " 0.

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Returning to general graphs of finite groups, we denote by L 2 pV Y, vol pY,G˚q q the Hilbert space of maps f : V Y Ñ C with finite norm }f } vol for the following scalar product:

xf, gy vol " ÿ xPV Y 1 |Γ x | f pxq gpxq .
We denote by L 2 pEY, Tvol pY,G˚q q the Hilbert space of maps φ : EY Ñ C with finite norm }φ} Tvol for the following scalar product:

xφ, ψy Tvol " 1 2 ÿ ePEY 1 |Γ e | φpeq ψpeq .
Proposition 6.2. Let pY, G ˚q be a graph of finite groups, whose map i : V Y Ñ R is bounded. Let c : EY Ñ R be a system of conductances on Y, and let

ppeq " e cpeq deg c popeqq for every e P EY.

(1) The Laplace operator ∆ c : L 2 pV Y, vol pY,G˚q q Ñ L 2 pV Y, vol pY,G˚q q is linear and bounded.

(2) The map d c : L 2 pV Y, vol pY,G˚q q Ñ L 2 pEY, Tvol pY,G˚q q defined by d c pf q : e Þ Ñ a ppeq `f ptpeqq ´f popeqq ȋs linear and bounded, and its dual operator

d c : L 2 pEY, Tvol pY,G˚q q Ñ L 2 pV Y, vol pY,G˚q q
is given by

d c pφq : x Þ Ñ ÿ ePEY, opeq"x ipeq 2 ´appeq φpeq ´appeq φpeq (3) Assume that c is reversible. Then ∆ c " d c d c
In particular, ∆ c is self-adjoint and nonnegative.

Proof. By the assumptions, there exists M P N such that ipxq ď M for every x P V Y, and hence ipeq ď M for every e P EY. Note that ipeq "

|Γ opeq |
|Γe| , that Γ e " Γ e and that ppeq ď 1 .

(1) For every f P L 2 pV Y, vol pY,G˚q q, using the Cauchy-Schwarz inequality and the change of 93 19/12/2016 variable e Þ Ñ e, we have

}∆ c f } 2 vol " ÿ xPV Y 1 |Γ x | ˇˇÿ ePEY, opeq"x ipeq ppeq `f pxq ´f ptpeqq ˘ˇˇ2 ď ÿ xPV Y 1 |Γ x | ´ÿ ePEY, opeq"x ipeq 2 ppeq 2 ¯´ÿ ePEY, opeq"x ˇˇf pxq ´f ptpeqq ˇˇ2 ď 2 M 2 ÿ xPV Y 1 |Γ x | ´ÿ ePEY, opeq"x ˇˇf pxq ˇˇ2 `ˇf ptpeqq ˇˇ2 ď 2 M 2 ÿ ePEY 1 |Γ e | ˇˇf popeqq ˇˇ2 `ˇf ptpeqq ˇˇ2 " 4 M 2 ÿ ePEY 1 |Γ e | ˇˇf popeqq ˇˇ2 " 4 M 2 ÿ xPV Y 1 |Γ x | ÿ ePEY, opeq"x ipeq |f pxq| 2 ď 4 M 3 ÿ xPV Y 1 |Γ x | |f pxq| 2 " 4 M 3 }f } 2 vol .
Hence the linear operator ∆ c is bounded.

(2) For every f P L 2 pV Y, vol pY,G˚q q, we have

}d c f } 2 Tvol " 1 2 ÿ ePEY ppeq |Γ e | ˇˇf ptpeqq ´f popeqq ˇˇ2 ď ÿ ePEY 1 |Γ e | ´ˇf ptpeqq ˇˇ2 `ˇf popeqq ˇˇ2 ¯" 2 ÿ ePEY 1 |Γ e | ˇˇf popeqq ˇˇ2 " 2 ÿ ePEY ipeq |Γ opeq | ˇˇf popeqq ˇˇ2 " 2 ÿ xPV Y ipxq |Γ x | |f pxq| 2 ď 2 M }f } 2 vol .
Hence the linear operator d c is bounded. For all f P L 2 pV Y, vol pY,G˚q q and φ P L 2 pEY, Tvol pY,G˚q q, using again the change of variable e Þ Ñ e, we have

xφ, d c f y Tvol " 1 2 ÿ ePEY 1 |Γ e | a ppeq φpeq `f ptpeqq ´f popeq " 1 2 ˆÿ ePEY a ppeq |Γ e | φpeq f popeqq ´ÿ ePEY a ppeq |Γ e | φpeq f popeqq " ÿ xPV Y 1 |Γ x | ÿ ePEY, opeq"x ipeq 2 ´appeq φpeq ´appeq φpeq ¯f pxq .
This gives the formula for d c .

(3) For all f, g P L 2 pV Y, vol pY,G˚q q, since ppeq " ppeq by reversibility, by making the change This proves the last claim in Proposition 6.2. l

The following result is an extension to anti-reversible systems of conductances of [START_REF] Coornaert | Récurrence de marches aléatoires et ergodicité du flot géodésique sur les graphes réguliers[END_REF]Prop. 3.3] (who treated the case of zero conductances), which is a discrete version of Sullivan's analogous result for hyperbolic manifolds (see [START_REF] Sullivan | Related aspects of positivity in Riemannian manifolds[END_REF]). Let r F c : T 1 X Ñ R be the potential for Γ associated with c, so that p r F c q ˘" r F c ˘, and let δ c be their critical exponent. Let C ˘: B 8 X ˆV X ˆV X Ñ R be the associated Gibbs cocycles. Let pµ x q xPV X be two Patterson densities on B 8 X for the pairs pΓ, F c ˘q. Proposition 6.3. Assume that X is pq `1q-regular, that the system of conductances c is anti-reversible and that the map deg c ˘: V X Ñ R is constant with value κ ˘. Then the total mass r φ ˘: x Þ Ñ }µ x } of the Patterson density is a positive eigenvector associated with the eigenvalue

1 ´eδc `qe ´δc κ ˘.
for the Laplace operator ∆ c ˘on C V X .

Proof. Note that c ˘: V X Ñ R is bounded, hence c ˘: EX Ñ R is bounded, hence p r F c q ˘" r F c ˘is bounded. As X is pq `1q-regular, the critical exponent δ Γ is finite and hence also the critical exponent δ c is finite by Assertions ( 6) and (7) of Lemma 3.7. Since

r φ ˘pxq " ż B8X dµ x " ż B8X e ´Cξ px, x 0 q dµ x0 ,
by Equation (4.2) and by linearity, we only have to prove that for every fixed ξ P B 8 X the map

f : x Þ Ñ e ´Cξ px, x 0 q is an eigenvector with eigenvalue 1 ´eδc `qe ´δc κ ˘for ∆ c ˘.
For every e P EX, recall4 that B e X is the set of points at infinity of the geodesic rays whose initial edge is e. By Equation (3.8) and by the definition of the potential associated with a system of conductances5 , for all e P EX and η P B e X, we have

C η ptpeq, opeqq " ż tpeq opeq p r F c ˘´δ c q " c ˘peq ´δc .

95

19/12/2016 Thus f ptpeqq " e ´Cξ ptpeq, opeqq´C ξ popeq, x 0 q " e ´c˘p eq`δc f popeqq if ξ P B e X, and otherwise f ptpeqq " e C ξ ptpeq, opeqq´C ξ popeq, x 0 q " e c ˘peq´δc f popeqq .

For every x P V X, let e ξ be the unique edge of X with origin x such that ξ P B e ξ X. Then,

∆ c ˘f pxq " f pxq ´1 c ˘pxq ÿ opeq"x e c ˘peq f ptpeqq " f pxq ´1 κ ˘ec ˘pe ξ q f ptpe ξ qq ´1 κ ˘ÿ e‰e ξ , opeq"x e c ˘peq f ptpeqq " ´1 ´eδc κ ˘´q e ´δc κ ˘¯f pxq .
This proves the result. l

Note that the anti-reversibility of the potential is used in an essential way in order to get the last equation in the proof of Proposition 6.3.

Patterson densities as harmonic measures for simplicial trees

In this Section, we define and study a Markov chain on the set of vertices of a simplicial tree endowed with a discrete group of automorphisms and with an appropriate system of conductances, such that the associated (nonsymmetric, nearest neighbour) random walk converges almost surely to points in the boundary of the tree, and we prove that the Patterson densities, once normalised, are the corresponding harmonic measures. We thereby generalise the zero potential case treated in [START_REF] Coornaert | Récurrence de marches aléatoires et ergodicité du flot géodésique sur les graphes réguliers[END_REF], which is also a special case of [CoM] when X is a tree under the additional restriction that the discrete group is cocompact. For other connections between harmonic measures and Patterson measures, we refer for instance to [CoM, BlHM, Tan, GouMM] and their references.

Let X be a pq `1q-regular simplicial tree, with q ě 2. Let Γ be a nonelementary discrete subgroup of AutpXq. Let r c : EX Ñ R be an anti-reversible system of conductances for Γ, such that the associated map r c : V X Ñ R on the vertices of X is constant. Let pµ x q xPV X be a Patterson density for pΓ, F c q, where F c is the potential associated with c. We denote by φ µ : x Þ Ñ }µ x } the associated total mass function on V X.

We start this Section by recalling a few facts about discrete Markov chains, for which we refer for instance to [START_REF] Revuz | Markov chains[END_REF][START_REF] Woess | Random walks on infinite graphs and groups -a survey on selected topics[END_REF]. A state space is a discrete and countable set I. A transition kernel on a I is a map P : I ˆI Ñ r0, 1s such that for every x P I,

ÿ yPI P px, yq " 1 .
Let λ be a probability measure on I. A (discrete) Markov chain on a state space I with initial distribution λ and transition kernel P is a sequence pZ n q nPN of random variables with values in I such that for all n P N and x 0 , . . . , x n`1 P I, the probability of events P satisfies 96 19/12/2016

(1) PrZ 0 " x 0 s " λptx 0 uq,

(2) PrZ n`1 " x n`1 | Z 0 " x 0 , Z 1 " x 1 , . . . , Z n " x n s " P px n , x n`1 q.
The associated random walk consists in choosing a point x 0 in I with law λ, and by induction, once x n is constructed, in choosing x n`1 in I with probability P px n , x n`1 q. Note that PrZ 0 " x 0 , Z 1 " x 1 , . . . , Z n " x n s " λptx 0 uq P px 0 , x 1 q . . . P px n´1 , x n q .

When the initial distribution λ is the unit Dirac mass ∆ x at x P I, the Markov chain is then uniquely determined by its transition kernel P and by x, and is denoted by pZ x n q nPN . For every n P N, we denote by P pnq the iterated matrix product of the transition kernel P : we have P p1q " P , P p0q px, yq is the Kronecker symbol δ x,y for all x, y P I, and by induction P pn`1q " P ¨P pnq where ¨is the matrix product of I ˆI matrices, that is, for all x, z P I, P pn`1q px, zq " ÿ yPI P px, yqP pnq py, zq .

Note that

P pnq px, yq " PrZ x n " ys is the probability for the random walk starting at time 0 from x of being at time n at the point y. The Green kernel of P is the map G P from I ˆI to r0, `8s defined by px, yq Þ Ñ G P px, yq "

ÿ nPN P pnq px, yq ,
and its Green function is the following power series in the complex variable z :

G P px, y | zq " ÿ nPN P pnq px, yq z n .
Recall that if G P px, yq ‰ 0 for all x, y P I, then the random walk is recurrent6 if G P px, yq " 8 for any (hence all) px, yq P I ˆI, and transient otherwise. Note that, using again matrix products of I ˆI matrices, G P " Id `P ¨GP . (6.2)

We will from now on consider as state space the set V X of vertices of X. If a Markov chain pZ x n q nPN starting at time 0 from x converges almost surely in V X Y B 8 X to a random variable Z x 8 , the law of Z x 8 is called the harmonic measure (or hitting measure on the boundary) associated with this Markov chain, and is denoted by

ν x " pZ x 8 q ˚pPq .
Note that ν x is a probability measure on B 8 X.

For instance, the transition kernel of the simple nearest neighbour random walk on X is defined by taking as transition kernel the map P where P px, yq " 1 q `1 Apx, yq 97 19/12/2016 for all x, y P V X, with A : V X ˆV X Ñ t0, 1u the adjacency matrix of the tree X, defined by Apx, yq " 1 for any two vertices x, y of X that are joined by an edge in X and Apx, yq " 0 otherwise. We denote by Gpx, y | zq "

ÿ kPN P pnq px, yq z n .
the Green function of P , whose radius of convergence is r " q`1 2 ?

q and which diverges at z " r, see for example [START_REF] Woess | Random walks on infinite graphs and groups -a survey on selected topics[END_REF], [START_REF] Woess | Denumerable Markov chains[END_REF]Ex. 9.82], [START_REF] Lyons | Probability on trees and networks[END_REF]§6.3].

The antireversible system of conductances r c : EX Ñ R defines a cocycle on the set of vertices of X, as follows. For every u, v in V X, let cpu, vq " 0 if u " v and otherwise let cpx, yq "

n ÿ i"1 r cpe i q ,
where pe 1 , e 2 , . . . , e m q is the geodesic edge path in X from u " ope 1 q to v " tpe n q. Lemma 6.4. (1) For every edge path pe 1

1 , e 1 2 , . . . , e 1 n 1 q from u to v, we have cpu, vq "

n 1 ÿ i"1 r cpe 1 i q .
(2) The map c : V X ˆV X Ñ R has the following cocycle property: for all u, v, w P V X, cpu, vq `cpv, wq " cpu, wq and hence cpv, uq " ´cpu, vq .

(3) We have cpu, vq "

ş v u r F c .
(4) For all ξ P B 8 X and u, v P V X, if C c ¨p¨, ¨q is the Gibbs cocycle associated with r F c , we have C c ξ pu, vq " cpv, uq `δc β ξ pu, vq . Proof. (1) Since X is a simplicial tree, any nongeodesic edge path from u to v has a back-andforth on some edge, which contributes to 0 to the sum defining cpx, yq by the anti-reversibility assumption on the system of conductances. Therefore, by induction, the sum in Assertion (1) indeed does not depend on the choice of the edge path from u to v.

Assertion (2) is immediate from Assertion (1). Assertion (3) follows from the definition of cp¨, ¨q by Proposition 3.11.

(4) For every ξ P B 8 X, if p P V X is such that ru, ξr X rv, ξr " rp, ξr , then using Equation (3.6) and Assertions (3) and (2), we have

C c ξ pu, vq " ż p v p r F c ´δc q ´ż p u p r F c ´δc q " cpv, pq ´cpu, pq `δc β ξ pu, vq " cpv, uq `δc β ξ pu, vq . l
We now define the transition kernel P c associated with 7 the (logarithmic) system of conductances c by, for all x, y P V X, P c px, yq " κ c φ µ pyq φ µ pxq e cpx, yq P px, yq .

7

The transition kernel also depends on the choice of the Patterson density if Γ is not of divergence type.

19/12/2016

From now on, we denote by pZ x n q nPN the Markov chain with initial distribution ∆ x and transition kernel P c . Let κ c " 1 `q e δc `q e ´δc , which belongs to s0, q`1 2 ? q s . Note that this constant κ c is less than the radius of convergence r of the Green function Gpx, y | zq if and only if coshpδ c ´1 2 ln qq ą 1, that is, if and only if δ c ‰ 1 2 ln q. The computation (due to Kesten) of the Green function of P is well known, and gives the following formula, see for instance [START_REF] Coornaert | Récurrence de marches aléatoires et ergodicité du flot géodésique sur les graphes réguliers[END_REF]Prop. 3.1]: If δ c ‰ 1 2 ln q, then there exists α ą 0 such that for all x, y P V X Gpx, y | κ c q " α e ´δc dpx, yq .

(6.3) Lemma 6.5. (1) The map P c is indeed a transition kernel on V X.

(

) The Green kernel G c " G Pc of P c is G c px, yq " e cpx, yq φ µ pyq φ µ pxq Gpx, y | κ c q . (6.4) 2
In particular, the Green kernel of P c is finite if δ c ‰ 1 2 ln q.

(3) Assume that δ c ‰ 1 2 ln q. For all x, y, z P V X, we have φ µ pyq G c py, zq φ µ pxq G c px, zq " e cpy, xq`δcpdpx, zq´dpy, zqq .

If furthermore z R rx, yr , then, for every ξ P O x pzq,

φ µ pyq G c py, zq φ µ pxq G c px, zq " e C c ξ px, yq .
Proof.

(1) By Proposition 6.3, the positive function φ µ is an eigenvector with eigenvalue e δc `q e ´δc for the operator

f Þ Ñ tx Þ Ñ ÿ ePEX, opeq"x
e r cpeq f ptpeqqu .

Since P popeq, tpeqq " 1 q`1 for every e P EX, we hence have ÿ yPV X P c px, yq " ÿ ePEX, opeq"x P c px, tpeqq " 1 `q pe δc `q e ´δc q φ µ pxq ÿ ePEX, opeq"x e r cpeq φ µ ptpeqq P px, tpeqq " 1 .

(2) Let us first prove that for all x, y P V X and n P N, we have P c px, x 1 q P c px 1 , x 2 q . . . P c px n´2 , x n´1 q P c px n´1 , yq " pκ c q n φ µ pyq φ µ pxq e cpx, yq ÿ x 1 ,..., x n´1 PV X P px, x 1 q P px 1 , x 2 q . . . P px n´2 , x n´1 q P px n´1 , yq " pκ c q n φ µ pyq φ µ pxq e cpx, yq P pnq px, yq .

P pnq c px, yq " pκ c q n φ µ pyq φ µ pxq e cpx
Equation (6.4) follows from Equation (6.5) by summation on n. As we have already seen, κ c ă r if and only if δ c ‰ 1 2 ln q. The last claim of Assertion (2) follows.

(3) Let x, y, z P V X. Using (twice) Assertion (2), the cocycle property of c and (twice) Equation ( 6.3), we have G c py, zq G c px, zq " e cpy, zq φ µ pzq φ µ pxq Gpy, z | κ c q e cpx, zq φµpyq φ µ pzq Gpx, z | κ c q " e cpy, xq φ µ pxq φ µ pyq α e ´δc dpy, zq α e ´δc dpx, zq " φ µ pxq φ µ pyq e cpy, xq`δcpdpx, zq´dpy, zqq .

This proves the first claim of Assertion (3). Under the additional assumptions on x, y, z, ξ, we have β ξ px, yq " dpx, zq ´dpy, zq .

The last claim of Assertion (3) hence follows from Lemma 6.4 (4). l

Using the criterion that the random walk starting from a given vertex of X with transition probabilities P c is transient if and only if the Green kernel G c px, yq of P c is finite (for any, hence for all, x, y P X), Lemma 6.5 (2) implies that if δ c ‰ 1 2 ln q, then pZ x n q nPN almost surely leaves every finite subset of V X. The following result strengthens this remark. Proposition 6.6. If δ c ‰ 1 2 ln q, then for every x P V X, the Markov chain pZ x n q nPN (with initial distribution ∆ x and transition kernel P c ) converges almost surely in V X Y B 8 X to a random variable with values in B 8 X. In particular the harmonic measure ν x of pZ x n q nPN is well defined if δ c ‰ 1 2 ln q. Proof. Since X is a tree, if px n q nPN is a sequence in V X such that dpx n , x n`1 q " 1 for every n P N and which does not converge to a point in B 8 X, then there exists a point y such that this sequence passes infinitely often through y, that is, tn P N : x n " yu is infinite. The result then follows from the fact that the Markov chain pZ x n q nPN is transient since

δ c ‰ 1 2 ln q. l
The following result, generalising [START_REF] Coornaert | Récurrence de marches aléatoires et ergodicité du flot géodésique sur les graphes réguliers[END_REF] when c " 0, says that the Patterson measures associated with a system of conductances c, once renormalised to probability measures, are exactly the harmonic measures for the random walk with transition probabilities P c . Theorem 6.7. Let pX, Γ, r c, pµ x q xPV X q be as in the beginning of Section 6.2. If δ c ‰ 1 2 ln q, then for every x P X, the harmonic measure of the Markov chain pZ x n q nPN is ν

x " µ x }µ x } . 100 19/12/2016
Proof. We fix x P X. Recall that given z P V X, the shadow O x pzq of z seen from x is the set of points at infinity of the geodesic rays from x through z.

For every n P N, we denote by Spx, nq and Bpx, nq the sphere and ball of centre x and radius n in V X, and we define two maps f 1 , f 2 : V X Ñ R with finite support by

f 1 pzq " µ x pO x pzqq }µ x } G c px, zq
and f 2 pzq " ν x pO x pzqq G c px, zq if z P Spx, nq, and f 1 pzq " f 2 pzq " 0 otherwise. Let us prove that f 1 " f 2 for every n P N.

Since tO x pzq : z P V Xu generates the Borel σ-algebra of B 8 X, this proves that the Borel measures ν x and µx }µx} coincide. We will use the following criterion: For all maps G : V X ˆV X Ñ R and f : V X Ñ R such that f has finite support, let us again denote by G ¨f the matrix product of G and f : for every y P V X, G ¨f pyq "

ÿ zPV X Gpy, zqf pzq . Lemma 6.8. For all f, f 1 : V X Ñ R with finite support, if G c ¨f " G c ¨f 1 , then f " f 1 .
Proof. By Equation (6.2), we have

f 1 " G c ¨f 1 ´Pc ¨Gc ¨f 1 " G c ¨f ´Pc ¨Gc ¨f " f . l
Let us hence fix n P N and prove that G c ¨f1 " G c ¨f2 . Theorem 6.7 then follows.

Step 1: For every y P Bpx, nq, since tO x pzq : z P Spx, nqu is a Borel partition of B 8 X, by Equation (4.2), since z R rx, yr if z P Spx, nq and y P Bpx, nq, and by the second claim of Lemma 6.5 (3), we have

1 " 1 φ µ pyq ż B8X dµ y " 1 φ µ pyq ÿ zPSpx,nq ż Oxpzq e ´Cc ξ py, xq dµ x pξq " 1 φ µ pyq ÿ zPSpx,nq ż Oxpzq φ µ pyq G c py, zq φ µ pxq G c px, zq dµ x pξq " ÿ zPSpx,nq G c py, zq µ x pO x pzqq }µ x } G c px, zq " G c ¨f1 pyq . (6.6)
Step 2: For all y, z P V X such that z R rx, yr , any random walk starting at time 0 from y and converging to a point in O x pzq goes through z. Let us denote by C x pzq the set of vertices different from z on the geodesic rays from z to the points in O x pzq. Partioning by the last time the random walk passes through z, using the Markov property saying that what happens before the random walk arrives at z and after it leaves z are independent, we have ν y pO x pzqq " PrZ y 8 P O x pzqs " G c py, zq Pr@ n ą 0, Z Chapter 7

Skinning measures with potential on CATp´1q spaces

In this Chapter, we introduce skinning measures as weighted pushforwards of the Patterson-Sullivan densities associated with a potential to the unit normal bundles of convex subsets of a CATp´1q space. The development follows [START_REF] Parkkonen | Skinning measure in negative curvature and equidistribution of equidistant submanifolds[END_REF] with modifications to fit the present context. Let X, x 0 , Γ, r F be as in the beginning of Chapter 4. Let pµ x q xPX be Patterson densities on B 8 X for the pairs pΓ, F ˘q.

Skinning measures

Let D be a nonempty proper closed convex subset of X. The outer skinning measure r σ where ρ P B 1 ˘D, using the endpoint homeomorphisms ρ Þ Ñ ρ ˘from B 1 ˘D to B 8 X ´B8 D, and noting that ρp0q " P D pρ ˘q depends continuously on ρ ˘.

When r

F " 0, the skinning measure has been defined by Oh and Shah [OhS2] for the outer unit normal bundles of spheres, horospheres and totally geodesic subspaces in real hyperbolic spaces. The definition was generalised in [START_REF] Parkkonen | Skinning measure in negative curvature and equidistribution of equidistant submanifolds[END_REF] to the outer unit normal bundles of nonempty proper closed convex sets in Riemannian manifolds with variable negative curvature.

Note that the Gibbs measure is defined on the space G X of geodesic lines, the potential is defined on the space T 1 X of germs at time t " 0 of geodesic lines, and since B 1 ˘D is contained in G ˘, 0 X (see Section 2.5), the skinning measures are defined on the spaces G ˘, 0 X of (generalised) geodesic rays. In the manifold case, all the above spaces are canonically identified with the unit tangent bundle, but in general, the natural restriction maps G X Ñ T 1 X and G X Ñ G ˘, 0 X have infinite (though compact) fibers. where ρ is a geodesic ray starting (at time t " 0) from x.

(2) When the potential r F is reversible (in particular when F " 0), we have C ´" C `, we may (and we will) take µ x " µ x for all x P X, hence ι ˚r m F " r m F and r σ D " ι ˚r σ D.

(3) The (normalised) Gibbs cocycle being unchanged when the potential F is replaced by the potential F `σ for any constant σ, we may (and will) take the Patterson densities, hence the Gibbs measure and the skinning measures, to be unchanged by such a replacement.

When D is a horoball in X, let us now related the skinning measures of D with previously known measures on B 8 X, constructed using techniques due to Hamenstädt. Let H be a horoball centred at a point ξ P B 8 X. Recall that P H : B 8 X ´tξu Ñ BH is the closest point map on H , mapping η ‰ ξ to the intersection with the boundary of H of the geodesic line from η to ξ. The following result is proved in [START_REF] Hersonsky | Counting orbit points in coverings of negatively curved manifolds and Hausdorff dimension of cusp excursions[END_REF]§2.3] The following results give the basic properties of the skinning measures analogous to those in [START_REF] Parkkonen | Skinning measure in negative curvature and equidistribution of equidistant submanifolds[END_REF]Sect. 3] when the potential is zero. (iv) The support of r σ D is

tv P B 1 ˘D : v ˘P ΛΓu " P D pΛΓ ´pΛΓ X B 8 Dqq .
In particular, r σ D is the zero measure if and only if ΛΓ is contained in B 8 D.

For future use, the version 1 of Assertion (iii) when F " 0 is

dpg ˘sq ˚r σ D d r σ NsD pg ˘swq " e ´δ s . (7.7)
As another particular case of Assertion (iii) for future use, consider the case when X " |X| λ is the geometric realisation of a metric tree pX, λq and when r F " r F c is the potential associated with a system of conductances c on X (see Equation (3.11) and Proposition 3.12). Then for 1 contained in [START_REF] Parkkonen | Skinning measure in negative curvature and equidistribution of equidistant submanifolds[END_REF]Prop. 4] 105 19/12/2016 all w P B 1 `D (respectively w P B 1 ´D), if e w is the first (respectively the last) edge followed by w, with length λpe ω q, then ż πpg ˘λpeω q wq πpwq r F ˘" cpe w q by Proposition 3.11, so that d r σ NsD ppg ˘λpeωq wq |˘r0,`8r q " e ´cpeωq`δλpeωq dr σ Dpwq .

(7.8)

Proof. We give details only for the proof of claim (iii) for the measure r σ D, the case of r σ D being similar, and the proofs of the other claims being straightforward modifications of those in [START_REF] Parkkonen | Skinning measure in negative curvature and equidistribution of equidistant submanifolds[END_REF]Prop. 4]. Since `pg s wq |r0,`8r ˘`" w `and since w P B 1 `D if and only if pg s wq |r0,`8r P B 1 `Ns D, we have, using the definition of the skinning measure and the cocycle property (3.7), for all s ě 0, d r σ ǸsD ppg s wq |r0,`8r q " e C ẁ`p x 0 , πpg s wqq dµ x0 pw `q " e C ẁ`p πpwq, πpg s wqq d r σ Dpwq .

This proves the claim (iii) for r σ D, using Equation (3.8). l

Given two nonempty closed convex subsets D and D 1 of X, let

A D,D 1 " B 8 X ´pB 8 D Y B 8 D 1 q
and let h D,D 1 : P D pA D,D 1 q Ñ P D1 pA D,D 1 q be the restriction of P D1 ˝pP D q ´1 to P D pA D,D 1 q. It is a homeomorphism between open subsets of B 1 ˘D and B 1 ˘D1 , associating to the element w in the domain the unique element w 1 in the range with w 1 ˘" w ˘. The proof of Proposition 5 of [START_REF] Parkkonen | Skinning measure in negative curvature and equidistribution of equidistant submanifolds[END_REF] generalises immediately to give the following result. Let w P G ˘X . The homeomorphisms W ˘pwq ˆR Ñ W 0˘p wq, defined by setting p , sq Þ Ñ 1 " g s , conjugate the actions of R by translation on the second factor of the domain and by the geodesic flow on the range, and the actions of Γ (trivial on the second factor of the domain). Let us consider the measures ν w on W 0˘p wq given, using the above homeomorphism, by dν w p 1 q " e C w˘p wp0q, p0qq dµ W ˘pwq p q ds .

(7.12)

They satisfy pg t q ˚νw " ν w for all t P R (since if 1 " g s , then g ´t 1 " g s´t ). Furthermore, γ ˚νw " ν γw for all γ P Γ. In general, they depend on w, not only on W ˘pwq. Furthermore, the support of ν w is t 1 P W 0˘p wq : 1 ¯P ΛΓu. These properties follow easily from the properties of the skinning measures on the strong stable or strong unstable leaves.

Lemma 7.5. (i) For every nonempty proper closed convex subset D 1 in X, there exists R 0 ą 0 such that for all R ě R 0 , η ą 0, and w P B 1 ˘D1 , we have ν w pV w, η, R q ą 0. (ii) For all w P G ˘X and t P R, the measures ν ḡt w and ν w are proportional: ν ḡt w " e C w˘p wptq, wp0qq ν w .

Proof. (i) By [PaP14a, Lem. 7], 2 there exists R 0 ą 0 (depending only on D 1 and on the Patterson densities) such that for all R ě R 0 , w P B 1 `D1 and w 1 P B 1 ´D1 , we have µ W `pwq pB `pw, Rqq ą 0 and µ W ´pw 1 q pB ´pw 1 , Rqq ą 0. The result hence follows by the definitions of ν w and V w, η, R .

(ii) For all w P G ˘X , s, t P R and P W ˘pwq, we have by Equations (7.12) and (7.11), and by the cocycle property of C ˘, dν ḡt w pg s q " dν ḡt w pg s´t g t q " e C pg t wq ˘pg t wp0q, g t p0qq dµ W ˘pg t wq pg t q dps ´tq " e C w˘p wptq, ptqq e ´Cw ˘p p0q, ptqq dµ W ˘pwq p q ds " e C w˘p wptq, p0qq e ´Cw ˘pwp0q, p0qq dν w pg s q " e C w˘p wptq, wp0qq dν w pg s q . l

The following disintegration result of the Gibbs measure over the skinning measures of any closed convex subset is a crucial tool for our equidistribution and counting results. Recall the definition in Equation (2.10) of the flow-invariant open sets U D and the definition of the fibrations f D :

U D Ñ B 1
˘D from Section 2.5. Proposition 7.6. Let D be a nonempty proper closed convex subset of X. The restriction to U D of the Gibbs measure r m F disintegrates by the fibration f D : U D Ñ B 1 ˘D, over the skinning measure r σ D of D, with conditional measure ν ρ on the fiber pf D q ´1pρq " W 0˘p ρq of ρ P B 1 ˘D: when ranges over U D , we have

d r m F |U D p q " ż ρPB 1
˘D dν ρ p q dr σ Dpρq .

Proof. In order to prove the claim for the fibration f D , let φ P C c pU D q. Using in the various steps below:

2 whose proof extends to the present situation, although the notation is different. • the positive endpoint homeomorphism w Þ Ñ w `from B 1 `D to B 8 X ´B8 D, and the negative endpoint homeomorphism v 1 Þ Ñ v 1 ´from W `pwq to B 8 X ´tw `u, with s P R the real parameter such that v 1 " g ´sv P W `pwq where v P W 0`p wq, noting that t ´s depends only on v `" w `and v ´" v 1

´,

• the definitions of the measures µ W `pwq (see Equation (7.9)) and r σ D (see Equation (7.1)) and the cocycle property of C ˘,

• Equation (3.8) and the cocycle properties of C `, we have

ż vPU D φpvq d r m F pvq " ż v`PB8X´B8D ż v´PB8X´tv`u ż tPR φpvq e C v´p x0, πpvqq`C v`p x0, πpvqq dt dµ x0 pv ´q dµ x0 pv `q " ż wPB 1 `D ż v 1 PW `pwq ż sPR φpg s v 1 q e C v1 ´px0, πpg s v 1 qq`C ẁ`p x0, πpg s v 1 qq ds dµ x0 pv 1 ´q dµ x0 pw `q " ż wPB 1 `D ż v 1 PW `pwq ż sPR φpg s v 1 q e C v1 ´pπpv 1 q, πpg s v 1 qq`C ẁ`p πpwq, πpg s v 1 qq ds dµ W `pwq pv 1 q dr σ D pwq " ż wPB 1 `D ż v 1 PW `pwq ż sPR φpg s v 1 q e C ẁ`p
πpwq, πpv 1 qq ds dµ W `pwq pv 1 q dr σ D pwq , which implies the claim for the fibration f D . The proof for the fibration f D is similar. l

In particular, for every u P G ´X , applying the above proposition and a change of variable to D " HB ´puq for which B 1 `D " N ú pW ´puqq and

U D " G X ´W 0`p ιuq " ď wPW ´puq W 0`p wq ,
the restriction to G X ´W 0`p ιuq of the Gibbs measure r m F disintegrates over the strong unstable measure µ W ´puq " ppN ú q ´1q ˚r σ D, with conditional measure on the fiber W 0`p wq of w P W ´puq the measure ν ẃ " ν Ń ú pwq : for every φ P C c pG X ´W 0`p ιuqq, we have

ż G X´W 0`p ιuq φpvq d r m F pvq " ż wPW ´puq ż v 1 PW `pwq ż sPR φpg s v 1 q e C ẁ`p πpwq, πpv 1 qq ds dµ Ẃ `pwq pv 1 q dµ W ´puq pwq . (7.13)
Note that if the Patterson densities have no atoms, then the stable and unstable leaves have measure zero for the associated Gibbs measure. This happens for instance if the Gibbs measure m F is finite, see Corollary 4.6 and Theorem 4.5.

Equivariant families of convex subsets and their skinning measures

Let I be an index set endowed with a left action of Γ. A family D " pD i q iPI of subsets of X or of p G X indexed by I is Γ-equivariant if γD i " D γi for all γ P Γ and i P I. We will denote 108 19/12/2016 by " " " D the equivalence relation on I defined by i " j if and only if D i " D j and there exists γ P Γ such that j " γi. This equivalence relation is Γ-equivariant: for all i, j P I and γ P Γ, we have γi " γj if and only if i " j. We say that D is locally finite if for every compact subset K in X or in p G X, the quotient set ti P I : D i X K ‰ Hu{ " is finite.

Examples.

(1) Fixing a nonempty proper closed convex subset D of X, taking I " Γ with the left action by translations pγ, iq Þ Ñ γi, and setting D i " iD for every i P Γ gives a Γ-equivariant family D " pD i q iPI . In this case, we have i " j if and only if i ´1j belongs to the stabiliser Γ D of D in Γ, and I{ " " Γ{Γ D . Note that γD depends only on the class rγs of γ in Γ{Γ D . We could also take I 1 " Γ{Γ D with the left action by translations pγ, rγ 1 sq Þ Ñ rγγ 1 s, and D 1 " pγDq rγsPI 1 , so that for all i, j P I 1 , we have i " D 1 j if and only if i " j, and besides, D 1 is locally finite if and only if D is locally finite. The following choices of D yield equivariant families with different characteristics:

(a) Let γ 0 P Γ be a loxodromic element with translation axis D " Ax γ 0 . The family pγDq γPΓ is locally finite and Γ-equivariant. Indeed, by Lemma 2.1, only finitely many elements of the family pγDq γPΓ{Γ D meet any given bounded subset of X.

(b) Let P G X be a geodesic line whose image under the canonical map G X Ñ ΓzG X has a dense orbit in ΓzG X under the geodesic flow, and let D " pRq be its image. Then the Γ-equivariant family pγDq γPΓ is not locally finite.

(c) More generally, let D be a convex subset such that Γ D zD is compact. Then the family pγDq γPΓ is a locally finite Γ-equivariant family.

(d) Let ξ P B 8 X be a bounded parabolic limit point of Γ, and let H be any horoball in X centred at ξ. Then the family pγH q γPΓ is a locally finite Γ-equivariant family.

(2) More generally, let pD α q αPA be a finite family of nonempty proper closed convex subsets of X, and for every α P A, let F α be a finite set. Define I " Ť αPA Γ ˆtαu ˆFα with the action of Γ by left translation on the first factor, and for every i " pγ, α, xq P I, let D i " γD α . Then I{ " " Ť αPA Γ{Γ D α ˆtαu ˆFα and the Γ-equivariant family D " pD i q iPI is locally finite if and only if the family pγD α q γPΓ is locally finite for every α P A. The cardinalities of F α for α P A contribute to the multiplicities (see Section 12.2).

Let D " pD i q iPI be a locally finite Γ-equivariant family of nonempty proper closed convex subsets of X. Let Ω " pΩ i q iPI be a Γ-equivariant family of subsets of p G X, where Ω i is a measurable subset of B 1 ˘Di for all i P I (the sign ˘being constant). Then

r σ Ω " ÿ iPI{" r σ Di | Ω i ,
is a well-defined Γ-invariant locally finite measure on p G X, whose support is contained in G ˘, 0 X. Hence, the measure r σ Ω induces a locally finite measure on Γz p G X, denoted by σ Ω , see for example [START_REF] Paulin | Equilibrium states in negative curvature[END_REF]§2.6], in particular for warnings concerning the fact that Γ does not always act freely on G X. When Ω " B 1 ˘D " pB 1 ˘Di q iPI , the measure r σ Ω is denoted by Example. Consider the Γ-equivariant family D " pγDq γPΓ{Γx with D " txu a singleton in X.

r σ D " ÿ iPI{" r σ Di .
With π ˘" pP D | B8X q ´1 : B 1 ˘D Ñ B 8 X the homeomorphism ρ Þ Ñ ρ ˘, we have pπ ˘q˚r σ D " µ
x by Remark (1) in Section 7, and

}σ D } " }µ x } |Γ x | . (7.14)
Chapter 8

Explicit measure computations for simplicial trees and graphs of groups

In this Chapter, we compute skinning measures and Bowen-Margulis measures for some highly symmetric simplicial trees X endowed with a nonelementary discrete subgroup Γ of AutpXq.

The potentials F are supposed to be 0 in this Chapter, and we assume that the Patterson densities pµ x q xPV X and pµ x q xPV X of Γ are equal, denoted by pµ x q xPV X . As the study of geometrically finite discrete subgroups of AutpXq mostly reduces to the study of particular (tree) lattices (see Remark 2.12), we will assume that Γ is a lattice in this Chapter.

The results of these computations will be useful when we state special cases of the equidistribution and counting results in regular and biregular trees and, in particular, in the arithmetic applications in Part III. The reader only interested in the continuous time case may skip directly to Chapter 9.

A rooted simplicial tree pX, x 0 q is spherically symmetric if X is not reduced to x 0 and has no terminal vertex, and if the stabiliser of x 0 in AutpXq acts transitively on each sphere of centre x 0 . The set of isomorphism classes of spherically symmetric rooted simplicial trees pX, x 0 q is in bijection with the set of sequences pp n q nPN in N ´t0u, where p n `1 is the degree of any vertex of X at distance n from x 0 .

x 0 p 0 `1 p 2 p 1 p 3 . . .
If pX, x 0 q is spherically symmetric, it is easy to check that the simplicial tree X is uniform if and only if the sequence pp n q nPN is periodic with palindromic period in the sense that there exists N P N ´t0u such that p n`N " p n and p N ´n " p n for all n P N (such that n ď N for 111 19/12/2016 the second property). If N " 1, then X " X p 0 is the regular tree of degree p 0 `1, and if N " 2, then X " X p 0 , p 1 is the biregular tree of degrees p 0 `1 and p 1 `1.

The Hausdorff dimension h X of B 8 X for any visual distance is then

h X " 1 N lnpp 0 . . . p N ´1q ,
see for example [START_REF] Lyons | Random walks and percolation on trees[END_REF]p. 935].

Computations of Bowen-Margulis measures for simplicial trees

The next result gives examples of computations of the total mass of Bowen-Margulis measures for lattices of simplicial trees having some regularity properties. Analogous computations can be performed for Riemannian manifolds having appropriate regularity properties. We refer for instance to [START_REF] Parkkonen | Counting arcs in negative curvature[END_REF]Prop. 10] and [START_REF] Parkkonen | Counting common perpendicular arcs in negative curvature[END_REF]Prop. 20 (1)] for computations of Bowen-Margulis measures for lattices in the isometry group of the real hyperbolic spaces, and to [START_REF] Parkkonen | Counting and equidistribution in Heisenberg groups[END_REF]Lem. 4.2 (iii)] for the computation in the complex hyperbolic case. In both cases, the main point is the computation of the proportionality constant between the Bowen-Margulis measure and Sasaki's Riemannian volume of the unit tangent bundle. When dealing now with simplicial trees, similar consequences of homogeneity properties will appear below.

We refer to Section 2.7 for the definitions of T π, Tvol, TVol appearing in the following result.

Proposition 8.1. Let pX, x 0 q be a spherically symmetric rooted simplicial tree with associated sequence pp n q nPN such that X is uniform, and let Γ be a lattice of X.

(1) For every x P V X, let r x " dpx, AutpXqx 0 q, and let

c x " pp rx ´1qe 2 rx h X pp 0 `1q 2 p 2 1 . . . p 2 rx´1 p rx `2p 0 pp 0 `1q 2 if r x ‰ 0 and c x " p 0 p 0 `1 if r x " 0. Then }m BM } " ÿ rxsPΓzV X 1 |Γ x | `}µ x } 2 ´ÿ ePEX : opeq"x µ x pB e Xq 2 " }µ x 0 } 2 ÿ rxsPΓzV X c x |Γ x | . (8.1)
(2) If X " X p, q is the biregular tree of degrees p `1 and q `1, with V X " V p X \ V q X the corresponding partition of the set of vertices of X, if the Patterson density pµ x q xPV X of Γ is normalised so that }µ x } " p`1 ? p for all x P V p X, then pT πq ˚mBM " Tvol Γz zX and }m BM } " TVolpΓz zXq "

ÿ rxsPΓzVpX p `1 |Γ x | `ÿ rxsPΓzVqX q `1 |Γ x | . (8.2) 112 19/12/2016
(3) If X " X q is the regular tree of degree q `1, if the Patterson density pµ x q xPV X of Γ is normalised to be a family of probability measures, then π ˚mBM " q q `1 vol Γz zX and in particular }m BM } " q q `1 VolpΓz zXq .

(8.3)

Proof. Let us first prove the first equality of Assertion (1). For every x P V X, we may partition the set of geodesic lines P G X with p0q " x according to the two edges starting from x contained in the image of . The only restriction for the edges is that they are required to be distinct.

For every e P EX, recall from Section 2.7 that B e X is the set of points at infinity of the geodesic rays whose initial edge is e. For all e P EX and x P V X, say that e points away from x if opeq P rx, tpeqs, and that e points towards x otherwise. In particular, all edges with origin x point away from x. Hence by Equation (4.11), and since µ x " µ x " µ x , we have

π ˚mBM " ÿ rxsPΓzV X 1 |Γ x | ÿ e, e 1 PEX : opeq"ope 1 q"x, e‰e 1 µ x pB e Xq µ x pB e 1 Xq ∆ rxs (8.4) " ÿ rxsPΓzV X 1 |Γ x | ´`ÿ ePEX : opeq"x µ x pB e Xq ˘2
´ÿ ePEX : opeq"x µ x pB e Xq 2 ¯∆rxs .

(8.5)

This gives the first equality of Assertion (1).

Let us prove the second equality of Assertion (1). By homogeneity, we assume that }µ x 0 } " 1 and we will prove that

}m BM } " ÿ rxsPΓzV X c x |Γ x | .
Let N P N ´t0u be such that p n`N " p n and p N ´n " p n for all n P N, which exists since X is assumed to be uniform. Then the automorphism group AutpXq of the simplicial tree X acts transitively on the set of vertices at distance a multiple of N from x 0 . Hence for every x P V X, the distance r x " dpx, AutpXqx 0 q belongs to t0, 1, . . . , t N 2 uu, and there exists γ x , γ 1

x P AutpXq such that dpx, γ x x 0 q " r x , x P rγ x x 0 , γ 1 x x 0 s and dpγ x x 0 , γ 1 x x 0 q " N . The map x Þ Ñ r x is constant on the orbits of Γ in V X (actually on the orbits of AutpXq) and hence the right hand side of Equation (8.1) is well defined.

Since the family pµ Haus x q xPV X of Hausdorff measures of the visual distances pB 8 X, d x q is invariant under any element of AutpXq, since Γ is a lattice and by Proposition 4.14, we have δ Γ " h X and γ ˚µx " µ γx for all x P V X and γ P AutpXq.

Since pX, x 0 q is spherically symmetric, and since µ x 0 is a probability measure, we have by induction, for every e P EX pointing away from x 0 with dpx 0 , opeqq " n,

µ x 0 pB e Xq " 1 pp 0 `1q p 1 . . . p n (8.6)
if n ‰ 0, and µ x 0 pB e Xq " 1 p 0 `1 otherwise. For every fixed x P V X, let us now compute µ x pB e Xq for every edge e of X with origin x. Let γ " γ x , γ 1 " γ 1

x P AutpXq be as above. By the spherical transitivity, we may assume that e or e belongs to the edge path from γx 0 to γ 1 x 0 . Case 1: Assume first that e points away from γx 0 . There are p 0 `1 such edges starting from x if r x " 0, and p rx otherwise. By Equation (8.6) and by invariance under AutpXq of pµ Haus x q xPV X , we have

µ γx 0 pB e Xq " 1 pp 0 `1q p 1 . . . p rx ,
with the convention that the denominator is p 0 `1 if r x " 0. Since the map ξ Þ Ñ β ξ px, γx 0 q is constant with value ´rx on B e X, and by the quasi-invariance property of the Patterson density (see Equation (4.2)), we have µ x pB e Xq " e ´δΓ p´rxq µ γx 0 pB e Xq " e rxh X pp 0 `1q p 1 . . . p rx , with the same convention as above.

Case 2: Assume now that e points towards γx 0 . This implies that r x ě 1, and there is one and only one such edge starting from x. Then as above we have

µ γ 1 x 0 pB e Xq " 1 pp N `1q p N ´1 . . . p rx , and
µ
x pB e Xq " e ´δΓ p´pN ´rxqq µ γ 1 x 0 pB e Xq " e pN ´rxqhX pp N `1q p N ´1 . . . p rx .

Therefore, if we set for every x P V X, C x " ´ÿ ePEX : opeq"x µ x pB e Xq ¯2 ´ÿ ePEX : opeq"x µ x pB e Xq 2 , (8.7)

we have if r x ‰ 0, since e N h X " p 0 p 1 . . . p N ´1 and p N " p 0 ,

C x " ´prx e rxh X pp 0 `1q p 1 . . . p rx `epN´rxqh X pp N `1q p N ´1 . . . p rx ¯2 ´´p rx `erxh X pp 0 `1q p 1 . . . p rx ˘2 ``e pN ´rxqhX pp N `1q p N ´1 . . . p rx ˘2 " pp rx 2 ´prx q e 2 rxh X pp 0 `1q 2 p 1 2 . . . p rx 2 `2 p rx e N h X pp 0 `1q p 1 . . . p rx p rx . . . p N ´1pp N `1q " pp rx ´1q e 2 rxh X pp 0 `1q 2 p 1 2 . . . p rx´1 2 p rx `2 p 0 pp 0 `1q 2 " c x ,
and, if r x " 0,

C x " ´pp 0 `1q 1 p 0 `1 ¯2 ´pp 0 `1q ´1 p 0 `1 ¯2 " p 0 p 0 `1 " c x . 114 19/12/2016
Assertion (1) of Proposition 8.1 now follows from Equation (8.5).

Let us prove Assertion (2) of Proposition 8.1. Note that X " X p, q is spherically symmetric with respect to any vertex of X, and that

h X " 1 2 lnppqq .
Let e be an edge of X, with x " opeq P V p X and y " tpeq P V q X. For every z P V X, we define C z as in Equation (8.7). Note that by homogeneity, we have C z " C x and }µ z } " }µ x } for all z P V p X, as well as C z " C y and }µ z } " }µ y } for all z P V q X. Hence the normalisation of the Patterson density as in the statement of Assertion (2) is possible. By the spherical symmetry at x, and the normalisation of the measure, we have µ x pB e Xq " 1 ? p and µ x pB e Xq " ? p. Therefore }µ y } " µ y pB e Xq `µy pB e Xq " e h X µ x pB e Xq `e´h X µ x pB e Xq " ? pq

1 ? p `1 ? pq ? p " q `1 ? q .
This symmetry in the values of }µ y } and }µ x } explains the choice of our normalisation. We have

C x " }µ x } 2 ´pp `1q ´}µ x } p `1 ¯2 " p p `1 }µ x } 2 " p `1
and similarly C y " q q`1 }µ y } 2 " q `1. This proves the second equality in Equation (8.2), by the first equation of Assertion (1).

In order to prove that pT πq ˚mBM " Tvol Γz zX , we now partition ΓzG X as ď resPΓzEX t P ΓzG X : p0q " πpopeqq, p1q " πptpeqqu .

Using on every element of this partition Hopf's decomposition with respect to the basepoint opeq, we have, by a ramified covering argument already used in the proof of the second part of Proposition 4.13, The first equality of Equation (8.2) follows.

pT
Finally, the last claim of Assertion (3) of Proposition 8.1 follows from Equation (8.1), since c x " q q`1 for every x P V X q (or by taking q " p in Equation (8.2) and by renormalising). The first claim of Assertion (3) follows from the first claim of Assertion (2), by using Equation (2.16). l 115 19/12/2016 Remark 8.2.

(1) In particular, when X " X q is regular, the Patterson density is normalised to be a family of probability measures and Γ is torsion free, then π ˚mBM is q q`1 times the counting measure on ΓzV X. In this case, Equation (8.3) is given by [START_REF] Coornaert | Upper and lower bounds for the mass of the geodesic flow on graphs[END_REF]Rem. 2].

(2) If X " X p, q is biregular with p ‰ q, then π ˚mBM is not proportional to vol Γz zX . In particular, if Γ is torsion free and the Patterson density is normalised to be a family of probability measures, then π ˚mBM is the sum of p p`1 times the counting measure on ΓzV p X and q q`1 times the counting measure on ΓzV q X. This statement is similar to the well-known fact that in pinched but variable curvature, the Bowen-Margulis measure is generally not absolutely continuous with respect to Sasaki's Riemannian measure on the unit tangent bundle (it would then be proportional by ergodicity of the geodesic flow in the lattice case).

Computations of skinning measures for simplicial trees

We now give examples of computations of the total mass of skinning measures (for zero potentials), after introducing some notation. Let X be a locally finite simplicial tree without terminal vertices, and let Γ be a discrete subgroup of AutpXq.

For every simplicial subtree D of X, we define the boundary BV D of V D in X as

BV D " tx P V D : D e P EX, opeq " x, tpeq R V Du .
The boundary BD of D is the maximal subgraph (which might not be connected) of X with set of vertices BV D. If Γ is a discrete subgroup of AutpXq, then the stabiliser Γ D of D acts discretely on BD.

For every x P V X, we define the codegree of x in D as codeg D pxq " 0 if x R D and otherwise codeg D pxq " deg X pxq ´deg D pxq .

Note that codeg D pxq " 0 if x R BV D, and that the codegree codeg N 1 D pxq of x P V X is 0 unless x lies in the boundary of the 1-neighbourhood of D, in which case it is constant equal to deg X pxq ´1. Let D " pD i q iPI be a locally finite Γ-equivariant family of simplicial subtrees of X, and let x P V X. We define the multiplicity 1 of x in (the boundary of) D as (see Section 7.2 for the definition of " D )

m D pxq " Cardti P I{ " D : x P BV D i u |Γ x | .
The numerator and the denominator are finite, by the local finiteness of the family D and the discreteness of Γ, and they depend only on the orbit of x under Γ. Note that if D is a simplicial subtree of X which is precisely invariant under Γ (that is, whenever γ P Γ is such that D X γD is nonempty, then γ belongs to the stabiliser Γ D of D in Γ), if D " pγDq γPΓ{Γ D , and if x P BV D, then

m D pxq " 1 |Γ x | .
In particular, if furthermore Γ is torsion free, then m D pxq " 1 if x P BV D, and m D pxq " 0 otherwise. We define the codegree of x in D as

codeg D pxq " ÿ iPI{" D codeg D i pxq ,
which is well defined as codeg D i pxq depends only on the class of i P I modulo " D . Note that

codeg D pxq " pdeg X x ´kq |Γ x | m D pxq (8.8)
if deg D i pxq " k for every x P BV D i and i P I. If every vertex of X has degree at least 3, this is in particular the case with k " 2 if D i is a line for all i P I and with k " 1 if D i is a horoball for all i P I.

We will say that a simplicial subtree D of X, with stabiliser Γ D in Γ, is almost precisely invariant if there exists N P N such that for every x P BV D, the number of γ P Γ{Γ D such that x P γBV D is at most N . It follows from this property that if D " pγDq γPΓ , then D is locally finite and codeg D pxq ď N codeg D pxq for every x P X.

Proposition 8.4. Assume that X is a regular or biregular simplicial tree with degrees at least 3, and that Γ is a lattice of X.

(1) For every simplicial subtree D of X, we have

π ˚r σ D " ÿ xPV X }µ x } codeg D pxq deg X pxq ∆ x .
(2) If D " pD i q iPI is a locally finite Γ-equivariant family of simplicial subtrees of X, then

π ˚σD " ÿ rxsPΓzV X }µ x } codeg D pxq |Γ x | deg X pxq ∆ rxs .
(3) Let k P N and let D be a simplicial subtree of X such that deg D pxq " k for every x P BV D and the Γ-equivariant family D " pγDq Γ{Γ D is locally finite. Then

π ˚σD " ÿ Γ D yPΓ D zBV D }µ y } pdeg X pyq ´kq |pΓ D q y | deg X pyq ∆ Γy .
(4) If D is a simplicial subtree of X such that the Γ-equivariant family D " pγDq Γ{Γ D is locally finite, then the skinning measure σ D is finite if and only if the graph of groups Γ D z zBD has finite volume.

Before proving Proposition 8.4, let us give some immediate consequences of Assertion (3). If X " X p, q is biregular of degrees p `1 and q `1, let V X " V p X \ V q X be the corresponding partition of the set of vertices of X and, for r P tp, qu, let B r D be the edgeless graph with set of vertices BV D X V r X.

117

19/12/2016 Corollary 8.5. Assume that pX, Γq is as in Proposition 8.4. Let D be a simplicial subtree of X such that the Γ-equivariant family D " pγDq Γ{Γ D is locally finite.

(1) If X " X p, q is biregular of degrees p `1 and q `1 and if the Patterson density pµ x q xPV X of Γ is normalised so that }µ x } " deg X pxq ?

deg X pxq´1 for all x P V X, then

' if D is a horoball, }σ D } " ? p VolpΓ D z zB p Dq `?q VolpΓ D z zB q Dq , ' if D is a line, }σ D } " p ´1 ? p VolpΓ D z zB p Dq `q ´1 ? q
VolpΓ D z zB q Dq . (8.9)

(2) If X " X q is the regular tree of degree q `1 and if the Patterson measures pµ x q xPV X are normalised to be probability measures, then

' if D is a horoball, }σ D } " q q `1 VolpΓ D z zBDq (8.10) ' if D is a line, }σ D } " q ´1 q `1 VolpΓ D z zDq . (8.11)
Proof of Proposition 8.4.

(1) We may partition the outer/inner unit normal bundle B 1 ˘D of D according to the first/last edge of the elements in B 1 ˘D. On each of the elements of this partition, for the computation of the skinning measures using its definition and its independence of the basepoint (see Section 7.1), we take as basepoint the initial/terminal point of the corresponding edge. Since D is a simplicial tree, note that for every e P EX such that opeq P V D, we have e P ED if and only if tpeq P V D. Thus, we have (2) By the definition 2 of the skinning measures associated with Γ-equivariant families, we have r σ D " ř iPI{" r σ Di , where " " " D . Hence by Assertion (1)

π ˚r σ D "
π ˚r σ D " ÿ iPI{" r π ˚σD i " ÿ iPI{" ÿ xPV X }µ x } codeg D i pxq deg X pxq ∆ x " ÿ xPV X ´ÿ iPI{" codeg D i pxq ¯}µ x } deg X pxq ∆ x " ÿ xPV X }µ x } codeg D pxq deg X pxq ∆ x .
By the definition of the measure induced in ΓzV X when Γ may have torsion (see for instance [START_REF] Paulin | Equilibrium states in negative curvature[END_REF]§2.6]), Assertion (2) follows.

(3) It follows from Assertion (2) and from Equation (8.8) that

π ˚σD " ÿ rxsPΓzV X deg X pxq ´k deg X pxq }µ x } m D pxq ∆ rxs .
For every x P V X, by the definition of m D pxq, we have, by partitioning BV D into its orbits under Γ D ,

m D pxq " 1 |Γ x | Cardtγ P Γ D zΓ : γx P BV Du " 1 |Γ x | ÿ Γ D yPΓ D zBV D Cardtγ P Γ D zΓ : Γ D γx " Γ D yu " 1 |Γ x | ÿ Γ D yPΓ D zBV D, Γx"Γy Cardtγ P Γ D zΓ : Γ D γy " Γ D yu " 1 |Γ x | ÿ Γ D yPΓ D zBV D, Γx"Γy rΓ y : pΓ D q y s " ÿ Γ D yPΓ D zBV D, Γx"Γy 1 |pΓ D q y | .
This proves Assertion (3), since ř rxsPΓzV X, Γx"Γy ∆ rxs " ∆ Γy . (4) It follows from Assertion (2) that

}σ D } " ÿ rxsPΓzV X }µ x } codeg D pxq |Γ x | deg X pxq .
Note that for every x P BV D, we have

|Γ x | m D pxq ď codeg D pxq ď deg X pxq |Γ x | m D pxq .
Let m " min xPV X }µ x } and M " max xPV X }µ x }, which are positive and finite, as the total mass of the Patterson measures takes at most two values, since Γ is a lattice and X is biregular. By arguments similar to those in the proof of Assertion (3), we hence have

m min xPV X deg X pxq VolpΓ D z zBDq ď }σ D } ď M VolpΓ D z zBDq
The result follows. l

2 See Section 7.2.

19/12/2016

We now give a formula for the skinning measure (with zero potential) of a geodesic line in the simplicial tree X, using Hamenstädt's distance d H and measure 3 µ H associated with a fixed horoball H in X. This expression for the skinning measure will be useful in Part III.

Lemma 8.6. Let H be a horoball in X centred at a point ξ P B 8 X. Let L be a geodesic line in X with endpoints L ˘P B 8 X ´tξu. Then for all ρ P B 1 `L such that ρ `‰ ξ,

dr σ L pρq " d H pL `, L ´qδ Γ d H pρ `, L ´qδ Γ d H pρ `, L `qδ Γ dµ H pρ `q .
Proof. By Equations (2.9) and (7.6), the power d H δ Γ of the distance and the measure µ H scale by the same factor when the horoball is replaced by another one centred at the same point. Thus, we can assume in the proof that L does not intersect the interior of H .

Fix ρ P B 1 `L such that ρ `‰ ξ. Let y be the closest point to ξ on L, let x 0 be the closest point to L on H , and let z be the closest point to ξ on ρpr0, `8rq. Let t Þ Ñ x t be the geodesic ray starting from x 0 at time t " 0 and converging to ξ. When t is big enough, the points ρ `, z, x t and ξ are in this order on the geodesic line sρ `, ξr.

We have, by the definition in Equation (7.1) of the skinning measure, dr σ L pρq " e δ Γ βρ `pxt, ρp0qq dµ xt pρ `q " e δ Γ βρ `pxt, zq`δ Γ βρ `pz, ρp0qq dµ xt pρ `q " e ´δΓ β ξ pxt, zq´δ Γ dpz, ρp0qq dµ xt pρ `q " e δ Γ t´δ Γ β ξ px 0 , zq´δ Γ dpz, ρp0qq dµ xt pρ `q , and by the definition of Hamenstädt's measure µ H (see Equation (7.5))

dµ H pρ `q " e δ Γ t dµ xt pρ `q . ρ ỳ L ´x0 L xt z " ρp0q BH ξ L ´Lx t BH ξ ρ ỳ " ρp0q
x 0 " z Case 1: Assume first that ρp0q ‰ y. We may assume that ρp0q P ry, L `r. Then z " ρp0q and z is the closest point to H on the geodesic line sL `, ρ `r. Thus

d H pL ´, L `q " d H pL ´, ρ `q
and d H pL `, ρ `q " e ´dpz, x 0 q " e β ξ px 0 , zq , and the claim follows.

Case 2: Assume now that y " ρp0q. Then ry, zs " ry, ξr X ry, ρ `r, and we may assume that x 0 " z up to adjusting the horoball H while keeping its point at infinity. Thus d H pL ´, L `q " e ´dpy, x 0 q " e ´dpz, ρp0qq and d H pL ´, ρ `q " d H pL `, ρ `q " 1, and the claim follows. l

3 See the definitions of Hamenstädt's distance and measure in Sections 2.3 and 7.1 respectively. 120 19/12/2016

Chapter 9

Rate of mixing for the geodesic flow

Let X, x 0 , Γ, r F , pµ x q xPX be as in the beginning of Chapter 7. In this Chapter, we start by collecting in Section 9.1 known results on the rate of mixing of the geodesic flow for manifolds. The main part of the Chapter then consists in proving analogous bounds for the discrete time and continuous time geodesic flow for quotient spaces of simplicial and metric trees respectively.

We define m F " m F }m F } when the Gibbs measure is finite. Recall that this measure is nonzero since Γ is nonelementary.

Let α P s0, 1s. 1 We will say that the (continuous time) geodesic flow on ΓzG X is exponentially mixing for the α-Hölder regularity or that it has exponential decay of α-Hölder correlations for the potential F if there exist C, κ ą 0 such that for all φ, ψ P C α b pΓzG Xq and t P R, we have

ˇˇż ΓzG X φ ˝g´t ψ dm F ´żΓzG X φ dm F ż ΓzG X ψ dm F ˇˇď C e ´κ|t| }φ} α }ψ} α ,
and that it is polynomially mixing or has polynomial decay of α-Hölder correlations if there exist C ą 0 and n P N ´t0u such that for all φ, ψ P C α b pΓzG Xq and t P R, we have

ˇˇż ΓzG X φ ˝g´t ψ dm F ´żΓzG X φ dm F ż ΓzG X ψ dm F ˇˇď C p1 `|t|q ´n }φ} α }ψ} α .

Rate of mixing for Riemannian manifolds

When X " Ă M is a complete Riemannian manifold with pinched negative sectional curvature with bounded derivatives, then the boundary at infinity of Ă M , the strong unstable, unstable, stable, and strong stable foliations of T 1 Ă M are only Hölder-smooth in general. 2 Hence Hölder regularity on functions on T 1 Ă M is appropriate. The geodesic flow is known to have exponential decay of Hölder correlations for compact manifolds M when • M is two-dimensional and F is any Hölder potential by [START_REF] Dolgopyat | On decay of correlation in Anosov flows[END_REF],

1 We refer to Section 3.1 for the definition of the Banach space C α b pZq of bounded α-Hölder-continuous functions on a metric space Z.

2 See for instance [Brin] when Ă M has a compact quotient (a result first proved by Anosov), and [START_REF] Paulin | Equilibrium states in negative curvature[END_REF]Theo. 7.3].

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• M is 1{9-pinched and F " 0 by [START_REF] Giulietti | Anosov flows and dynamical zeta functions[END_REF]Coro. 2.7],

• m F is the Liouville measure by [Live], see also [Tsu], [START_REF] Nonnenmacher | Decay of correlations for normally hyperbolic trapping[END_REF]Coro. 5] who give more precise estimates, • M is locally symmetric and F is any Hölder potential by [Sto], see also [MO].

When Ă

M is a symmetric space, then the boundary at infinity of Ă M , the strong unstable, unstable, stable, and strong stable foliations of T 1 Ă M are smooth. Hence talking about leafwise C -smooth functions on T 1 Ă M makes sense. We will denote by C c pN q the vector space of realvalued C -smooth functions on the orbifold T 1 M " ΓzT 1 Ă M (that is, the maps induced on T 1 M by the C -smooth Γ-invariant functions on T 1 Ă M ), with compact support in T 1 M , and by ||ψ|| the Sobolev W ,2 -norm of any ψ P C c pT 1 M q.

Given P N, we will say that the geodesic flow on T 1 M is exponentially mixing for the -Sobolev regularity (or that it has exponential decay of -Sobolev correlations) for the potential F if there exist c, κ ą 0 such that for all φ, ψ P C c pT 1 M q and all t P R, we have ˇˇż

T 1 M φ ˝g´t ψ dm F ´żT 1 M φ dm F ż T 1 M
ψ dm F ˇˇď c e ´κ|t| }ψ} }φ} .

When F " 0 and Γ is an arithmetic lattice in the isometry group of Ă M (the Gibbs measure then coincides, up to a multiplicative constant, with the Liouville measure), this property, for some P N, follows from [KM1, Theorem 2.4.5], with the help of [START_REF] Clozel | Démonstration de la conjecture τ[END_REF]Theorem 3.1] to check its spectral gap property, and of [KM2, Lemma 3.1] to deal with finite cover problems.

Rate of mixing for simplicial trees

Let X be a locally finite simplicial tree without terminal vertices, with geometric realisation X " |X| 1 . Let Γ be a nonelementary discrete subgroup of AutpXq and let r c : EX Ñ R be a system of conductances for Γ on X.

In this Section, building on the end of Section 4.4 concerning the mixing properties themselves, we now study the rates of the mixing properties of the discrete time geodesic flow on ΓzG X for the Gibbs measure m c " m Fc , when it is mixing.

Let pZ, m, T q be a dynamical system with pZ, mq a probability space and T : Z Ñ Z a (not necessarily invertible) measure preserving map. For all n P N and φ, ψ P L 2 pmq, the (well-defined) n-th correlation coefficient of φ, ψ is

cov m, n pφ, ψq " ż Z φ ˝T n ψ dm ´żZ φ dm ż Z ψ dm .
Let α P s0, 1s and assume that Z is a metric space (endowed with its Borel σ-algebra).

Similarly as for the case of flows in the beginning of Chapter 9, we will say that the dynamical system pZ, m, T q is exponentially mixing for the α-Hölder regularity or that it has exponential decay of α-Hölder correlations if there exist C, κ ą 0 such that for all φ, ψ P C α b pZq and n P N, we have

| cov m, n pφ, ψq| ď C e ´κ n }φ} α }ψ} α .

Note that this property is invariant under measure preserving conjugations of dynamical systems by bilipschitz homeomorphisms.

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The main result of this Section is a simple criterion for the exponential decay of correlation for the discrete time geodesic flow on ΓzG X.

We define m c " mc }mc} when the Gibbs measure m c on ΓzG X is finite, and we use the dynamical system pΓzG X, m c , g 1 q in the definition of the correlation coefficients.

Given a finite subset E of ΓzV X, we denote by τ E : ΓzG X Ñ N Y t`8u the first return time to E of the discrete time geodesic flow: τ E p q " inftn P N ´t0u : g n p0q P Eu , with the usual convention that inf H " `8.

Theorem 9.1. Let X, Γ, r c be as above, with δ c finite. Assume that the Gibbs measure m c is finite and mixing for the discrete time geodesic flow on ΓzG X. Assume moreover that there exist a finite subset E of ΓzV X and C 1 , κ 1 ą 0 such that for all n P N, we have m c `t P ΓzG X : p0q P E and τ E p q ě nu ˘ď C 1 e ´κ1 n .

(9.1)

Then the discrete time geodesic flow on ΓzG X has exponential decay of α-Hölder correlations for the system of conductances c.

A similar statement holds for the square of the discrete time geodesic flow on ΓzG even X when m c is finite, C ΛΓ is a uniform simplicial tree with degrees at least 3 and L Γ " 2Z.

Note that the crucial Hypothesis (9.1) of Theorem 9.1 is in particular satisfied if ΓzX is finite, by taking E " ΓzV X. But the result is quite well-known in this case: when Γ is torsion free, it follows from Bowen's result [START_REF] Bowen | Equilibrium states and the ergodic theory of Anosov diffeomorphisms[END_REF]1.26] that a mixing subshift of finite type is exponentially mixing.

Proof. Let X 1 " C ΛΓ. Using the coding introduced in Section 5.2, we first reduce this statement to a symbolic dynamics one.

Step 1 : Reduction to two-sided symbolic dynamics Let pΣ, σq be the (two-sided) topological Markov shift with alphabet A and transition matrix A constructed in Section 5.2, conjugated to pΓzG X 1 , g 1 q by the homeomorphism Θ : ΓzG X 1 Ñ Σ (see Theorem 5.1). Let P " Θ ˚mc }mc} , which is a mixing σ-invariant probability measure on Σ. Let E " tpe ´, h, e `q P A : tpe ´q " ope `q P Eu .

The set E is finite since the degrees and the vertex stabilisers of X are finite. For all x P Σ and k P Z, we denote by x k the k-th component of x " px n q nPZ . Let τ E pxq " inftn P N ´t0u : x n P E u be the first return time to E of x under iteration of the shift σ.

Let π `: Σ Ñ A N be the natural extension px n q nPZ Þ Ñ px n q nPN . Theorem 9.1 will follow from the following two-sided symbolic dynamics result. 3 Theorem 9.2. Let pΣ, σq be a locally compact transitive two-sided topological Markov shift with alphabet A and transition matrix A, and let P be a mixing σ-invariant probability measure with full support on Σ. Assume that 3 Assumption (1) of Theorem 9.2 is far from being optimal, but will be sufficient for our purpose.

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(1) for every A-admissible finite sequence pw 0 , . . . , w n q in A , the Jacobian of the map from tpx k q kPN P π `pΣq : x 0 " w n u to tpy k q kPN P π `pΣq : y 0 " w 0 , . . . , y n " w n u defined by px 0 , x 1 , x 2 , . . . q Þ Ñ pw 0 , . . . , w n , x 1 , x 2 , . . . q, with respect to the restrictions of the pushforward measure pπ `q˚P , is constant;

(2) there exist a finite subset E of A and C 1 , κ 1 ą 0 such that for all n P N, we have P `tx P Σ : x 0 P E and τ E pxq ě nu ˘ď C 1 e ´κ1 n .

(9.2)

Then pΣ, P, σq has exponential decay of α-Hölder correlations.

Proof that Theorem 9.2 implies Theorem 9.1. Since r m c is supported on G X 1 , up to replacing X by X 1 , we may assume that B 8 Γ " ΛΓ.

By the construction of Θ just before the statement of Theorem 5.1, for every " Γ r P ΓzG X, we have pΘ q 0 " pe 0 p r q, h 0 p r q, e 0 p r qq with e 0 p r qq " pp r pr0, 1sqq where p : X Ñ ΓzX 1 is the canonical projection, so that o `e0 p r qq ˘" p0q. Since Θ conjugates g 1 to σ, we have pΘ q n " pσ n pΘ qq 0 " pΘpg n qq 0 P E if and only if g n p0q P E, and τ E pΘ q " τ E p q .

Therefore Theorem 9.1 will follow from Theorem 9.2 by conjugation since Θ is bilipschitz, once we have proved that Hypothesis (1) of Theorem 9.2 is satisfied for the two-sided topological Markov shift pΣ, σq conjugated by Θ to pΓzG X, g 1 q, which is the main point in this proof.

We hence fix an A-admissible finite sequence w " pw 0 , . . . , w n q in A . We denote by rw n s " tpx k q kPN P π `pΣq : x 0 " w n u , rws " tpy k q kPN P π `pΣq : y 0 " w 0 , . . . , y n " w n u and f w : px 0 , x 1 , x 2 , . . . q Þ Ñ pw 0 , . . . , w n , x 1 , x 2 , . . . q the sets and map appearing in Hypothesis (1). We denote by r w and Ă w n the discrete generalised geodesic lines in X associated with w and w n (see the proof of Theorem 5.1 just after Equation (5.3)). Since w ends with w n , by the construction of Θ, there exists γ P Γ sending the two consecutive edges of r w n to the last two consecutive edges of w. We denote by x " r wp0q and y " Ă w n p0q the footpoints of r w and Ă w n respectively. For every discrete generalised geodesic line ω P p G X which is isometric exactly on an interval I containing 0 in its interior (as for ω " r w, Ă w n ), let

G ω X " t P G X : |I " ω |I u
be the space of extensions of ω |I to geodesic lines. With B ω X " t ˘: P G ω Xu its set of points at ˘8, we have a homeomorphism G ω X Ñ pB ώ X ˆBὼ Xq defined by Þ Ñ p ´, `q, using Hopf's parametrisation with respect to the point ωp0q, since all the geodesic lines in G ω X are at the point ωp0q at time t " 0. Using as basepoint x 0 " ωp0q in the definition of the Gibbs measure (see Equation (4.10)), this homeomorphism sends the restriction to G ω X of the Gibbs measure d r m c p q to the product measure dµ ώp0q p ´q dµ ὼp0q p `q. Hence the pushforward of r m c|G ω X by the positive endpoint map e `: Þ Ñ `is µ ώp0q pB ώ Xq dµ ὼp0q p `q, and note that µ ώp0q pB ώ Xq is a positive constant.

Since π `: Σ Ñ Σ `is the map which forgets about the past, there exist measurable maps u w : B r w X Ñ rws and u wn : B Ằ wn X Ñ rw n s such that the following diagrams commute:

G r w X π `˝Θ ˝p
ÝÝÝÝÑ rws

e `OE Õ uw B r w X and G Ă wn X π `˝Θ ˝p ÝÝÝÝÑ rw n s e `OE Õ uw n B Ằ wn X
.

Furthermore, the map u w (respectively u wn ) is surjective, and has constant finite order fibers given by the orbits of the finite stabiliser Γ r w (respectively Γ Ă wn ). Since P " Θ ˚mc }mc} , the pushforward by the map u w (respectively u wn ) of the measure µ x (respectively µ ỳ ) is a constant time the restriction of pπ `q˚P to rws (respectively rw n s). Finally, by the construction of the (inverse of the) coding in the proof of Theorem 5.1, the following diagram is commutative:

B Ằ wn X uw n ÝÑ rw n s γ Ó Ó fw B r w X uw ÝÑ rw n s .
Recall that the pushforwards of measures µ, ν, which are absolutely continuous one with respect to the other, by a measurable map f are again absolutely continuous one with respect to the other, and satisfy (almost everywhere)

d f ˚µ d f ˚ν ˝f " d µ d ν .
Hence in order to prove that Hypothesis (1) in the statement of Theorem 9.2 is satisfied, we only have to prove that the map γ : B Ằ wn X Ñ B r w X has a constant Jacobian for the measures µ ỳ on B Ằ wn X and µ x on B r w X respectively. For all ξ, η P B r w X, by the properties of the Patterson densities (see Equations (4.1) and (4.2)), and since γy belongs to the geodesic ray from x to ξ and η (see Equation (3.6) and the above picture), we have This proves that Hypothesis (1) in Theorem 9.2 is satisfied, and concludes the proof of Theorem 9.1.

l

We now indicate how to pass from a one-sided version of Theorem 9.2 to the two-sided one, as was communicated to us by J. Buzzi.

Step 2 : Reduction to one-sided symbolic dynamics Let pΣ `, σ `q be the one-sided topological Markov shift with alphabet A and transition matrix A, that is, Σ `is the closed subset of the topological product space A N defined by Σ `" x " px n q nPN P A N : @ n P N, A xn,x n`1 " 1u , and σ `: Σ `Ñ Σ `is the (one-sided) shift4 defined by pσ `pxqq n " x n`1 for all x P Σ `and n P N. We endow Σ `with the distance dpx, x 1 q " e ´max nPN : @ i P t0,...,nu, x i " x 1 i ( .

Note that the distances on Σ and Σ `are bounded by 1.

Let π `: Σ Ñ Σ `be the natural extension px n q nPZ Þ Ñ px n q nPN , which satisfies π `˝σ " σ `˝π `and is 1-Lipschitz. Note that Σ is transitive (respectively locally compact) if and only if Σ `is transitive (respectively locally compact).

In the one-sided case, we always assume that the cylinders start at time t " 0: given an admissible sequence w " pw 0 , w 1 , . . . , w n´1 q, the cylinder of length |w| " n it defines is rws " rw 0 , . . . , w n´1 s " tpx n q nPN P Σ `: @ i P t0, . . . , n ´1u, x i " w i u .

We first explain how to relate the decay of correlations for the two-sided and one-sided systems. This is well-known since the works of Sinai [START_REF] Sinai | Gibbs measures in ergodic theory[END_REF]§3] and Bowen [Bowe2,Lem. 1.6], see for instance [START_REF] Young | Statistical properties of dynamical systems with some hyperbolicity[END_REF]§4], and the following proof has been communicated to us by J. Buzzi. We fix α P s0, 1s. For all metric space Z and bounded α-Hölder-continuous function f :

Z Ñ R, let }f } 1 α " sup x, yPZ 0ădpx,yqď1
|f pxq ´f pyq| dpx, yq α , so that5 }f } α " }f } 8 `}f } 1 α . Lemma 9.3. For every a P A , let us fix z a P Σ such that pz a q 0 " a. Let φ : Σ Ñ R be a bounded α-Hölder-continuous map and N P N. Define φ pN q : Σ `Ñ R by: @x P Σ `, φ pN q pxq " φpyq where

" y i " x i`N if i ě ´N y i " z x 0 i`N otherwise.
Then φ pN q is bounded and α-Hölder-continuous on Σ `, with |φ ˝σN ´φpNq ˝π`| ď }φ} 1 α e ´αN . Moreover, }φ pN q } 1 α ď e αN }φ} 1 α and }φ pN q } 8 ď }φ} 8 .

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Proof. For all x " px n q nPZ P Σ, with y associated with π `pxq as in the statement, we have pσ N pxq ˘n " y n if |n| ď N , hence |φ ˝σN pxq ´φpNq pπ `pxqq| " |φpσ N pxqq ´φpyq| ď }φ} 1 α dpσ N pxq, yq α ď }φ} 1 α e ´αN .

Moreover, if y, y 1 P Σ are associated with x " px n q nPN , x 1 " px 1 n q nPN P Σ `respectively, then dpy, y 1 q " e N dpx, x 1 q if dpx, x 1 q ă e ´N and otherwise dpy, y 1 q ď 1 ď e N dpx, x 1 q, so that |φ pN q pxq ´φpNq px 1 q| " |φpyq ´φpy 1 q| ď }φ} 1 α dpy, y 1 q α ď }φ} 1 α e αN dpx, x 1 q α . l Proposition 9.4. Let µ be a σ-invariant probability measure on Σ. Assume that the dynamical system pΣ `, σ `, pπ `q˚µ q has exponential decay of α-Hölder correlations. Then pΣ, σ, µq has exponential decay of α-Hölder correlations.

Proof. Let C, κ ą 0 be such that for all bounded α-Hölder-continuous maps φ 1 , ψ 1 : Σ `Ñ R and n P N, we have

| cov pπ `q˚µ, n pφ 1 , ψ 1 q| ď C }φ} α }ψ} α e ´κn .
Let φ, ψ : Σ Ñ R be bounded α-Hölder-continuous maps and n P N. Denoting by ˘t any value in r´t, ts for any t ě 0, we have, by the first part of the above lemma and for any N P N (to be chosen appropriately later on),

ż Σ φ ˝σn ψ dµ " ż Σ φ ˝σn`N ψ ˝σN dµ " ż Σ
pφ pN q ˝π`˘} φ} 1 α e ´α N q ˝σn pψ pN q ˝π`˘} ψ} 1 α e ´α N q dµ " ż Σ `φpN q ˝σn `ψpN q dpπ `q˚µ ˘}φ} α }ψ} α e ´α N .

A similar estimate holds for the second term in the definition of the correlation coefficients.

Hence by the second part of the above lemma | cov µ, n pφ, ψq| ď | cov pπ `q˚µ, n pφ pN q , ψ pN q q| `2 }φ} α }ψ} α e ´αN ď C p}φ} 8 `}φ} 1 α e α N q p}ψ} 8 `}ψ} 1 α e α N q e ´κ n `2 }φ} α }ψ} α e ´α N ď }φ} α }ψ} α pC e 2α N ´κ n `2 e ´α N q .

Taking N " t κ n 4 α u, C 1 " C `2 e α and κ 1 " κ 4 , we have

| cov µ, n pφ, ψq| ď C 1 }φ} α }ψ} α e ´κ1 n ,
and the result follows. l

In order to conclude Step 2, we now state the one-sided version of Theorem 9.2 and prove how it implies Theorem 9.2. 6 Theorem 9.5. Let pΣ `, σ `q be a locally compact transitive one-sided topological Markov shift with alphabet A and transition matrix A, and let P `be a mixing σ `-invariant probability measure with full support on Σ `. Assume that 6 Assumption (1) of Theorem 9.5 is far from being optimal, but will be sufficient for our purpose.

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(1) for every A-admissible finite sequence w " pw 0 , . . . , w n q in A , the Jacobian of the map from rw n s to rws defined by pw n , x 1 , x 2 , . . . q Þ Ñ pw 0 , . . . , w n , x 1 , x 2 , . . . q with respect to the restrictions of the measure P `is constant;

(2) there exist a finite subset E of A and C 1 , κ 1 ą 0 such that for all n P N, we have P ``tx P Σ `: x 0 P E and τ E pxq ě nu ˘ď C 1 e ´κ1 n .

(9.3)

Then pΣ `, P `, σ `q has exponential decay of α-Hölder correlations.

Proof that Theorem 9.5 implies Theorem 9.2. Let pΣ, σ, P, E q be as in the statement of Theorem 9.2. Let P `" pπ `q˚P , which is a mixing σ `-invariant probability measure on Σ `. Note that Hypothesis (1) in Theorem 9.5 follows from Hypothesis (1) of Theorem 9.2. Similarly, Equation (9.3) follows from Equation (9.2). Hence Theorem 9.2 follows from Theorem 9.5 and Proposition 9.4. l

Let us now consider Theorem 9.5. The scheme of its proof, using inducing and Young tower arguments, was communicated to us by O. Sarig.

Step 3 : Proof of Theorem 9.5

In this final Step, using inducing of the dynamical system pΣ `, σ `q on the subspace tx P Σ `: x 0 P E u " Ť aPE ras (a finite union of 1-cylinders), we present pΣ `, σ `q as a Young tower to which we will apply the results of [START_REF] Young | Recurrence times and rates of mixing[END_REF].

Note that since σ `is mixing with full measure, there exists a σ `-invariant measurable subset ∆ of Σ `such that the orbit under σ `of every element of ∆ passes infinitely many times inside the nonempty open subset Ť aPE ras. We again denote by τ E : ∆ Ñ N ´t0u the restriction to ∆ of the first return time in Ť aPE ras, so that if ∆ 0 " tx P ∆ :

x 0 P E u " ď aPE ∆ X ras ,
then τ E pxq " mintn P N ´t0u : σ n `x P ∆ 0 u for all x P ∆. We denote by F : ∆ Ñ ∆ 0 the first return map to ∆ 0 under iteration of the one-sided shift, that is

F : x Þ Ñ σ τ E pxq `pxq .
Let W be the set of admissible sequences w of length |w| at least 2 such that if w " pw 0 , . . . , w n q then w 0 , w n P E and w 1 , . . . , w n´1 R E .

We have the following properties: ' the sets ∆ a " ∆ X ras for a P E form a finite measurable partition of ∆ 0 and for every a P E , the sets ∆ w " ∆ X rws for w P W and w 0 " a form a countable measurable partition of ∆ a ;

' for every w P W , the first return time τ E is constant (equal to |w| ´1) on each ∆ w , and if w |w|´1 " b, then the first return map F is a bijection from ∆ w to ∆ b ;

' for all w P W and x, y P ∆ w , since ' for every w P W , the Jacobian of the first return map F : ∆ w Ñ ∆ w |w|´1 for the restrictions to ∆ w and ∆ w |w|´1 of P `is constant.7

By an easy adaptation of [START_REF] Young | Recurrence times and rates of mixing[END_REF]Theo. 3] (see also [START_REF] Melbourne | Rapid decay of correlations for nonuniformly hyperbolic flows[END_REF]§2.1]) which considers the case when E is a singleton, we have the following noneffective8 exponential decay of correlation: there exists κ ą 0 such that for every φ, ψ P C α b pΣ `q, there exists a constant C φ,ψ ą 0 such that

| cov P `, n pφ, ψq| ď C φ,ψ e ´κn
By an elegant argument using the Principle of Uniform Boundedness, it is proved in [ChCS, Appendix B] that this implies that there exists C, κ ą 0 such that for every φ, ψ P C α b pΣ `q, we have

| cov P `, n pφ, ψq| ď C }φ} α }ψ} α e ´κ n .
This concludes the proof of Theorem 9.5, hence the proof of Theorem 9.1. l l

The next result gives examples of applications of Theorem 9.1 when ΓzX is infinite. It strengthens [START_REF] Athreya | Ultrametric Logarithm Laws[END_REF]Theo. 2.1] that applies only to arithmetic lattices and only for the locally constant regularity (see Section 15.4), see also [BekL] for an approach using spectral gaps. It was claimed in [Kwo], but was retracted by the author.

Corollary 9.6. Let X be a locally finite simplicial tree without terminal vertices. Let Γ be a geometrically finite subgroup of AutpXq such that the smallest nonempty Γ-invariant subtree of X is uniform without vertices of degree 2. Let α P s0, 1s.

(1) If L Γ " Z, then the discrete time geodesic flow on ΓzG X has exponential decay of α-Hölder correlations for the system of conductances c " 0.

(2) If L Γ " 2Z, then the square of the discrete time geodesic flow on ΓzG even X has exponential decay of α-Hölder correlations for the 0 system of conductances, that is, there exist C, κ ą 0 such that for all φ, ψ P C α b pΓzG even Xq and n P Z, we have

ˇˇż ΓzGevenX φ ˝g´2n ψ d m BM ´1 m BM pΓzG even Xq ż ΓzGevenX φ d m BM ż ΓzGevenX ψ d m BM ˇď C e ´κ|n| }φ} α }ψ} α .
The main point of this corollary is to prove the exponential decay of volumes of geodesic lines going high in the cuspidal rays of ΓzX, stated as Assumption (9.1) in Theorem 9.1. There is a long history of similar results, starting from the exponential decay of volumes of small cusp neighbourhoods in noncompact finite volume hyperbolic manifolds (based on the description of their ends) used by Sullivan to deduce Diophantine approximation results (see [START_REF] Sullivan | Disjoint spheres, approximation by imaginary quadratic numbers, and the logarithm law for geodesics[END_REF]§9]). 9 These results were extended to the case of locally symmetric Riemannian 129 19/12/2016 manifolds by Kleinbock-Margulis [START_REF] Kleinbock | Logarithm laws for flows on homogeneous spaces[END_REF] (based on the description of their ends using Siegel sets). Note that the geometrically finite lattice assumption on Γ is here in order to obtain similar descriptions of the ends of ΓzX.

Proof. Up to replacing X by C ΛΓ, we assume that X is a uniform simplicial tree with degrees at least 3 and that Γ is a geometrically finite lattice of X. We use the 0 system of conductances.

(1) By [START_REF] Paulin | Groupes géométriquement finis d'automorphismes d'arbres et approximation diophantienne dans les arbres[END_REF] and as recalled in Section 2.7, the graph ΓzX is the union of a finite graph Y and finitely many geodesic rays R i for i P t1, . . . , ku, such that if px i, n q nPN is the sequence of vertices in increasing order along R i for i " 1, . . . , k, then the vertex group G x i, n of x i, n in the quotient graph of groups Γz zX satisfies G x i, n Ă G x i, n`1 for all n P N, and the edge group of the edge e i,n with origin x i,n and endpoint x i, n`1 is equal to G x i, n . 10 Note that since the degrees of X are at least 3, we have rG x i, n`1 : G x i,n s ě 2 and |G x i, 0 | ě 1, so that

|G x i, n | ě 2 n . (9.4)
Let E be the (finite) set of vertices V Y of Y. Note that for all n P N ´t0u and P ΓzG X, if p0q P E and τ E p q ě 2n, then needs to leave Y after time 0 and it travels (geodesically) inside some cuspidal ray for a time at least n, so that there exists i P t1, . . . , ku such that pnq " x i, n . Hence for all n P N, using ' the invariance of m BM under the (discrete time) geodesic flow in order to get the third term, ' Equation (8.4) where r x i, n is a fixed lift of x i, n in V X for the fifth term, and ' Equation (9.4) since |Γ r x i, n | " |G x i, n | and the fact that the degrees of the uniform simplicial tree X are uniformly bounded and that the total mass of the Patterson measures of the lattice Γ are uniformly bounded (see Proposition 4.14) for the last term, we have m BM `t P ΓzG X : p0q P E and τ E p q ě 2nu ď

k ÿ i"1 m BM `t P ΓzG X : pnq " x i,n u " k ÿ i"1 m BM `t P ΓzG X : p0q " x i,n u ˘" k ÿ i"1 π ˚mBM ptx i,n uq " k ÿ i"1 1 |Γ r x i, n | ÿ e, e 1 PEX : opeq"ope 1 q" r x i, n , e‰e 1 µ r x i, n pB e Xq µ r x i, n pB e 1 Xq ď k 1 2 n max xPV X degpxq 2 max xPV X }µ x } 2 .
The result then follows from Theorem 9.1 using the above finite set E which satisfies Assumption (9.1) as we just proved, and using Propositions 4.14 and 4.15 in order to check that under the assumption that L Γ " Z, the Bowen-Margulis measure m BM of Γ is finite and mixing under the discrete time geodesic flow on ΓzG X.

(2) The proof of Assertion (2) of Corollary 9.6 is similar to the one of Assertion (1). l 10 identifying the edge group of an edge e with its image by the structural map Ge Þ Ñ G opeq 130 19/12/2016

Remark. The techniques introduced in the above proof in order to check the main hypothesis of Theorem 9.1 may be applied to numerous other examples. For instance, let X be a locally finite simplicial tree without terminal vertices. Let Γ be a nonelementary discrete subgroup of AutpXq such that the smallest nonempty Γ-invariant subtree of X is uniform without vertices of degree 2, and such that L Γ " Z. Let α P s0, 1s. Assume that ΓzX is the union of a finite graph A and finitely many trees T 1 , . . . , T k meeting A in one and exactly one vertex ˚1, . . . , ˚n such that for every edge e in T i pointing away from the root ˚i of T i , the canonical morphism G e Ñ G opeq between edge and vertex groups of the quotient graph of groups Γz zX is an isomorphism. Assume that there exists C, κ ą 0 such that for all n P N,

ÿ i"1,...,k, xPV T i : dpx,˚iq"n 1 |G x | ď Ce ´κ n .
Then the discrete time geodesic flow on ΓzG X has exponential decay of α-Hölder correlations for the 0 system of conductances. This is in particular the case for every k, q P N such that k ě 2, q ą 2k `1 and q ´k is odd, when the quotient graph of groups Γz zX has underlying edge-indexed graph 11 a loop-edge with both indices equal to q´k`1 2 glued to the root of a regular k-ary rooted tree, with indices 1 for the edges pointing towards the root and q ´k `1 for the edges pointing away from the root (see the picture below with k " 2). Note that X is then the pq `1q-regular tree, and that the loop edge is here in order to ensure that L Γ " Z. For instance, the vertex group of a point at distance n from the root may be chosen to be Z{p q´k`1 2 qZ ˆ`Z{pq ´k `1qZ ˘n.

q ´1 1 1 q ´1 q ´1 1 1 q ´1 q ´1 1 1 q ´1 q ´1 1 1 q ´1 q ´1 q ´1 1 1 q´1 2 1 q ´1 1 q ´1 1 q ´1 q ´1 1 q´1 2

Rate of mixing for metric trees

Let pX, λq, X, Γ, r F , pµ x q xPV X and m F be as in the beginning of Section 4.4. The aim of this Section is to study the problem of finding conditions on these data under which the (continuous time) geodesic flow on ΓzG X is polynomially mixing for the Gibbs measure m F .

We will actually prove a stronger property, though it applies only to observables which are smooth enough along the flow. Let us fix α P s0, 1s. Let pZ, µ, pφ t q tPR q be a topological space Z endowed with a continuous one-parameter group pφ t q tPR of homeomorphisms preserving a (Borel) probability measure µ on Z. For all k P N, let C k, α b pZq be the real vector space of maps f : Z Ñ R such that for all z P Z, the map t Þ Ñ f pφ t xq is C k -smooth, and such 11 See definition in Section 2.7.

131

19/12/2016 that the maps B i t f : Z Ñ R defined by x Þ Ñ d i dt i |t"0 f pφ t xq for 0 ď i ď k are bounded and α-Hölder-continuous. It is a Banach space when endowed with the norm

}f } k, α " k ÿ i"0 }B i t f } α ,
and it is contained in L 2 pZ, µq by the finiteness of µ. We denote by C k, α c pZq the vector subspace of elements of C k, α b pZq with compact support. For all ψ, ψ 1 P L 2 pZ, µq and t P R, let

cov µ, t pψ, ψ 1 q " ż Z ψ ˝φt ψ 1 dµ ´żZ ψ dµ ż Z ψ 1 dµ
be the correlation coefficient of the observables ψ, ψ 1 at time t under the flow pφ t q tPR for the measure µ. We say12 that the (continuous time) dynamical system pZ, µ, pφ t q tPR q has superpolynomial decay of α-Hölder correlations if for every n P N there exist C " C n ą 0 and k " k n P N such that for all ψ, ψ 1 P C k, α b pZq and t P R, we have

| cov µ, t pψ, ψ 1 q| ď C p1 `|t|q ´n }ψ} k, α }ψ 1 } k, α .
Following Dolgopyat, we say that the dynamical system pZ, µ, pφ t q tPR q is rapidly mixing if there exists α ą 0 such that pZ, µ, pφ t q tPR q has superpolynomial decay of α-Hölder correlations

The two assumptions on our data that we will use are the following ones, introduced respectively in [START_REF] Dolgopyat | Prevalence of rapid mixing in hyperbolic flows[END_REF] and [START_REF] Melbourne | Rapid decay of correlations for nonuniformly hyperbolic flows[END_REF]. Recall that the Gibbs measure m F , when finite, is mixing if and only the length spectrum L Γ is dense in R (see Theorem 4.8). The rapid mixing property will require stronger assumptions on L Γ .

We say that the length spectrum L Γ of Γ is 2-Diophantine if there exists a ratio of two translation lengths of elements of Γ which is Diophantine. Recall that a real number x is Diophantine if there exist α, β ą 0 such that |x ´p q | ě α q ´β for all p, q P Z with q ą 0.

Let E be a finite subset of vertices of ΓzX, and let r E be the set of vertices of X mapping to E. We denote by T E the set of triples pλpγq, dpγq, qpγ, pqq where γ P Γ has translation length λpγq ą 0, has dpγq vertices on its translation axis Axpγq modulo γ Z and if the first return time of a vertex p in r E X Axpγq in r E X Axpγq under the discrete time geodesic flow along the translation axis has period qpγ, pq. We say that the length spectrum L Γ of Γ is 4-Diophantine with respect to E if for all sequences pb k q kPN in r1 `8r converging to `8 and pω k q kPN , pϕ k q kPN in r0, 2πr , there exists N P N such that for all a ě N and C, β ě 1, there exist k ě 1 and pτ, d, qq P T E such that

d `pb k τ `ωk dqtβ ln b k u `qϕ k , 2πZq ě C q b ´a k .
We define the first return time after time on a finite subset E of vertices of ΓzX as the map τ ą E : ΓzG X Ñ r0, `8s defined by τ ą E p q " inftt ą : ptq P Eu. Theorem 9.7. Assume that the critical exponent δ F is finite, that the Gibbs measure m F is finite and mixing, and that the lengths of the edges of pX, λq have a finite upper bound. (1) there exist C, κ ą 0 and P s0, min λr such that for all t ě 0, m c pt P ΓzG X : dp p0q, Eq ď and τ ą E p q ě tuq ď C e ´κ t ,

(2) the length spectrum of Γ is 4-Diophantine with respect to E.

Then the (continuous time) geodesic flow on ΓzG X has superpolynomial decay of α-Hölder correlations for the normalised Gibbs measure m F }m F } .

Note that the existence of E satisfying the exponentially small tail Hypothesis (1) is in particular satisfied if Γ is geometrically finite with E the set of vertices of a finite subgraph of ΓzX whose complement in ΓzX is the underlying graph of a union of cuspidal rays in Γz zX : see the proof of Corollary 9.6 and use the hypothesis on the lengths of edges.

Note that the exponentially small tail Hypothesis (1) might be weakened to a superpolynomially small tail hypothesis while keeping the same conclusion, see [START_REF] Melbourne | Decay of correlations for slowly mixing flows[END_REF]. Since the former is easier to check than the latter, we prefer to state Theorem 9.7 as it is.

We will follow a scheme of proof analogous to the one in Section 9.2 for simplicial trees, by reducing the study to a problem of suspensions of Young towers, and then apply results of [START_REF] Dolgopyat | Prevalence of rapid mixing in hyperbolic flows[END_REF] and [START_REF] Melbourne | Rapid decay of correlations for nonuniformly hyperbolic flows[END_REF] for the rapid mixing property of suspensions of hyperbolic and nonuniformly hyperbolic dynamical systems.

Proof. Since the Gibbs measure normalised to be a probability measure depends only on the cohomology class of the potential (see Equation (4.9)), we may assume by Proposition 3.12 that F " F c is the potential on ΓzT 1 X associated with a system of conductances r c : EX Ñ R for Γ. We denote by δ c the critical exponent of pΓ, F c q, and by m c the Gibbs measure m Fc .

Step 1 : Reduction to a suspension of a two-sided symbolic dynamics

We refer to the paragraphs before the statement of Theorem 5.9 for the definitions of ' the system of conductances 7 c for Γ on the simplicial tree X, ' the (two-sided) topological Markov shift pΣ, σ, Pq on the alphabet A , conjugated to the discrete time geodesic flow `ΓzG X, 7 g 1 , m 7 c }m 7 c } ˘by the homeomorphism Θ : ΓzG X Ñ Σ, ' the roof function r : Σ Ñ s0, `8r ' and the suspension pΣ, σ, a Pq r " pΣ r , pσ t r q tPR , a P r q over pΣ, σ, a Pq with roof function r, where a " 1 }Pr} . This suspension is conjugated to the continuous time geodesic flow `ΓzG X, mc }mc} , pg t q tPR ˘by the bilipschitz homeomorphism Θ r : ΓzG X Ñ Σ r defined at the end of the proof of Theorem 5.9. We will always (uniquely) represent the elements of Σ r as rx, ss with x P Σ and 0 ď s ă rpxq.

Note that since Θ ´1 r conjugates pσ t r q tPR and pg t q tPR , we have for all f : ΓzG X Ñ R and x P Σ r , when defined,

B i t pf ˝Θ´1 r qpxq " d i dt i |t"0 f ˝Θ´1 r pσ t r xq " d i dt i |t"0 f pg t Θ ´1 r pxqq " pB i t f q ˝Θ´1 r pxq . 133 19/12/2016
Hence if f : ΓzG X Ñ R is C k -smooth along the orbits of pg t q tPR , then f ˝Θ´1 r is C k -smooth along the orbits of pσ t r q tPR . Furthermore, since Θ r is bilipschitz, the precomposition map by Θ ´1 r is a continuous linear isomorphism from C k, α b pΓzG Xq to C k, α b pΣ r q. Note that since Θ r conjugates pg t q tPR and pσ t r q tPR , and sends mc }mc} to Pr }Pr} , we have, for all ψ, ψ 1 P L 2 pΓzG Xq and t P R, cov mc }mc} , t pψ, ψ 1 q " cov Pr }Pr } , t pψ ˝Θ´1 r , ψ 1 ˝Θ´1 r q .

Therefore we only have to prove that the suspension pΣ r , pσ t r q tPR , Pr }Pr} q is rapid mixing.

Step 2 : Reduction to a suspension of a one-sided symbolic dynamics

In this Step, we explain the rather standard reduction concerning mixing rates from suspensions of two-sided topological Markov shifts to suspensions of one-sided topological Markov shifts. We use the obvious modifications of the notation and constructions concerning the suspension of a noninvertible transformation to a semiflow, given for invertible transformations at the beginning of Section 5.3.

We consider the one-sided topological Markov shift pΣ `, σ `, P `q over the alphabet A constructed at the beginning of Step 2 of the proof of Theorem 9.1, with the system of conductances c now replaced by 7 c. Let π `: Σ Ñ Σ `be the natural extension so that P `" pπ `q˚P and π `˝σ " σ `˝π `.

We are going to construct in Step 2, as the suspension of pΣ `, σ `, P `q with an appropriate roof function r `, a semiflow ppΣ `qr `, `pσ `qt r `˘tě0 , pP `qr `q, and prove that the flow pΣ r , pσ t r q tPR , Pr }Pr} q is rapid mixing if the semiflow ppΣ `qr `, `pσ `qt r `˘tě0 , pP `qr }pP `qr `} q is rapid mixing.

We start by introducing the notation that will be used in Step 2.

Let r `: Σ `Ñ s0, `8r be the map r `: x Þ Ñ λpe 0 q (9.5) if x " px n q nPN P Σ `and x 0 " pe 0 , h 0 , e 0 q P A . Note that this map has a positive lower bound, and a finite upper bound, and that it is locally constant (and even constant on the 1-cylinders of Σ `). By Equation (5.10), we have r `˝π `" r .

(9.6)

We denote by ppΣ `qr `, `pσ `qt r `˘tě0 , pP `qr `q the suspension semiflow over pΣ `, σ `, P `q with roof function r `. We (uniquely) represent the points of the suspension space pΣ `qr `as rx, ss for x P Σ `and 0 ď s ă r `pxq. For all t ě 0, we have pσ `qt r `prx, ssq " rσ n `x, s 1 s where n P N and s 1 P R are such that t `s " ř n´1 i"0 r `pσ i `xq `s1 and 0 ď s 1 ď r `pσ n `xq.

We define the suspended natural extension as the map π r `: Σ r Ñ pΣ `qr `by π r `: rx, ss Þ Ñ rπ `pxq, ss , which is well defined by Equation (9.6). Note that π r `is 1-Lipschitz for the Bowen-Walters distance on Σ r and pΣ `qr `(see Proposition 5.11).

19/12/2016

For all ψ : Σ r Ñ R and T ě 0, let us construct a function ψ pT q : pΣ `qr `Ñ R as follows. For every rx, ss P pΣ `qr `, let N P N and s 1 ě 0 be such that pσ `qT r `rx, ss " rσ N `x, s 1 s, with 0 ď s 1 ă r `pσ N `xq and s `T "

N ´1 ÿ i"0 r `pσ i `xq `s1 .
Let ψ pT q prx, ssq " ψpry, s 1 sq where y " py n q nPZ is such that y i " x i`N if i ě ´N and y i " z x 0 i`N otherwise. Note that y 0 " x N , hence rpyq " r `pσ N `pxqq, and the above map is well defined. Finally, for every

ψ P C k, α b pΣ r q or ψ P C k, α b ppΣ `qr `q, let }ψ} k, 8 " k ÿ i"0 }B i t ψ} 8 and }ψ} 1 k, α " k ÿ i"0 }B i t ψ} 1 α , so that }ψ} k, α " }ψ} k, 8 `}ψ} 1 k, α .
Lemma 9.8. Let T ě 0 and ψ P C k, α b pΣ r q.

(1) For all t ě 0, we have pσ `qt r `˝π r `" π r `˝σ t r .

(2) With α 1 " α sup λ , there exists a constant C 1 ą 0 (independent of k, T and ψ) such that |ψ ˝σT

r ´ψpT q ˝π r `| ď C 1 }ψ} 1 α e ´α1 T .
(3) We have ψ pT q P C k, α b ppΣ `qr `q and }ψ pT q } 8 ď }ψ} 8 . With α 2 " α inf λ , there exists a constant C 2 ą 0 (independent of k, T and ψ) such that

}ψ pT q } 1 k, α ď C 2 e α 2 T }ψ} 1 k, α . (9.7)
Proof.

(1) For every rx, ss P Σ r , let n P N and s 1 ě 0 be such that t `s "

n´1 ÿ i"0 r `pσ i `π`p xqq `s1 and 0 ď s 1 ď r `pσ n `π`p xqq .
Since r `˝σ i `˝π `" r ˝σi for all i P N, these two equalities are equivalent to (2) By Proposition 5.11, we may assume that, in the formula of the Hölder norms, the Bowen-Walters distance is replaced by the function d BW , as this will only change C 1 by C α BW C 1 . For every rx, ss P Σ r , with ry, s 1 s and N associated with π r `prx, ssq " rπ `pxq, ss as in the definition of ψ pT q prπ `pxq, ssq, we have d BW pσ T r rx, ss, ry, s 1 sq " d BW prσ N x, s 1 s, ry, s 1 sq ď dpσ N x, yq ď e ´N .

Since the positive roof function r is bounded from above by the least upper bound sup λ of the lengths of the edges, we have

N ě N ´1 ÿ i"0 rpσ i xq sup λ " 1 sup λ ps `T ´s1 q ě T sup λ ´1 .
Hence |ψ ˝σT r prx, ssq ´ψpT q pπ r `pxqq| " |ψpσ T r rx, ssq ´ψpry, s 1 sq| ď }ψ} 1 α d BW pσ T r rx, ss, ry, s 1 sq α ď }ψ} 1 α e ´α sup λ T `α .

(3) The inequality }ψ pT q } 8 ď }ψ} 8 is immediate by construction. Let us prove that ψ pT q is C k along semiflow lines. Fix rx, ss P pΣ `qr `. Let us consider ą 0 small enough, so that ă r `pxq ´s and ă r `pσ N `xq ´s1 , with ry, s 1 s and N as in the construction of ψ pT q prx, ssq. Then ψ pT q ˝pσ `q r `prx, ssq " ψ pT q prx, s ` sq " ψpry, s 1 ` sq " ψ ˝σ r pry, s 1 sq .

Therefore by taking derivatives with respect to in this formula, ψ pT q is indeed C k along semiflow lines, and, for i " 0, . . . , k, we have

B i t `ψpT q ˘" `Bi t ψ ˘pT q . (9.8)
Let us prove that there exists a constant C 2 ą 0 (independent of T and ψ) such that

}ψ pT q } 1 α ď C 2 e α 2 T }ψ} 1 α .
(9.9)

Let rx, ss P Σ r , with ry, s 1 s and N as in the definition of ψ pT q prx, ssq. Let rx, s s P Σ r , with ry, s 1 s and N as in the definition of ψ pT q prx, s sq. Up to exchanging rx, ss and rx, s s, we assume that N ď N .

By Proposition 5.11, we may assume that, in the Hölder norms formulas, the Bowen-Walters distance is replaced by the function d BW , as this will only change

C 2 by C 2 α BW C 2 . Let C 3 " minte ´1, inf λu .
Note that the map d BW on Σ r ˆΣr is bounded from above by 1 `sup λ, since the distance on Σ is at most 1 and since the roof function r is bounded from above by sup λ.

We have |ψ pT q prx, ssq ´ψpT q prx, s sq| " |ψpry, s 1 sq ´ψpry, s 1 sq| ď }ψ}

N ´1 ÿ i"0 r `pσ i `xq `s1
and since the roof function r `is bounded from below by inf λ, we have T ě N inf λ ´inf λ, or equivalently N ď T inf λ `1. Hence dpx, xq ă C 3 e ´T inf λ ď e ´N by the definition of C 3 . In particular the sequences x and x indexed by N have the same N `1 first coefficients. Since r `pzq depends only on z 0 for all z P Σ `, we thus have r `pσ i `xq " r `pσ i `xq for i " 0, . . . , N . Note that we have

s `T " N ´1 ÿ i"0 r `pσ i `xq `s1 .
If N " N , then by taking the difference of the last two centred equations, we have s ´s " s 1 ´s1 , and by construction, the sequences y and y indexed by Z satisfy y i " pyq i if i ď 0 and if 0 ď i ď ´ln dpx, xq ´N . Therefore dpy, yq ď e N dpx, xq and Therefore Equation (9.9) follows from Equation (9.10) whenever C 2 ě e α .

If on the contrary N ą N , then again by difference s ´s "

N ´1 ÿ i"N `1 r `pσ i `xq `r`p σ N `xq ´s1 `s1 .
Note that s 1 ě 0, that r `pσ N `xq ´s1 ě 0, and that |s ´s| ă C 3 e ´T inf λ ď inf λ by the definition of C 3 . Hence we have N " N `1 and s´s " r `pσ N `xq´s 1 `s1 . By construction, the sequences σy and y indexed by Z satisfy pσyq i " pyq i if i ď 0 and if 0 ď i ď ´ln dpx, xq ´N ´1. Hence by the definition of d BW and since rpyq " r `pσ N `xq, we have d BW pry, s 1 s, ry, s 1 sq ď dpσy, yq `rpyq ´s1 `s1 ď e N `1 dpx, xq `|s ´s| ď e N `1 d BW prx, ss, rx, s sq ď e T inf λ `2 d BW prx, ss, rx, s sq .

Therefore Equation (9.9) follows from Equation (9.10) whenever C 2 ě e 2α . This ends the proof of Equation (9.9). Now note that Equations (9.8) and (9.9) imply Equation (9.7) by summation (using the independence of C 2 on ψ), thus concluding the proof of Lemma 9.8. l 137 19/12/2016 Proposition 9.9. Let µ be a pσ t r q tPR -invariant probability measure on Σ r . Assume that the dynamical system ppΣ `qr `, ppσ `qt r `qtPR , pπ r `q˚µ q has superpolynomial decay of α-Hölder correlations. Then pΣ r , pσ t r q tPR , µq has superpolynomial decay of α-Hölder correlations.

Proof. We fix n P N. Let N " 1 `2r sup λ inf λ s. Let k P N and C 4 ą 0 (depending on n) be such that for all ψ, ψ 1 P C k, α b ppΣ `qr `q, we have for all t ě 1

| cov pπ r `q˚µ, t pψ, ψ 1 q| ď C 4 }ψ} k, α }ψ 1 } k, α t N n . (9.11)
Now let ψ, ψ 1 P C k, α b pΣ r q. We again denote by ˘t any value in r´t, ts for any t ě 0. By invariance of µ under pσ t r q tPR , by Lemma 9.8 (2) and by Lemma 9.8 (1), we have, for any T ě 0 (to be chosen appropriately later on),

ż Σr ψ ˝σt r ψ 1 dµ " ż Σr ψ ˝σT `t r ψ 1 ˝σT r dµ " ż Σr pψ pT q ˝π r `˘C 1 }ψ} 1 α e ´α1 T q ˝σt r pψ 1 pT q ˝π r `˘C 1 }ψ 1 } 1 α e ´α1 T q dµ " ż pΣ `qr `ψpT q ˝pσ `qt r `ψ1 pT q dpπ r `q˚µ ˘C2 1 }ψ} α }ψ 1 } α e ´α1 T .
A similar estimate holds for the second term in the definition of the correlation coefficients. Hence, applying Equation (9.11) to the observables ψ pT q and ψ 1 pT q , by Lemma 9.8 (3), we have

| cov µ, t pψ, ψ 1 q| ď | cov pπ r `q˚µ, t pψ pT q , ψ 1 pT q q| `2 C 2 1 }ψ} α }ψ 1 } α e ´α1 T ď C 4 p}ψ} k, 8 `C2 }ψ} 1 k, α e α 2 T q p}ψ 1 } k, 8 `C2 }ψ 1 } 1 k, α e α 2 T q t ´N n `2 C 2 1 }ψ} α }ψ 1 } α e ´α1 T ď }ψ} k, α }ψ 1 } k, α pC 4 C 2 2 e 2α 2 T t ´N n `2 C 2 1 e ´α1 T q .
Take T " n α 1 ln t ě 0. Since N " 1 `2r α 2 α 1 s, we have 2α 2 n α 1 ´N n ď ´n. Hence with

C 5 " C 4 C 2 2 `2 C 2 1 , we have for all t ě 1 | cov µ, t pψ, ψ 1 q| ď C 5 }ψ} k, α }ψ 1 } k, α t ´n .
This concludes the proof of Proposition 9.9. l

Step 3 : Conclusion of the proof of Theorem 9.7

In this Step, we prove that the semiflow ppΣ `qr `, `pσ `qt r `˘tě0 , pP `qr }pP `qr `} q is rapid mixing, which concludes the proof of Theorem 9.7, using Proposition 9.9 with µ " Pr }Pr} . Recall (see the proof of Theorem 5.9) that Y " t P ΓzG X : p0q P V Xu is a cross-section of the geodesic flow on ΓzG X, and that if R : Y Ñ ΓzG X is the reparametrisation map of P Y to a discrete geodesic line 7 P ΓzG X with the same origin, then the measure µ Y , induced by the Gibbs measure m c on the cross-section Y by disintegration along the flow, maps by R ˚to a constant multiple of m7 c (see Lemma 5.10 (2)). Hence for all n P N, by 138 19/12/2016 Assumption (b) (1) in the statement of Theorem 9.7, we have m7 c

´! 7 P ΓzG X :

7 p0q P E @ k P t1, . . . , n ´1u, 7 pkq R E )ď }m7 c } }µ Y } µ Y ´! R ´1p 7 q P Y : R ´1p 7 qp0q P E @ t P s0, n inf λr , R ´1p 7 qptq R E )ď }m7 c } }µ Y } m c ´! g s R ´1p 7 q P ΓzG X : 0 ď s ď , dpg s R ´1p 7 qp0q, Eq ď @ t P s , n inf λ ´ r , g s R ´1p 7 qptq R E )ď }m7 c } }µ Y } C e ´κ pinf λq n`κ .
Therefore Equation (9.1) (where c is replaced by 7c) is satisfied, with C 1 "

}m 7 c } C e κ }µ Y }
and κ 1 " κ inf λ. As seen in the proof of Theorem 9.1, this implies that there exists a finite subset E of the alphabet A such that Equation (9.3) is satisfied.

We now apply [Mel1, Theo. 2.3] with the dynamical system pX, m 0 , T q " pΣ `, P `, σ `q (using the system of conductances 7 c) and the roof function h " r `. This dynamical system is presented as a Young tower in Step 3 of the proof of Theorem 9.1. Equation ( 9.3) for the first return map τ E and the 4-Diophantine hypothesis are exactly the hypothesis needed in order to apply [START_REF] Melbourne | Rapid decay of correlations for nonuniformly hyperbolic flows[END_REF]Theo. 2.3]. Thus the semiflow ppΣ `qr `, `pσ `qt r `˘tě0 , pP `qr }pP `qr `} q has superpolynomial decay of α-Hölder correlations.

When ΓzX is compact, the alphabet A is finite and pΣ `, σ `, P `q is a (one-sided) subshift of finite type, hence we do not need the exponentially small tail assumption, and only the 2-Diophantine hypothesis, and we may apply [START_REF] Dolgopyat | Prevalence of rapid mixing in hyperbolic flows[END_REF]. l Corollary 9.10. Assume that the critical exponent δ F is finite, that the Gibbs measure m F is finite and mixing, that the lengths of the edges of pX, λq have a finite upper bound, and that ΓzX is geometrically finite. There exists a full measure subset A of R 4 (for the Lebesgue measure) such that if Γ has a quadruple of translation lengths in A, then the (continuous time) geodesic flow on ΓzG X has superpolynomial decay of α-Hölder correlations for the Bowen-Margulis measure m BM .

Proof. The exponentially small tail Assumption (b) (1) is checked as in the proof of Corollary 9.6. The deduction of Corollary 9.10 from Theorem 9.7 then proceeds, by an argument going back in part to Dolgopyat, as for the deduction of Corollary 2.4 from Theorem 2.3 in [START_REF] Melbourne | Rapid decay of correlations for nonuniformly hyperbolic flows[END_REF]. l

Note that under the general assumptions of Theorem 4.8, the geodesic flow on ΓzG X might not be exponentially mixing, see for instance [START_REF] Pollicott | Meromorphic extensions of generalised zeta functions[END_REF]page 162] or [START_REF] Ruelle | Flows which do not exponentially mix[END_REF]

Equidistribution of equidistant level sets to Gibbs measures

Let X be a geodesically complete proper CATp´1q space, let Γ be a nonelementary discrete group of isometries of X, let r F be a continuous Γ-invariant map on T 1 X such that δ " δ Γ, F ȋs finite and positive and that the triple pX, Γ, r F q satisfies the HC-property, 1 and let pµ x q xPX be Patterson densities for the pairs pΓ, F ˘q.

In this Chapter, we prove that the skinning measure on (any nontrivial piece of) the outer unit normal bundle of any properly immersed nonempty proper closed convex subset of X, pushed a long time by the geodesic flow, equidistributes towards the Gibbs measure, under finiteness and mixing assumptions. This result gives four important extensions of [PaP14a, Thm. 1], one for general CATp´1q spaces with constant potentials, one for Riemannian manifolds with pinched negative curvature and Hölder potentials, one for R-trees with general potentials, and one for simplicial trees.

A general equidistribution result

Before stating this equidistribution result, we start by a technical construction which will also be useful in the following Chapter 11. We refer to Section 2.5 for the notation concerning the dynamical neighbourhoods (including V w, η 1 , R ) and to Chapter 7 for the notation concerning the skinning measures (including ν w ).

Technical construction of bump functions. Let D ˘be nonempty proper closed convex subsets of X, and let R ą 0 be such that ν w pV w, η 1 , R q ą 0 for all η 1 ą 0 and w P B 1 ¯D˘. Let η ą 0 and let Ω ˘be measurable subsets of B 1 ¯D˘. We now construct functions φ η, R, Ω ˘:

G X Ñ r0, `8r whose supports are contained in dynamical neigbourhoods of Ω ˘. If X " Ă M is a manifold and r F " 0, we recover the same bump functions after the standard identifications. For all η 1 ą 0, let h η, η 1 : G ¯X Ñ r0, `8r be the Γ-invariant measurable maps defined by h η, η 1 pwq " 1 ν w pV w, η, η 1 q (10.1) if ν w pV w, η, η 1 q ą 0 (which is for instance satisfied if w ˘P ΛΓ) and h η, η 1 pwq " 0 otherwise.

1 See Definition 3.4.

19/12/2016

These functions h η, η 1 have the following behaviour under precomposition by the geodesic flow. By Lemma 7.5 (2), by Equation (2.13), and by the invariance of ν w under the geodesic flow, we have, for all t P R and w P G ˘X , h η, η 1 pg ¯twq " e C w˘p wp0q, wp¯tqq h η, e ´tη 1 pwq .

(10.2)

Let us also describe the behaviour of h η, η 1 when η 1 is small. Let w P G ˘X be such that w is isometric at least on ˘r0, `8r, which is for instance the case if w P B 1 ˘D¯. For all η 1 ą 0 and P B ˘pw, η 1 q, let p w be an extension of w such that d W ˘pwq p , p wq ă η 1 . Then p wp0q " wp0q by the assumption on w, and using [START_REF] Paulin | Equilibrium states in negative curvature[END_REF]Lem. 2.4], 2 we have dp p0q, wp0qq " dp p0q, p wp0qq ď d W ˘pwq p , p wq ă η 1 .

Hence, with κ 1 and κ 2 the constants in the Definition 3.4, if

c 1 " κ 1 `δ `sup π ´1pBpwp0q, η 1 qq | r F | ,
we have, by Proposition 3.10 (2),

| C w¯p wp0q, p0qq | ď c 1 pη 1 q κ 2 .
Using the defining Equation (7.12) of ν w , for all s P R, η 1 ą 0 and P B ˘pw, η 1 q, we have e ´c1 pη 1 q κ 2 ds dµ W ˘pwq p q ď dν w pg s q ď e c 1 pη 1 q κ 2 ds dµ W ˘pwq p q .

It follows that for all η 1 P s0, 1s and w P B 1 ˘D¯s uch that w ˘P ΛΓ, we have the following control of h η, η 1 pwq: e ´c1 pη 1 q κ 2 2η µ W ˘pwq pB ˘pw, η 1 qq ď h η, η 1 pwq ď e c 1 pη 1 q κ 2 2η µ W ˘pwq pB ˘pw, η 1 qq .

(10.3)

Note that when X is an R-tree, we may take κ 2 " 1 and c 1 " } r F ´δ} 8 in this equation, as we saw in the proof of Proposition 3.5.

Recall that 1 A denotes the characteristic function of a subset A. We now define the test functions φ η, R, Ω ˘: G X Ñ r0, `8r by

φ η, R, Ω ¯" h η, R ˝f D¯1 V η, R pΩ ¯q , (10.4)
where V η, R pΩ ¯q and f D¯a re as in Section 2.5. Note that if P V η, R pΩ ¯q, then ˘R B 8 D by convexity. Thus, belongs to the domain of definition

U D¯o f f D¯; hence φ η, R, Ω ¯p q " h η, R
˝f D¯p q is well defined. By convention, φ η, R, Ω ¯p q " 0 if R V η, R pΩ ¯q. The following property of the bump functions is proved as in [START_REF] Parkkonen | Skinning measure in negative curvature and equidistribution of equidistant submanifolds[END_REF]Prop. 18], using the disintegration result of Proposition 7.6.

Lemma 10.1. For every η ą 0, the functions φ η, R, Ω ˘are measurable, nonnegative and satisfy

ż G X φ η, R, Ω ˘d r m F " r σ ˘pΩ ¯q . l
2 Although it is stated for Riemannian manifolds, the argument is valid in general CATp´1q spaces.

19/12/2016

We now state and prove the aforementioned equidistribution result. Note that as elements of the outer unit normal bundles are now only geodesic rays on r0, `8r, their pushforwards by the geodesic flow at time t are geodesic rays on r´t, `8r and the convergence towards geodesic lines (defined on s ´8, `8r) does take place in the full space of generalised geodesic lines p G X. This explains why it is important not to forget to consider the negative timesskinning measures, supported on geodesic rays, pushed by the geodesic flow, have a chance to weak-star converge to Gibbs measures, supported on geodesic lines, up to renormalisation.

The proof of the following result has similarities with that of [PaP14a, Theo. 1], but the computations do not apply in the present context because the proof in loc. cit. does not keep track of the past: here we can no longer reduce our study to the outer unit normal bundle of the t-neighbourhood of the elements of D.

Theorem 10.2. Let pX, Γ, r F q be as in the beginning of Chapter 10. Assume that the Gibbs measure m F on ΓzG X is finite and mixing for the geodesic flow. Let D " pD i q iPI be a locally finite Γ-equivariant family of nonempty proper closed convex subsets of X with ΓzI finite. Let Ω " pΩ i q iPI be a locally finite Γ-equivariant family of measurable subsets of p G X, with

Ω i Ă B 1
˘Di for all i P I. Assume that σ Ω is finite and nonzero. Then, as t Ñ `8, for the weak-star convergence of measures on Γz p G X,

1 }pg ˘tq ˚σΩ } pg ˘tq ˚σΩ á 1 }m F } m F .
Proof. We only give the proof when ˘" `, the other case is treated similarly. Given three numbers a, b, c (depending on some parameters), we write a " b ˘c if |a ´b| ď c. Let η P s0, 1s. As in the proof of [START_REF] Parkkonen | Skinning measure in negative curvature and equidistribution of equidistant submanifolds[END_REF]Thm. 19], we may assume that ΓzI is finite. Hence, using Lemma 7.5 (i), we may fix R ą 0 such that ν ẃ pV ẁ, η, R q ą 0 for all w P B 1 `Di and i P I.

Using the notation introduced in the above construction of the bump functions φ ή, R, Ω i , we may hence consider the global test functions r Φ η : G X Ñ r0, `8r,

r Φ η pvq " ÿ iPI{" φ ή, R, Ω i " ÿ iPI{" h ή, R ˝f Di 1 V ὴ, R pΩ i q .
As in loc. cit., the function r Φ η : G X Ñ r0, `8r is well defined (independent of the representatives of i), measurable and Γ-equivariant. Hence it defines, by passing to the quotient, a measurable function Φ η : ΓzG X Ñ r0, `8r . By Lemma 10.1, the function Φ η is integrable and satisfies ż

ΓzG X Φ η dm F " }σ Ὼ } . (10.5) Fix ψ P C c pΓz p G Xq. Let us prove that lim tÑ`8 1 }pg t q ˚σῺ } ż ΓzG X ψ dpg t q ˚σῺ " 1 }m F } ż ΓzG X ψ dm F .
Consider a fundamental domain ∆ Γ for the action of Γ on p G X, such that the boundary of ∆ Γ has zero measure, the interiors of its translates are disjoint and any compact subset of p G X meets only finitely many translates of ∆ Γ : see [START_REF] Roblin | Ergodicité et équidistribution en courbure négative[END_REF]p. 13] (or the proof of [START_REF] Parkkonen | Skinning measure in negative curvature and equidistribution of equidistant submanifolds[END_REF]145 19/12/2016 Prop. 18]), using the fact that r m F has no atoms, since m F is finite and according to Corollary 4.6(1) and to Theorem 4.5.

By a standard argument of finite partition of unity and up to modifying ∆ Γ , we may assume that there exists a function r ψ :

p G X Ñ R whose support has a small neighbourhood contained in ∆ Γ such that r ψ " ψ ˝p, where p : p G X Ñ Γz p G X is the canonical projection (which is Lipschitz). Fix ą 0. Since r ψ is uniformly continuous, for every η ą 0 small enough and for every t ě 0 large enough, for all w P G `X and P V ẁ, η, e ´tR , we have

r ψp q " r ψpwq ˘ 2 . (10.6)
If t is big enough and η small enough, we have, using respectively • the definition of the global test function r Φ η , since the support of r ψ is contained in ∆ Γ and the support of φ ή, R, Ω i is contained in U Di , for the second equality,

• the disintegration property of f Di in Proposition 7.6 for the third equality,

• the fact that if is in the support of ν ρ , then f Di pg ´t q " f Di p q " ρ and the change of variables by the geodesic flow w " g t ρ for the fourth equality, • the fact that the support of ν ǵ´t w is contained in W 0`p g ´twq, and that W 0`p g ´twq X g t V ὴ, R pΩ i q " g t `W 0`p g ´twq X V ὴ, R pΩ i qq " g t V g´t w, η, R " V ẁ, η, e ´tR for the fifth equality, • Equation (10.6) for the sixth equality, and • the definition of h ´, the invariance of the measure ν ǵ´t w and the Gibbs measure r m F under the geodesic flow, and the definition of the measure σ Ὼ for the last two equalities.

ż ΓzG X ψ Φ η ˝g´t dm F " ż ∆ Γ X G X r ψ r Φ η ˝g´t d r m F " ÿ iPI{" ż PU Di r ψp q φ ή, R, Ω i pg ´t q d r m F p q " ÿ iPI{" ż ρPB 1 `Di ż PU Di r ψp q h ή, R pf Di pg ´t qq1 V ὴ,R pΩ i q pg ´t q dν ρ p q dr σ Di pρq " ÿ iPI{" ż wPg t B 1 `Di ż Pg t V ὴ,R pΩ i q
r ψp q h ή, R pg ´twqq dν ǵ´t w p q dpg t q ˚r σ Di pwq "

ÿ iPI{" ż wPg t B 1 `Di ż PV ẁ, η, e ´tR r ψp q h ή, R pg ´twq dν ǵ´t w p q dpg t q ˚r σ Di pwq " ÿ iPI{" ż wPg t B 1 `Di r ψpwq h ή, R pg ´twq ν ǵ´t w pg t V g´t w, η, R q dpg t q ˚r σ Di pwq ˘ 2 ż ∆ Γ X G X r Φ η ˝g´t d r m F " ÿ iPI{" ż p G X r ψ dpg t q ˚r σ Di ˘ 2 ż ΓzG X Φ η ˝g´t dm F " ż Γz p G X ψ dpg t q ˚σῺ ˘ 2 ż ΓzG X Φ η dm F (10.7) 146 19/12/2016
We then conclude as in the end of the proof of [START_REF] Parkkonen | Skinning measure in negative curvature and equidistribution of equidistant submanifolds[END_REF]Thm. 19]. By Equation (10.5), we have }pg t q ˚σῺ } " }σ Ὼ } " ş ΓzG X Φ η dm F . By the mixing property of the geodesic flow on ΓzG X for the Gibbs measure m F , for t ě 0 large enough (while η is small but fixed), we hence have

ş Γz p G X ψ dpg t q ˚σῺ }pg t q ˚σῺ } " ş ΓzG X Φ η ˝g´t ψ dm F ş ΓzG X Φ η dm F ˘ 2 " ş ΓzG X ψ dm F }m F } ˘ .
This proves the result. l

Recall that by Proposition 3.5, Theorem 10.2 applies to Riemannian manifolds with pinched negative curvature and for R-trees for which the geodesic flow is mixing and which satisfy the finiteness requirements of the Theorem.

Since pushforwards of measures are weak-star continuous and preserve total mass, we have, under the assumptions of Theorem 10.2, the following equidistribution result in X of the immersed t-neighbourhood of a properly immersed nonempty proper closed convex subset of X: as t Ñ `8, 1

}σ Ὼ } π ˚pg t q ˚σῺ á 1 }m F } π ˚mF .
(10.8)

Rate of equidistribution of equidistant level sets for manifolds

If X " Ă M is a simply connected Riemannian manifold of pinched negative curvature and if the geodesic flow of Γz Ă M is mixing with exponentially decaying correlations, we get a version of Theorem 10.2 with error bounds. See Section 9.1 for conditions on Γ and r F that imply exponential mixing.

Theorem 10.3. Let Ă M be a complete simply connected Riemannian manifold with negative sectional curvature. Let Γ be a nonelementary discrete group of isometries of Ă M . Let r F : T 1 Ă M Ñ R be a bounded Γ-invariant Hölder-continuous function with positive critical exponent δ " δ Γ, F . Let D " pD i q iPI be a locally finite Γ-equivariant family of nonempty proper closed convex subsets of Ă M , with finite nonzero skinning measure σ D . Let M " Γz Ă M and let F : T 1 M Ñ R be the potential induced by r F .

(i) If M is compact and if the geodesic flow on T 1 M is mixing with exponential speed for the Hölder regularity for the potential F , then there exist α P s0, 1s and κ 2 ą 0 such that for all ψ P C α c pT 1 M q, we have, as t Ñ `8,

1 }σ D } ż ψ dpg t q ˚σD " 1 }m F } ż ψ dm F `Ope ´κ2 t }ψ} α q .
(ii) If Ă M is a symmetric space, if D i has smooth boundary for every i P I, if m F is finite and smooth, and if the geodesic flow on T 1 M is mixing with exponential speed for the Sobolev regularity for the potential F , then there exists P N and κ 2 ą 0 such that for all ψ P C c pT 1 M q, we have, as t Ñ `8,

1 }σ D } ż ψ dpg t q ˚σD " 1 }m F } ż ψ dm F `Ope ´κ2 t }ψ} q . 147 19/12/2016
Note that if Ă M is a symmetric space and M has finite volume, then M is geometrically finite. Theorem 4.7 implies that m F is finite if F is small enough. The maps Op¨q depend on Ă M , Γ, F, D, and the speeds of mixing.

Proof. Up to rescaling, we may assume that the sectional curvature is bounded from above by ´1. The critical exponent δ and the Gibbs measure m F are finite in all the cases we consider here.

The deduction of this result from the proof of Theorem 10.2 by regularisations of the global test function Φ η introduced in the proof of Theorem 10.2 is analogous to the deduction of [START_REF] Parkkonen | Skinning measure in negative curvature and equidistribution of equidistant submanifolds[END_REF]Theo. 20] from [START_REF] Parkkonen | Skinning measure in negative curvature and equidistribution of equidistant submanifolds[END_REF]Theo. 19] when F " 0. The doubling property of the Patterson densities and the Gibbs measure for general F , required by this deduction in the Hölder regularity case, is given by [START_REF] Paulin | Equilibrium states in negative curvature[END_REF]Prop. 3.12]. For the assertion (ii), the required smoothness of m F (that is, the fact that m F is absolutely continuous with respect to the Lebesgue measure with smooth Radon-Nikodym derivative) allows to use the convolution approximation. l 10.3 Equidistribution of equidistant level sets on simplicial graphs and random walks on graphs of groups Let X, X, Γ, r c, Ă F c , δ c be as in the beginning of Section 9.2. Let r F c : T 1 X Ñ R be its associated potential, and let δ " δ c be the critical exponent of c. Let pµ x q xPV X be two Patterson densities on B 8 X for the pairs pΓ, F c q, and let m c " m Fc be the associated Gibbs measure on ΓzG X.

In this Section, we state an equidistribution result analogous to Theorem 10.2, which now holds in the space of generalised discrete geodesic lines Γz p G X, but whose proof is completely analogous.

Theorem 10.4. Let X, Γ, r c, pµ x q xPV X be as above, with δ c finite. Assume that the Gibbs measure m c on ΓzG X is finite and mixing for the discrete time geodesic flow. Let D " pD i q iPI be a locally finite Γ-equivariant family of nonempty proper simplicial subtrees of X. Let Ω " pΩ i q iPI be a locally finite Γ-equivariant family of measurable subsets of p G X, with Ω i Ă B 1 `Di for all i P I. Assume that σ Ὼ is finite and nonzero. Then, as n Ñ `8, for the weak-star convergence of measures on Γz p G X,

1 }pg n q ˚σῺ } π ˚pg n q ˚σῺ á 1 }m F } m F . l
We leave to the reader the analog of this result when the restriction to ΓzG even X of the Gibbs measure is finite and mixing for the square of the discrete time geodesic flow.

Using Propositions 4.14 and 4.15 in order to check that the Bowen-Margulis measure m BM on ΓzG X is finite and mixing, we have the following consequence of Theorem 10.4, using the system of conductances r c " 0.

Corollary 10.5. Let X be a uniform simplicial tree. Let Γ be a lattice of X such that the graph ΓzX is not bipartite. Let D " pD i q iPI be a locally finite Γ-equivariant family of nonempty proper simplicial subtrees of X. Let Ω " pΩ i q iPI be a locally finite Γ-equivariant family of measurable subsets of p G X, with Ω i Ă B 1 `Di for all i P I. Assume that the skinning measure 148 19/12/2016

σ Ὼ (with vanishing potential) is finite and nonzero. Then, as n Ñ `8, for the weak-star convergence of measures on Γz p G X,

1 }pg n q ˚σῺ } π ˚pg n q ˚σῺ á 1 }m BM } m BM . l
When furthermore X is regular, we have the following corollary, using Proposition 8.1 (3).

Corollary 10.6. Let X be a regular simplicial tree of degree at least 3. Let Γ be a lattice of X such that the graph ΓzX is not bipartite. Let D " pD i q iPI be a locally finite Γ-equivariant family of nonempty proper simplicial subtrees of X. Let Ω " pΩ i q iPI be a locally finite Γequivariant family of measurable subsets of p G X, with Ω i Ă B 1 `Di for all i P I. Assume that the skinning measure σ Ὼ (with vanishing potential) is finite and nonzero. Then, as n Ñ `8, for the weak-star convergence of measures on Γz p G X,

1 }pg n q ˚σῺ } π ˚pg n q ˚σῺ á 1 VolpΓz zXq vol Γz zX . l
Let us give an application of Corollary 10.6 in terms of random walks on graphs of groups, which might also be deduced from general result on random walks, as indicated by M. Burger and S. Mozes.

Let pY, G ˚q be a connected graph of finite groups with finite volume, and let pY 1 , G 1 ˚q be a connected subgraph of subgroups. 3 Note that pY 1 , G 1 ˚q also has finite volume, less than or equal to the volume of pY, G ˚q. We say that pY, G ˚q is homogeneous if ř ePEY, opeq"x |Gx| |Ge| is constant at least 3 for all x P V Y. We say that a connected graph of groups is 2-acylindrical if the action of its fundamental group on its Bass-Serre tree is 2-acylindrical (see Remark 5.4). In particular, this action is faithful if the graph has at least two edges.

The non-backtracking simple random walk on pY, G ˚q starting transversally to pY 1 , G 1 ˚q is the following Markovian random process pX n " pf n , γ n qq nPN where f n P EY and γ n P G opfnq for all n P N. Choose at random a vertex y 0 of Y 1 for the probability measure

1 VolpY 1 ,G 1 ˚q vol Y 1 ,G 1
˚(we will call y 0 the origin of the random path). Then choose uniformly at random X 0 " pf 0 , γ 0 q where f 0 P EY is such that opf 0 q " y 0 and γ 0 is a double coset in G 1 y 0 zG y 0 {ρ f 0 pG f 0 q such that if f 0 P EY 1 then γ 0 R G 1 y 0 ρ f 0 pG f 0 q. 4 Assuming X n " pf n , γ n q constructed, choose uniformly at random X n`1 " pf n`1 , γ n`1 q where f n`1 P EY is such that opf n`1 q " tpf n q and γ n`

1 P G opf n`1 q {ρ f n`1 pG f n`1 q is such that if f n`1 " f n then γ n`1 R ρ f n`1 pG f n`1 q. The n-th vertex of pX n " pf n , γ n qq nPN is opf n q.
Corollary 10.7. Let pY, G ˚q be a homogeneous 2-acylindrical nonbipartite connected graph of finite groups with finite volume, and let pY 1 , G 1 ˚q be a homogeneous nonempty proper connected subgraph of subgroups. Then the n-th vertex of the non-backtracking simple random walk on pY, G ˚q starting transversally to pY 1 , G 1 ˚q converges in distribution to 1 VolpY,G˚q vol Y,G˚a s n Ñ `8.

Proof. Let Γ be the fundamental group of pY, G ˚q (with respect to a choice of basepoint in V Y 1 ), which is a lattice of the Bass-Serre tree X of pY, G ˚q, since Γ acts faithfully on X 3 See Section 2.7 for definitions and background. 4 This last condition says that γ0 is not the double coset of the trivial element.

149

19/12/2016 and pY, G ˚q has finite volume. Note that X is regular since pY, G ˚q is homogeneous. Let p : X Ñ Y " ΓzX be the canonical projection. Let Γ 1 be the fundamental group of pY 1 , G 1 ˚q (with respect to the same choice of basepoint). By [START_REF] Bass | Covering theory for graphs of groups[END_REF]2.15], there exists a simplicial subtree X 1 whose stabiliser in Γ is Γ 1 , such that the quotient graph of groups Γ 1 z zX 1 identifies with pY 1 , G 1 ˚q. Similarly, X 1 is regular since pY 1 , G 1 ˚q is homogeneous. Let D " pγX 1 q γPΓ{Γ 1 , which is a locally finite family of nonempty proper simplicial subtrees of X.

Using the notation of Example 2.10 for the graph of groups Γz zX (which identifies with pY, G ˚q), we fix lifts r f and r y in X by p of every edge f and vertex y of Y such that r f " r f , and elements g f P Γ such that g f Ą tpf q " tp r f q. We may assume that r

f P EX 1 if f P EY 1 , that r y P V X 1 if y P V Y 1 , and that g f P Γ 1 if f P EY 1 .
Let pΩ, Pq be the (canonically constructed) probability space of the random walk pX n " pf n , γ n qq nPN . For all n P N, let y n " opf n q be the random variable (with values in the discrete space Y " ΓzX) of the n-th vertex of the random walk pX n q nPN . Let us define a measurable map Θ : Ω Ñ Γz p G X, with image contained in the image of

Ω Θ ÝÝÝÝÑ Γz p G X yn OE Ö π˝g n Y " ΓzX .
(10.9)

Assuming that we have such a map, we have py n q ˚P " pπ ˚˝pg n q ˚˝Θ ˚qP " π ˚pg n q ˚σD }σ D } " 1

}pg n q ˚σD } π ˚pg n q ˚σD so that the convergence of the law of y n to 1 VolpY,G˚q vol Y,G˚f ollows from Corollary 10.6 applied to B

1 `D . Y X p f n f n`1 y n`1 y n y n`2 α n Ă y n γ n α n`1 Ć y n`1 e n e n`1 g ´1 fn Ă f n g ´1 fn Ă f n γ n`1 g ´1 fn`1 Ć f n`1 g ´1 fn g fn y 0 f 0 r y 0 r f 0 g f0 e 0 g ´1 f0 r f 0 γ 0 150 19/12/2016
Let pX n " pf n , γ n qq nPN be a random path with origin y 0 P Y 1 , corresponding to ω P Ω. Fix a representative of γ n in its right class for every n ě 1, and a representative of γ 0 in its double class, that we still denote by γ n and γ 0 respectively. Using ideas used for the coding introduced in Section 5.2, let us construct by induction an infinite geodesic edge path pe n q nPN with origin ope 0 q " r y 0 and a sequence pα n q nPN in Γ such that

e n " α n γ n g ´1 fn Ă f n .
(10.10) Let α 0 " id and e 0 " γ 0 g ´1 f 0 r f 0 . Since op r f 0 q " g f 0 r y 0 by the construction of the lifts and since γ 0 P G y 0 " Γ Ă y 0 , we have ope 0 q " r y 0 . Since the stabiliser of g ´1 f 0 r f 0 is ρ f 0 pG f 0 q, the edge e 0 does not depend on the choice of the representative γ 0 modulo ρ f 0 pG f 0 q on the right, but depends on the choice of the representative γ 0 modulo G 1 y 0 " Γ 1 Ă y 0 on the left. The hypothesis that if f 0 P EY 1 then γ 0 R G 1 y 0 ρ f 0 pG f 0 q ensures that the edge e 0 does not belong to EX 1 . Indeed, assume otherwise that e 0 belongs to EX 1 . Then f 0 " ppe 0 q P EY 1 , and by the assumptions on the choice of lifts, the edges g ´1 f 0 r f 0 and e 0 both belong to EX 1 . Since they are both mapped to f 0 by the map X 1 Ñ Y 1 " Γ 1 zX 1 , they are mapped one to the other by an element of Γ 1

Ă y 0 " G 1 y 0 . Let γ 1 0 P Γ 1 Ă y 0 be such that γ 1 0 e 0 " g ´1 f 0 r f 0 . Then γ 1 0 ´1γ 0 belongs to the stabiliser in Γ of the edge g ´1 f 0 r f 0 , which is equal to g ´1 f 0 Γ Ă f 0 g f 0 " ρ f 0 pG f 0 q. Therefore γ 0 P G 1 y 0 ρ f 0 pG f 0 q, a
contradiction. Assume by induction that e n and α n are constructed. Define

α n`1 " α n γ n g ´1 fn g fn and e n`1 " α n`1 γ n`1 g ´1 f n`1 Ć f n`1 ,
so that the induction formula (10.10) at rank n `1 is satisfied. By the construction of the lifts, since y n`1 " tpf n q " opf n`1 , we have

Ć y n`1 " g ´1 fn tp Ă f n q " g ´1 f n`1 op Ć f n`1 q .
Hence, since γ n`1 P G y n`1 fixes Ć y n`1 , using the induction formula (10.10) at rank n for the last equality,

ope n`1 q " α n`1 γ n`1 g ´1 f n`1 op Ć f n`1 q " α n`1 γ n`1 Ć y n`1 " α n`1 Ć y n`1 " α n`1 g ´1 fn tp Ă f n q " α n γ n g ´1 fn tp Ă f n q " tpe n q .
In particular, the sequence pe n q nPN is an edge path in X.

Since the stabiliser of g ´1

f n`1 Ć f n`1 is ρ f n`1 pG f n`1 q, the edge γ n`1 g ´1 f n`1 Ć f n`1
does not depend on the choice of the representative of the right coset γ n`1 . Let us prove that the length

2 edge path pg ´1 fn Ă f n , γ n`1 g ´1 f n`1 Ć f n`1 q is geodesic.
Otherwise, the two edges of this path are opposite one to another, hence f n`1 " f n by using the projection p : X Ñ Y, therefore

g f n`1 " g fn . Thus γ n`1 maps g ´1 fn Ă f n to g ´1 fn Ă
f n , hence belongs to ρ fn pG fn q " ρ f n`1 pG f n`1 q, a contradiction by the assumptions on the random walk.

By construction, the element α n`1 of Γ sends the above length 2 geodesic edge path pg ´1 fn Ă f n , γ n`1 g ´1 f n`1 Ć f n`1 q to pe n , e n`1 q. This implies on the one hand that the edge path 151 19/12/2016 pe n , e n`1 q is geodesic, and on the other hand that α n`1 is uniquely defined, since the action of Γ on X is 2-acylindrical.

In particular, pe n q nPN is the sequence of edges followed by a (discrete) geodesic ray in X, starting from a point of X 1 but not by an edge of X 1 , that is, an element of B 1 `X1 . Furthermore, this ray is well defined up to the action of Γ 1 Ă y 0 , hence its image, that we denote by Θpωq, is well defined in Γz p G X. Since ppope n qq " ppĂ y n q " y n for all n P N, the commutativity of the diagram (10.9) is immediate.

For every x P V X 1 , let B 1 `X1 pxq be the subset of B 1 `X1 consisting of the elements w with wp0q " x. By construction, the above map from the subset of random paths in Ω starting from y 0 to Γ 1 Ă y 0 zB 1 `X1 p r y 0 q, which associates to pX n q nPN the Γ 1 Ă y 0 -orbit of the geodesic ray with consecutive edges pe n q nPN , is clearly a bijection. This bijection maps the measure P to the normalised skinning measure σ D }σ D } , since by homogeneity, the restriction to B 1 `X1 p r y 0 q of r σ X1

to B 1 `X1 p r y 0 q, normalized to be a probability measure, is the restriction to B 1 `X1 p r y 0 q of the AutpXq x -homogeneous probability measure on the space of geodesic rays with origin x in the regular tree X. This proves the result. l

When Y is finite, all the groups G y for y P V Y are trivial and Y 1 is reduced to a vertex,5 the above random walk is the non-backtracking simple random walk on the nonbipartite regular finite graph Y, and 1 VolpY,G˚q vol Y,G˚i s the uniform distribution on V Y. Hence this result (stated as Corollary 1.3 in the Introduction) is classical. See for instance [START_REF] Ortner | Non-backtracking random walks and cogrowth of graphs[END_REF]Thm. 1.2] and [ABLS], which under further assumptions on the spectral properties of Y gives precise rates of convergence, and also the book [START_REF] Lyons | Probability on trees and networks[END_REF], including its Section 6.3 and its references.

Rate of equidistribution for metric and simplicial trees

In this Section, we give error terms for the equidistribution results stated in Theorem 10.2 for metric trees, and in Theorem 10.4 for simplicial trees, under additional assumptions required in order to get the error terms for the mixing property discussed in Chapter 9.

We first consider the simplicial case, for the discrete time geodesic flow. Let X, X, Γ, r c, Ă F c , δ c , pµ x q xPV X , m c " m Fc be as in Section 10.3.

Theorem 10.8. Assume that δ c is finite and that the Gibbs measure m c on ΓzG X is finite. Assume furthermore that

(1) the families pΛΓ, µ x , d x q xPV C ΛΓ and pΛΓ, µ x , d x q xPV C ΛΓ of metric measure spaces are uniformly doubling,6

(2) there exists α P s0, 1s such that the discrete time geodesic flow on pΓzG X, m c q is exponentially mixing for the α-Hölder regularity.

Let D " pD i q iPI be a locally finite Γ-equivariant family of nonempty proper simplicial subtrees of X with ΓzI finite. Let Ω " pΩ i q iPI be a locally finite Γ-equivariant family of measurable subsets of p G X, with Ω i Ă B 1 ˘Di for all i P I. Assume that σ Ω is finite and nonzero. Then there exists κ 1 ą 0 such that for all ψ P C α c pΓz p G Xq, we have, as n Ñ `8,

1 }pg ˘nq ˚σΩ } ż ψ d π ˚pg ˘nq ˚σΩ " 1 }m c } ż ψ d m c `Op}ψ} α e ´κ1 n q .
152 19/12/2016

Remarks.

(1) If r c " 0, if the simplicial tree X 1 with |X 1 | 1 " C ΛΓ is uniform, if L Γ " Z and if Γ is a lattice of X 1 , then we claim that δ c " δ Γ is finite, m c " m BM is finite and mixing, and pΛΓ, µ x " µ x , d x q xPV C ΛΓ is uniformly doubling. Indeed, the above finiteness and mixing properties follow from the results of Section 4.4. Since X 1 is uniform, it has a cocompact discrete sugbroup Γ 1 whose Patterson density (for the vanishing potential) is uniformly doubling on ΛΓ 1 " ΛΓ, by Lemma 4.2 (4). Since r c " 0 and Γ is a lattice, the Patterson densities of Γ and of Γ 1 coincide (up a scalar multiple) by Proposition 4.14 (2).

(2) Assume that r c " 0, that the simplicial tree X 1 with |X 1 | 1 " C λΓ is uniform without vertices of degree 2, that L Γ " Z and that Γ is a geometrically finite lattice of X 1 . Then all assumptions of Theorem 10.8 are satisfied by the first remark and by Corollary 9.6. Therefore we have an exponentially small error term in the equidistribution of the equidistant levels sets.

Proof. We only give the proof when ˘" `, the other case is treated similarly. We follow the proof of Theorem 10.2, concentrating on the new features. We now have ΓzI finite by assumption. Let η P s0, 1s and ψ P C α c pΓz p G Xq. We consider the constant R ą 0, the test function Φ η , the fundamental domain ∆ Γ and the lift r ψ " ψ ˝p as in the proof of Theorem 10.2.

For all n P N, all w P G `X isometric on r´n, `8r and all P V ẁ, η, e ´nR " B `pw, e ´nRq, 7 by Lemma 2.7 where we can take η " 0, we have dp , wq " Ope ´nq. Since p is Lipschitz, the map r ψ is α-Hölder with α-Hölder norm at most }ψ} α . Hence for all n P N, all w P G `X isometric on r´n, `8r and all P V ẁ, η, e ´nR , we have r ψp q " r ψpwq `Ope ´n α }ψ} α q .

(10.11)

As in the proof of Theorem 10.2 with t replaced by n, using Equation (10.11) instead of Equation (10.6) in the series of equations (10.7), since the symbols w that appear in them are indeed generalised geodesic lines isometric on r´n, `8r , we have

ş Γz p G X ψ dpg n q ˚σῺ }pg n q ˚σῺ } " ş ΓzG X ψ Φ η ˝g´n dm c ş ΓzG X Φ η dm c
`Ope ´n α }ψ} α q .

(10.12)

Let us now apply the assumption on the decay of correlations. In order to do that, we need to regularise our test functions Φ η . We start by an independent lemma. Lemma 10.9. There are universal constants 0 ą 0, c 0 ě 1 such that for all P s0, 0 r and P G X, the ball B d p , q is contained in t 1 P G X : 1 p0q " p0q, 1 p˘8q P B d p0q p p˘8q, c 0 ? qu and contains t 1 P G X : 1 p0q " p0q, 1 p˘8q P B d p0q p p˘8, 1 c 0 ? qu.

Proof. If , 1 P G X have distinct footpoints, then dp p0q, 1 p0qq ě 1, so that dp ptq, 1 ptqq ě 1 4 if |t| ď 1 4 , so that dp , 1 q ě ş 1

4

´1 4

1 4 e ´2|t| " 0 .

7 We have V ẁ, η, s " B `pw, sq for every s ą 0 since X is simplicial and η ă 1.

19/12/2016

Conversely, assume that , 1 P G X have equal footpoints, so that they coincide on r´N, N 1 s for some N, N 1 P N. By the definition of the visual distances (see Equation (2.1)), we have d p0q p p`8q, 1 p`8qq " e ´N 1 and similarly d p0q p p´8q, 1 p´8qq " e ´N . By the definition of the distance on p G X (see Equation (2.4)), we have, by an easy change of variables, dp , 1 q " ż `8

N 1 2 |t ´N 1 | e ´2t dt `ż ´N ´8 2 | ´N ´t| e 2t dt " pe ´2N 1 `e´2N q ż `8 0 2 u e ´2u du " 1 2 pe ´2N 1 `e´2N q .
The result follows. l

By the definition of the Gibbs measures, 8 this lemma implies that µ ´ p0q `Bd p0q p p´8q,

1 c 0 ? q ˘µ` p0q `Bd p0q p p`8q, 1 c 0 ? q ď r m c pB d p , qq ď µ ´ p0q `Bd p0q p p´8q, c 0 ? q ˘µ` p0q `Bd p0q p p`8q, c 0 ? q ˘.
Since the Patterson densities are uniformly doubling for basepoints in C λΓ, since the footpoints of the geodesic lines in the support of m c belong to C λΓ, the Gibbs measure m c is hence doubling on its support. As in the proof of [START_REF] Parkkonen | Skinning measure in negative curvature and equidistribution of equidistant submanifolds[END_REF]Thm. 20], using discrete convolution approximation (see for instance [START_REF] Semmes | Finding curves on general spaces through quantitative topology, with applications to Sobolev and Poincaré inequalities[END_REF] or [KinKST]), there exists κ 2 ą 0 and, for every η ą 0, a nonnegative function

RΦ η P C α c pΓzG Xq such that (1) ş ΓzG X RΦ η d m c " ş ΓzG X Φ η d m c , (2) ş ΓzG X | RΦ η ´Φη | d m c " O `η ş ΓzG X Φ η d m c ˘, (3) } RΦ η } α " O `η´κ 2 ş ΓzG X Φ η d m c ˘.
By Equation (10.5), the integral ş ΓzG X Φ η dm c " }σ Ὼ } is constant (in particular independent of η). Let m c " mc }mc} . All integrals below besides the first one being over ΓzG X, and using ' Equation (10.12) and the above property (2) of the regularised map RΦ η for the first equality,

' the assumption of exponential decay of correlations for the second one, involving some constant κ ą 0, for the second equality, ' the above properties (1) and (3) of the regularised map RΦ η for the last equality,

ş Γz p G X ψ dpg n q ˚σῺ }pg n q ˚σῺ } " ş ψ RΦ η ˝g´n d m c ş Φ η d m c `Ope ´n α }ψ} α `η }ψ} 8 q " ş RΦ η d m c ş ψ d m c ş Φ η d m c `Ope ´n α }ψ} α `η }ψ} 8 `e´κ n } RΦ η } α }ψ} α q " ż ψ d m c `O `pe ´n α `η `e´κ n η ´κ2 q}ψ} α ˘.
Taking η " e ´λ n with λ " κ 2κ 2 , the result follows with κ 1 " mintα, κ 2κ 2 , κ 2 u. l

Let us now consider the metric tree case, for the continuous time geodesic flow, where the main change is to assume a superpolynomial decay of correlations and hence get a superpolynomial error term, for observables which are smooth enough along the flow lines. Let pX, λq, X, Γ, r F , δ F pµ x q xPX and m F be as in the beginning of Section 4.4.

Theorem 10.10. Assume that δ F is finite and that the Gibbs measure m F on ΓzG X is finite. Assume furthermore that

(1) the families pΛΓ, µ x , d x q xPC ΛΓ and pΛΓ, µ x , d x q xPC ΛΓ of metric measure spaces are uniformly doubling, 9 , and r F is bounded on T 1 C ΛΓ,

(2) there exists α P s0, 1s such that the (continuous time) geodesic flow on pΓzG X, m F q has superpolynomial decay of α-Hölder correlations.

Let D " pD i q iPI be a locally finite Γ-equivariant family of nonempty proper closed convex subsets of X with ΓzI finite. Let Ω " pΩ i q iPI be a locally finite Γ-equivariant family of measurable subsets of p G X, with Ω i Ă B 1 ˘Di for all i P I. Assume that σ Ω is finite and nonzero. Then for every n P N, there exists k P N such that for all ψ P C k, α c pΓz p G Xq, we have, as t Ñ `8,

1 }pg ˘tq ˚σΩ } ż ψ d π ˚pg ˘tq ˚σΩ " 1 }m F } ż ψ d m F `Op}ψ} k, α t ´nq .
Remarks. (1) If F " 0, if the metric subtree X 1 " C ΛΓ of X is uniform, if the length spectrum of Γ on X is not contained in a discrete subgroup of R and if Γ is a lattice of X 1 , then we claim that δ F " δ Γ is finite, m F " m BM is finite and mixing, and pΛΓ, µ x " µ x , d x q xPCΛΓ is uniformly doubling. Indeed, the above finiteness and mixing properties follow from Proposition 4.14 and Theorem 4.8. Since X 1 is uniform, it has a cocompact discrete sugbroup of isometries Γ 1 whose Patterson density (for the vanishing potential) is uniformly doubling on ΛΓ 1 " ΛΓ, by Lemma 4.2 (4). Since F " 0 and Γ is a lattice, the Patterson densities of Γ and of Γ 1 coincide (up to a scalar multiple) by Proposition 4.14.

(2) Assume that F " 0, that the metric subtree X 1 " C ΛΓ of X is uniform, that the length spectrum of Γ on X is 4-Diophantine and that Γ is a geometrically finite lattice of X 1 . Then all assumptions of Theorem 10.10 are satisfied by the first remark and by Corollary 9 See Section 4.1 for definitions.

155

19/12/2016 9.10. Therefore we have a superpolynomially small error term in the equidistribution of the equidistant levels sets.

Proof. The proof is similar to the one of Theorem 10.8, except for the doubling property of the Gibbs measure on its support and the conclusion of the proof. Let X 1 " C ΛΓ. The modification of Lemma 10.9 is the third assertion of the following result, which will also be useful in Section 12.6. Its second claim strengthens for R-trees the Hölder-continuity of the footpoint projection stated in Section 2.3. If a and b are positive functions of some parameters, we write ab if there exists a universal constant C ą 0 such that 1 C b ď a ď C b. Lemma 10.11. Let Y be an R-tree.

(1) There exists a universal constant c 1 ą 0 such that for all , 1 P G Y , if dp , 1 q ď c 1 , then 1 p0q P pRq, the intersection pRq X 1 pRq is not reduced to a point, the orientations of and 1 coincide on this intersection, and dp , 1 q -d p0q p p´8q, 1 p´8qq 2 `d p0q p p`8q, 1 p`8qq 2 `dp p0q, 1 p0qq .

(2) The footpoint map π : G Y Ñ Y defined by Þ Ñ p0q is (uniformly locally) Lipschitz.

(3) There are universal constants 0 ą 0, c 0 ě 1 such that for all P s0, 0 r and P G Y , the ball B d p , q in G Y is contained in t 1 P G Y : 1 p0q P pRq, dp 1 p0q, p0qq ď c 0 , 1 p˘8q P B d p0q p p˘8q, c 0 ? qu and contains t 1 P G Y : 1 p0q P pRq, dp 1 p0q, p0qq ď 1 c 0 , 1 p˘8q P B d p0q p p˘8q, 1 c 0 ? qu .

Proof.

(1) Let , 1 P G Y . If 1 p0q R pRq, then 1 ptq R pRq for all t ě 0 or 1 ptq R pRq for all t ď 0, since Y is an R-tree. In the first case, we hence have dp ptq, 1 ptqq ě t for all t ě 0, thus dp , 1 q is at least c 2 " ş `8 0 te ´2 t dt " 1 4 ą 0. The same estimate holds in the second case. This argument furthermore shows that if the geodesic segment (or ray or line) pRq X 1 pRq is reduced to a point, then dp , 1 q is at least c 2 ą 0.

If dp 1 p0q, p0qq ě 1, then dp ptq, 1 ptqq ě 1 4 for |t| ď 1 4 , thus

dp , 1 q ě ż 1 4 ´1 4 1 4 e ´2|t| dt ,
which is a positive universal constant.

If dp 1 p0q, p0qq ď 1, if pRq X 1 pRq contains 1 p0q and is not reduced to a point, but if the orientations of and 1 do not coincide on this intersection, then dp ptq, 1 ptqq ě 2t dp p0q, 1 p0qq ě 2t ´1 for all t ě 1, so that dp , 1 q is at least ş `8 1 p2t ´1q e ´2t dt, which is a positive constant.

Assume now that 1 p0q P pRq, that dp 1 p0q, p0qq ď 1, that pRq X 1 pRq is not reduced to a point and that the orientations of and 1 coincide on this intersection. Then there exists s P R such that 1 p0q " psq, so that |s| " dp p0q, 1 p0qq. Assume for instance that s ě 0, the other case being treated similarly. Then there exist S, S 1 ě 0 maximal such that 1 ptq " pt `sq for all t P r´S, S 1 s, with the convention that S " `8 if 1 p´8q " p´8q, that S 1 " `8 if 1 p`8q " p`8q, and that e ´8 " 0.

156 19/12/2016 p0q pS 1 `sq psq p´S `sq p´8q p`8q

1 p´8q 1 p`8q 1 p´Sq 1 p0q 1 pS 1 q
By the definition of the visual distances (see Equation (2.1)), we have, for t big enough, d p0q p p`8q, 1 p`8qq " e 1 2 rpt´S 1 q`pt´S 1 ´sqs´t -e ´S1 .

Similarly d p0q p p´8q, 1 p´8qq -e ´S .

As can be seen in the above picture, we have

dp ptq, 1 ptqq " $ & % ´2t ´2S `s if t ď ´S s if ´S ď t ď S 1 `s 2t ´2S 1 ´s if t ě S 1 `s .
By the definition of the distance on p G Y (see Equation (2.4)), by easy changes of variables, assuming that at least one of S, S 1 is larger than some positive constant for the last line, we have

dp , 1 q " ż ´S ´8 p´2t ´2S `sq e ´2|t| dt `ż S 1 `s ´S s e ´2|t| dt `ż `8 S 1 `sp2t ´2S 1 ´sq e ´2t dt " e ´2S`s ż `8 s u e ´u du `s ż S 1 `s ´S e ´2|t| dt `e´2S 1 ´s ż `8
s u e ´u du e ´2S `e´2S 1 `s .

Assertion (1) of Lemma 10.11 follows.

(2) The second assertion follows immediately from the first one.

(3) The first inclusion in the third assertion follows easily from the first one, and the second one follows from the argument of its proof, and the fact that if d p0q p p`8q, 1 p`8qq, d p0q p p`8q, 1 p`8qq and dp p0q, 1 p0qq are at most some small positive constant, then pRq X 1 pRq contains 1 p0q and is not reduced to a point. l

If the footpoints of , 1 P G X 1 are at distance bounded by c 0 0 , then by Proposition 3.10 (2), since | r F | is bounded on T 1 X 1 by assumption, the quantities |C ξ p p0q, 1 p0qq| for ξ P ΛΓ are bounded by the constant c 1 0 " c 0 0 pmax T 1 X 1 | r F ´δF |q. By the definition of the Gibbs measures (see Equation (4.3)), Assertion (3) of the above lemma hence implies that if ď 0 then e ´2c 1 0 µ ´ p0q `Bd p0q p p´8q,

1 c 0 ? q ˘µ` p0q `Bd p0q p p`8q, 1 c 0 ? q ď r m c pB d p , qq ď e 2c 1 0 µ ´ p0q `Bd p0q p p´8q, c 0 ? q ˘µ` p0q `Bd p0q p p`8q, c 0 ? q ˘.
As in the simplicial case, since the Patterson densities are uniformly doubling for basepoints in X 1 , the Gibbs measure m c is hence doubling on its support. 157 19/12/2016

Fix n P N. As in the end of the proof of Theorem 10.8, using the assumption of superpolynomial decay of correlations, involving some degree of regularity k in order to have polynomial decay in t ´N n where N " rκ 2 s `1, instead of the exponential one, we have for all t ě 1 ş Γz p G X ψ dpg t q ˚σῺ }pg t q ˚σῺ } "

ż ψ d m c `O `pe ´t α `η `t´Nn η ´κ2 q}ψ} k, α ˘.
Taking η " t ´n, by the definition of N , we hence have

ş Γz p G X ψ dpg t q ˚σῺ }pg t q ˚σῺ } " ż ψ d m c `O `t´n }ψ} k, α ˘.
This proves Theorem 10.10. l 158 19/12/2016

Chapter 11

Equidistribution of common perpendicular arcs

In this Chapter, we prove the equidistribution of the initial and terminal vectors of common perpendiculars of convex subsets in the universal covering space level for Riemannian manifolds and for metric and simplicial trees. The results generalise [START_REF] Parkkonen | Counting common perpendicular arcs in negative curvature[END_REF]Thm. 8].

In Sections 11.1 to 11.3, we consider the continuous time situation where the CAT(´1)space X is either a proper R-tree without terminal point or a complete Riemannian manifold with pinched negative curvature at most ´1, and x 0 is any basepoint in X. In Section 11.4, X is the geometric realisation of a simplicial tree X, with the discrete time geodesic flow, and x 0 is any vertex of X.

Let Γ be a nonelementary discrete group of isometries of X. Let r F be a continuous Γinvariant potential on T 1 X, which is Hölder-continuous if X is a manifold. Assume that δ " δ Γ, F ˘is positive and let pµ x q xPX be Patterson densities for the pairs pΓ, F ˘q, with associated Gibbs measure m F . Let D ´" pD í q iPI ´and D `" pD j q jPI `be locally finite Γ-equivariant families of nonempty proper closed convex subsets of X.

For every pi, jq in I ´ˆI `such that the closures D í and D j of D í and D j in X X B 8 X have empty intersection, let λ i,j " dpD í , D j q be the length of the common perpendicular from D í to D j , and α í, j P p G X its parametrisation: it is the unique map from R to X such that α í, j ptq " α í, j p0q P D í if t ď 0, α í, j ptq " α í, j pλ i, j q P D j if t ě λ i, j , and α í,j |r0, λ i,j s " α i, j is the shortest geodesic arc starting from a point of D í and ending at a point of D j . Let α ì,j " g λ i,j α í,j . In particular, we have g λ i,j 2 α í,j " g ´λi,j 2 α ì,j .

We now state our main equidistribution result of common perpendiculars between convex subsets in the continuous time and upstairs settings. We will give the discrete time version in Section 11.4, and the downstairs version in Chapter 12.

Theorem 11.1. Let X be a proper R-tree without terminal points or a complete Riemannian manifold with pinched negative curvature at most ´1. Let Γ be a nonelementary discrete group of isometries of X and let r F be a bounded Γ-invariant potential on X as above. Assume that the critical exponent δ is finite and positive, and that the Gibbs measure m F is finite and or the weak-star and narrow convergences of measures on the locally compact space p G X ˆp G X.

Recall that the narrow topology 1 on the set M f pY q of finite measures on a Polish space Y is the smallest topology such that the map from M f pY q to R defined by µ Þ Ñ µpgq is continuous for every bounded continuous function g : Y Ñ R.

The proof of Theorem 11.1 follows that of [START_REF] Parkkonen | Counting common perpendicular arcs in negative curvature[END_REF]Thm.8], which proves this result when X is a Riemannian manifold with pinched negative curvature at most ´1 and F " 0. The first two and a half steps work for both trees and manifolds and are given in Section 11.1. The differences begin in Step 3T. After this, the steps for trees are called 3T and 4T and are given in Section 11.2 and the corresponding steps for manifolds are 3M and 4M , given in Section 11.3.

In the special case of D ´" pγxq γPΓ and D `" pγyq γPΓ for some x, y P X, this statement gives the following version with potentials of Roblin's double equidistribution theorem [START_REF] Roblin | Ergodicité et équidistribution en courbure négative[END_REF]Theo. 4.1.1] when F " 0, see [START_REF] Paulin | Equilibrium states in negative curvature[END_REF]Theo. 9.1] for general F when X is a Riemannian manifold with pinched sectional curvature at most ´1.

Corollary 11.2. Assume that r F is bounded and that the Gibbs measure m F is finite and mixing for the geodesic flow on G X. Then Here is a version of Theorem 11.1 without the assumption that the critical exponent of δ " δ Γ, F is positive.

Theorem 11.3. Assume that r F is bounded and that the Gibbs measure m F is finite and mixing for the geodesic flow on G X. Then for every τ ą 0, we have

lim tÑ`8 δ }m F } 1 ´e´τ δ e ´δ t ÿ iPI ´{", jPI `{", γPΓ D í XD γj "H, t´τ ăλ i, γj ďt e ş α i,γj r F ∆ α í, γj b ∆ α γ´1 i, j " r σ D´b r σ Df
or the weak-star convergence of measures on the locally compact space p G X ˆp G X.

Proof. The claim follows from Theorem 11.1 by replacing F by F `κ for κ large enough so that δ Γ, F `κ " δ Γ, F `κ ą 0, and by using a classical geometric series type of arguments, see for instance [START_REF] Paulin | Equilibrium states in negative curvature[END_REF]Lem. 9.5] for more details. l

1 also called weak topology see for instance [START_REF] Dellacherie | Probabilities and Potential[END_REF] or [Bil, Part] 160

F ∆ α í, γj b ∆ α γ´1 i, j " r σ Dí b r σ Dj (11.1)
for the weak-star convergence of measures on p G X ˆp G X. Let Ω ´be a Borel subset of B 1 `Dí and let Ω `be a Borel subset of B 1 ´Dj . To simplify the notation, let

D ´" D í , D `" D j , α γ " α í, γj , α γ " α γ´1 i, j , λ γ " λ i, γj , r σ ˘" r σ D¯. (11.2)
Assume that Ω ´and Ω `have positive finite skinning measures and that their boundaries in B 1 `D´a nd B 1 ´D`h ave zero skinning measures (for r σ `and r σ ´respectively). Let

I Ω ´, Ω `ptq " δ }m F } e ´δ t ÿ γPΓ : 0ăλγ ďt α γ | s0,λγ s PΩ ´|s0,λγ s , α γ | s´λγ ,0s PΩ `|s´λγ ,0s e ş α γ r F . (11.3)
We will prove the stronger statement that, for every such Ω ˘, we have lim tÑ`8

I Ω ´, Ω `ptq " r σ `pΩ ´q r σ ´pΩ `q .

(11.4)

Step 2: First upper and lower bounds. Using Lemma 7.5 (1), we may fix R ą 0 such that ν w pV w, η, R q ą 0 for all η P s0, 1s and w P B 1 ¯D˘. Let φ η " φ η, R, Ω ˘be the test functions defined in Equation (10.4).

For all t ě 0, let

a η ptq " ÿ γPΓ ż PG X φ ή pg ´t{2 q φ ὴ pg t{2 γ ´1 q d r m F p q . (11.5)
As in [START_REF] Parkkonen | Counting common perpendicular arcs in negative curvature[END_REF], the heart of the proof is to give two pairs of upper and lower bounds, as T ě 0 is big enough and η P s0, 1s is small enough, of the (Césaro-type) quantity

i η pT q " ż T 0 e δ t a η ptq dt . (11.6)
By passing to the universal cover, the mixing property of the geodesic flow on ΓzG X for the Gibbs measure m F gives that, for every ą 0, there exists T " T ,η ě 0 such that for all t ě T , we have

e ´ }m F } ż G X φ ή d r m F ż G X φ ὴ d r m F ď a η ptq ď e }m F } ż G X φ ή d r m F ż G X φ ὴ d r m F .
Hence by Lemma 10.1, for all ą 0 and η P s0, 1s, there exists c " c ,η ą 0 such that for every T ě 0, we have

e ´ e δ T δ }m F } r σ `pΩ ´q r σ ´pΩ `q ´c ď i η pT q ď e e δ T δ }m F } r σ `pΩ ´q r σ ´pΩ `q `c . (11.7) 161 19/12/2016

Step 3: Second upper and lower bounds.

Let T ě 0 and η P s0, 1s. By Fubini's theorem for nonnegative measurable maps, the definition 2 of the test functions φ η and the flow-invariance 3 of the fibrations f D¯, we have (11.8) We start the computations by rewriting the product term involving the functions h η, R . For all γ P Γ and P U D´X U γD `, define (using Equation (2.11))

i η pT q " ÿ γPΓ ż T 0 e δ t ż G X h ή, R ˝f D´p q h ὴ, R ˝f D`p γ ´1 q 1 g t{2 V ὴ, R pΩ ´qp q 1 g ´t{2 V ή, R pγΩ `qp q d r m F dt .
w

´" f D´p q P G `, 0 X and w `" f γD `p q " γf D`p γ ´1 q P G ´, 0 X . (11.9)

This notation is ambiguous (w ´depends on , and w `depends on and γ), but it makes the computations less heavy. By Equations ( 10.2) and (3.8), we have, for every t ě 0,

h ή, R pw ´q " h ή, R ˝g´t{2 pg t{2 w ´q " e ş w ´pt{2q w ´p0q p r F ´δq h ή, e ´t{2 R pg t{2 w ´q .
Similarly,

h ὴ, R pγ ´1w `q " e ş w `p0q w `p´t{2q p r F ´δq h ὴ, e ´t{2 R pg ´t{2 w `q .
Hence,

h ή, R ˝f D´p q h ὴ, R ˝f D`p γ ´1 q " e ´δ t e ş w ´pt{2q w ´p0q r F `şw `p0q
w `p´t{2q r F h ή, e ´t{2 R pg t{2 w ´q h ὴ, e ´t{2 R pg ´t{2 w `q . (11.10)

11.2 Part II of the proof of Theorem 11.1: the metric tree case

In this Section, we assume that X is an R-tree and we will consider the manifold case separately in Section 11.3.

Step 3T. Consider the product term in Equation (11.8) involving the characteristic functions. By Lemma 2.8 (applied by replacing D `by γD `), there exists t 0 ě 2 ln R `4 such that for all η P s0, 1s and t ě t 0 , for all P G X, if 1 g t{2 V ὴ, R pΩ ´qp q 1 g ´t{2 V ή, R pγΩ `qp q ‰ 0, then the following facts hold.

(i) By the convexity of D ˘, we have P U D´X U γD `.

(ii) By the definition 4 of w ˘, we have w ´P Ω ´and w `P γΩ `. The notation pw ´, w

`q here coincides with the notation pw ´, w `q in Lemma 2.8.

2 See Equation (10.4).

3 See Equation (2.11).

4 See Equation (11.9).

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(iii) There exists a common perpendicular α γ from D ´to γD `, whose length λ γ satisfies

| λ γ ´t | ď 2η ,
whose origin is α γ p0q " w ´p0q, whose endpoint is α γ p0q " w `p0q, such that the points w ´p t 2 q and w `p´t 2 q are at distance at most η from p0q P α γ .

Hence, by Lemma 3. (11.11) For all η P s0, 1s, γ P Γ and T ě t 0 , let A η,γ pT q " pt, q P rt 0 , T s ˆG X :

P V ὴ, e ´t{2 R pg t{2 Ω ´q X V ή, e ´t{2 R pγg ´t{2 Ω `q(
and j η, γ pT q " ij pt, qPAη,γ pT q h ή, e ´t{2 R pg t{2 w ´q h ὴ, e ´t{2 R pg ´t{2 w `q dt d r m F p q .

By the above, since the integral of a function is equal to the integral on any Borel set containing its support, and since the integral of a nonnegative function is nondecreasing in the integration domain, there hence exists c 4 ą 0 such that for all T ě 0 and η P s0, 1s, we have

i η pT q ě ´c4 `e´2η}F }8 ÿ γPΓ : t 0 `2ďλγ ďT ´2η α γ | r0, λγ s PΩ ´|r0, λγ s , α γ | r´λγ , 0s PγΩ `|r´λγ , 0s e ş αγ r F j η, γ pT q ,
and similarly, for every T 1 ě T (later on, we will take T 1 to be T `4η),

i η pT q ď c 4 `e2η }F }8 ÿ γPΓ : t 0 `2ďλγ ďT `2η α γ | r0, λγ s PΩ ´|r0, λγ s , α γ | r´λγ , 0s PγΩ `|r´λγ , 0s e ş αγ r F j η, γ pT 1 q .
Step 4T: Conclusion. Let ą 0. Let γ P Γ be such that D ´and γD `do not intersect and the length of their common perpendicular satisfies λ γ ě t 0 `2. Let us prove that if η is small enough and λ γ is large enough (with the enough's independent of γ), then for every T ě λ γ `2η, we have

1 ´ ď j η, γ pT q ď 1 ` . (11.12)

This estimate proves the claim (11.4), as follows. For every ą 0, if η ą 0 is small enough, we have I Ω ´,Ω `pT q ď r σ `pΩ ´q r σ ´pΩ `q.

i η pT `2ηq ě ´c4 `e´2η}F }8 p1 ´ q ´IΩ ´,Ω `pT q δ }m F }e ´δ T ´IΩ ´,Ω `pt 0 `2q δ }m F }e ´δpt 0 `2q
The similar estimate for the lower limit proves the claim (11.4).

To prove the claim (11.12), let η P s0, 1s and T ě λ γ `2η. To simplify the notation, let r t " e ´t{2 R, w t " g t{2 w ´and w t " g ´t{2 w `.

By the definition of j η, γ , using the inequalities (10.3) (and the comment following them), we hence have j η, γ pT q " ij pt, qPAη,γ pT q h ή, rt pw t q h ὴ, rt pw t q dt d r m F p q " e Ope ´ γ {2 q p2ηq 2 ij pt, qPAη,γ pT q dt d r m F p q µ W `pw t q pB `pw t , r t qq µ W ´pw t q pB ´pw t , r t qq .

(11.13) Let x γ be the midpoint of the common perpendicular α γ . Let us use the Hopf parametrisation of G X with basepoint x γ , denoting by s its time parameter. When pt, q P A η,γ pT q, we have, by Proposition 3.10 where we may take κ 2 " 1, d r m F p q " e C ´ ´pxγ , p0qq `C`

`pxγ , p0qq dµ xγ p ´q dµ xγ p `q ds " e Opηq dµ xγ p ´q dµ xγ p `q ds . (11.14)

Let P γ be the plane domain of pt, sq P R 2 such that there exist s ˘P s ´η, ηr with s ¯" λγ ´t 2 ˘s. It is easy to see that P γ is a rhombus centred at pλ γ , 0q whose area is p2ηq 2 . Let ξ γ be the point at infinity of any fixed geodesic ray from x γ through α γ p0q. If A is a subset of G X, we denote by A ˘the subset t ˘: P Au of B 8 X.

Lemma 11.4. For every t ě t 0 , we have

pB ˘pw t , r t qq ¯" B dx γ pξ γ , R e ´λγ 2 q .
Proof. We prove the statement for the negative endpoints, the proof of the claim for positive endpoints is similar. Let 1 P B ˘pw t , r t q, with 1 ´‰ ξ γ . Let p P X be such that r 1 p0q, ξ γ r X r 1 p0q, 1 ´r " r 1 p0q, ps. p ξ γ w ´p0q " α γ p0q

1 ´xγ

w ´p t 2 q " 1 p0q 164 19/12/2016
Since t ě t 0 ą 2 ln R, we have r t ă 1, hence 1 p0q " w t p0q " w ´pt{2q and p P s 1 p0q, ξ γ r . Since t ě t 0 ě 1 ě η, we have p P rx γ , ξ γ r . Hence

d xγ p 1
´, ξ γ q " e ´dpp, xγ q " e ´dpp,w ´p t

2 qq´p λγ 2 ´t 2 q ă r t e ´λγ 2 `t 2 " R e ´λγ 2 .
Conversely, if ξ P B dx γ pξ γ , R e ´λγ 2 q with ξ ‰ ξ γ , let 1 P G X be such that 1 p0q " w ´pt{2q and 1

´" ξ. Let p w t be the extension of w t such that p p w t q ´" ξ γ . Then as above, we have d W ´pw t q p p w t , 1 q ă r t . l

It follows from this lemma that, for all t ě t 0 , s ˘P s ´η, ηr and P G X, we have g ¯s¯ P B ˘pw t , r t q if and only if dp p0q, α γ p0qq " s ˘`t 2 (or equivalently by the definition of the time parameter s in Hopf's parametrisation, s ˘`t 2 " λγ 2 ˘s), and ˘P B dx γ pξ γ , R e ´λγ 2 q. Thus, A η,γ pT q " P γ ˆBdx γ pξ γ , R e ´λγ 2 q ˆBdx γ pξ γ , R e ´λγ 2 q .

To finish Step 4T and the proof of the theorem for R-trees, note that by the definition of the skinning measure (using again the Hopf parametrisation with basepoint x γ ), by the above Lemma 11.4 and by Proposition 3.10 where we may take κ 2 " 1, we have (11.16) which gives the inequalities (11.12). l

µ W ˘pw t q pB ˘pw t ,
The effective control on j η, γ pT q given by Equation (11.16) is stronger than what is needed in order to prove Equation (11.12) in Step 4T. We will use it in Section 12.6 in order to obtain error terms.

Part III of the proof of Theorem 11.1: the manifold case

The proof of Theorem 11.1 for manifolds is the same as for trees until Equation (11.10). The rest of the proof that we give below is more technical than for trees but the structure of the proof is the same. In this Section, X " Ă M is a Riemannian manifold, and we identify G X with T 1 Ă M .

Step 3M. Consider the product term in Equation (11.8) involving the characteristic functions.

The quantity 1 V ὴ, R pΩ ´qpg ´t{2 vq 1 V ή, R pΩ `qpγ ´1g t{2 vq is different from 0 (hence equal to 1) if and only if

v P g t{2 V ὴ, R pΩ ´q X γg ´t{2 V ή, R pΩ `q " V ὴ, e ´t{2 R pg t{2 Ω ´q X V ή, e ´t{2 R pγg ´t{2 Ω `q ,
165 19/12/2016 see Section 2.5 and in particular Equation (2.15). By [START_REF] Parkkonen | Counting common perpendicular arcs in negative curvature[END_REF]Lem. 7] (applied by replacing D `by γD `and w by v), there exists t 0 , c 0 ą 0 such that for all η P s0, 1s and t ě t 0 , for all v P T 1 Ă M , if 1 V ὴ, R pΩ ´qpg ´t{2 vq 1 V ή, R pΩ `qpγ ´1g t{2 vq ‰ 0, then the following facts hold:

(i) by the convexity of D ˘, we have v P U D´X U γD `,

(ii) by the definition of w ˘(see Equation (11.9)), we have w ´P Ω ´and w `P γΩ `. The notation pw ´, w `q here coincides with the notation pw ´, w `q in Lemma 2.8, (iii) there exists a common perpendicular α γ from D ´to γD `, whose length λ γ satisfies

| λ γ ´t | ď 2η `c0 e ´t{2 ,
whose origin πpv γ q is at distance at most c 0 e ´t{2 from πpw ´q, whose endpoint πpv γ q is at distance at most c 0 e ´t{2 from πpw `q, such that both points πpg t{2 w ´q and πpg ´t{2 w `q are at distance at most η `c0 e ´t{2 from πpvq, which is at distance at most c 0 e ´t{2 from some point p v of α γ .

In particular, using (iii) and the uniform continuity property of the r F -weighted length (see Proposition 3.10 (3) which introduces a constant c 2 P s0, 1s), and since r F is bounded, for all η P s0, 1s, t ě t 0 and v P T

1 Ă M for which 1 V ὴ, R pΩ ´qpg ´t{2 vq 1 V ή, R pΩ `qpγ ´1g t{2 vq ‰ 0, we have e ş πpg t{2 w ´q πpw ´q r F `şπpw `q πpg ´t{2 w `q r F " e ş pv πpv γ q r F `şπpv γ q pv r F `Oppη`e ´t{2 q c 2 q " e ş αγ r
F e Oppη`e ´λγ {2 q c 2 q . (11.17) For all η P s0, 1s, γ P Γ and T ě t 0 , define A η,γ pT q as the set of pt, vq P rt 0 , T s ˆT 1 Ă M such that v P V ὴ, e ´t{2 R pg t{2 Ω ´q X V ή, e ´t{2 R pγg ´t{2 Ω `q, and j η, γ pT q " ij pt, vqPAη,γ pT q h ή, e ´t{2 R pg t{2 w ´q h ὴ, e ´t{2 R pg ´t{2 w `q dt d r m F pvq .

By the above, since the integral of a function is equal to the integral on any Borel set containing its support, and since the integral of a nonnegative function is nondecreasing in the integration domain, there hence exists c 4 ą 0 such that for all T ě 0 and η P s0, 1s, we have i η pT q ě ´c4 `ÿ γPΓ : t 0 `2`c 0 ďλγ ďT ´Opη`e ´λγ {2 q

v γ P N ´Opη`e ´λγ {2 q
Ω ´, v γ P γN ´Opη`e ´λγ {2 q Ω `eş αγ r F j η, γ pT q e ´Oppη`e ´λγ {2 q c 2 q , and similarly, for every T 1 ě T , i η pT q ď c 4 `ÿ γPΓ : t 0 `2`c 0 ďλγ ďT `Opη`e ´λγ {2 q v γ P N

Opη`e ´λγ {2 q Ω ´, v γ P γN

Opη`e ´λγ {2 q Ω èş αγ r F j η, γ pT 1 q e Oppη`e ´λγ {2 q c 2 q .

We will take T 1 to be of the form T `Opη `e´λγ{2 q, for a bigger Op¨q than the one appearing in the index of the above summation. 166 19/12/2016

Step 4M: Conclusion. Let γ P Γ be such that D ´and γD `have a common perpendicular with length λ γ ě t 0 `2 `c0 . Let us prove that for all ą 0, if η is small enough and λ γ is large enough, then for every T ě λ γ `Opη `e´λγ{2 q (with the enough's and Op¨q independent of γ), we have 1 ´ ď j η, γ pT q ď 1 ` . (11.18)

Note that r σ ˘pN ε pΩ ¯qq and r σ ˘pN ´εpΩ ¯qq tend to r σ ˘pΩ ¯q as ε Ñ 0 (since r σ ˘pBΩ ¯q " 0 as required in Step 1). Using Steps 2, 3M and 4M, this will prove Equation (11.4), hence will complete the proof of Theorem 11.1.

We say that p Ă M , Γ, r F q has radius-continuous strong stable/unstable ball masses if for every ą 0, if r ą 1 is close enough to 1, then for every v P T 1 Ă M , if B ´pv, 1q meets the support of µ Ẁ ´pvq , then µ Ẁ ´pvq pB ´pv, rqq ď e µ Ẁ ´pvq pB ´pv, 1qq and if B `pv, 1q meets the support of µ Ẃ `pvq , then µ Ẃ `pvq pB `pv, rqq ď e µ Ẃ `pvq pB `pv, 1qq .

We say that p Ă M , Γ, r F q has radius-Hölder-continuous strong stable/unstable ball masses if there exists c P s0, 1s and c 1 ą 0 such that for every P s0, 1s, if B ´pv, 1q meets the support of µ Ẁ ´pvq , then

µ Ẁ ´pvq pB ´pv, rqq ď e c 1 c µ Ẁ ´pvq pB ´pv, 1qq
and if B `pv, 1q meets the support of µ Ẃ `pvq , then µ Ẃ `pvq pB `pv, rqq ď e c 1 c µ Ẃ `pvq pB `pv, 1qq .

Note that when F " 0 and M has locally symmetric with finite volume, the conditional measures on the strong stable/unstable leaves are homogeneous. Hence p Ă M , Γ, r F q has radius-Hölder-continuous strong stable/unstable ball masses.

When the sectional curvature has bounded derivatives and when p Ă M , Γ, r F q has Hölder strong stable/unstable ball masses, we will prove a stronger statement: with a constant c 7 ą 0 and functions Op¨q independent of γ, for all η P s0, 1s and T ě λ γ `Opη `e´λγ{2 q, we have j η, γ pT q " ´1 `O ´e´λγ{2 2η ¯¯2 e Oppη`e ´λγ {2 q c 7 q . (11.19) This stronger version will be needed for the error term estimate in Section 12.3. In order to obtain Theorem 11.1, only the fact that j η, γ pT q tends to 1 as firstly λ γ tends to `8, secondly η tends to 0 is needed. A reader not interested in the error term may skip many technical details below.

Given a, b ą 0 and a point x in a metric space X (with a, b, x depending on parameters), we will denote by Bpx, a e Opbq q any subset Y of X such that there exists a constant c ą 0 (independent of the parameters) with Bpx, a e ´c b q Ă Y Ă Bpx, a e c b q .

Let η P s0, 1s and T ě λ γ `Opη `e´λγ{2 q. To simplify the notation, let r t " e ´t{2 R, w t " g t{2 w ´and w t " g ´t{2 w `.

19/12/2016

By the definition of j η, γ , using the inequalities (10.3), we hence have j η, γ pT q " ij pt,vqPAη,γ pT q h ή, rt pw t q h ὴ, rt pw t q dt d r m F pvq " e Ope ´c2 λγ {2 q p2ηq 2 ij pt,vqPAη,γ pT q dt d r m F pvq µ Ẃ `pw t q pB `pw t , rtqq µ Ẁ ´pw t q pB ´pw t , rtqq .

(11.20)

We start the proof of Equation (11.18) by defining parameters s `, s ´, s, v 1 , v 2 associated with pt, vq P A η,γ pT q.

v 0 γ v 2 v 1 v g t{2 w ´g´t{2 w s s ´´s x0 W ´pg ´t{2 w `q W `pg t{2 w ´q W `pv 1 q W ´pv 2 q W `pv 0 γ q W ´pv 0 γ q
We have pt, vq P A η,γ pT q if and only if there exist s ˘P s ´η, ηr such that

g ¯s¯v P B ˘pg ˘t{2 w ¯, e ´t{2 Rq .
The notation s ˘coincides with the one in the proof of Lemma 2.8 (where pD `, wq has been replaced by pγD `, vq).

In order to define the parameters s, v 1 , v 2 , we use the well known local product structure of the unit tangent bundle in negative curvature. If v P T 1 M is close enough to v 0 γ (in particular, v ´‰ pv 0 γ q `and v `‰ pv 0 γ q ´), then let v 1 " f HB ´pv 0 γ q pvq be the unique element of W ´pv 0 γ q such that v 1 `" v `, let v 2 " f HB `pv 0 γ q pvq be the unique element of W `pv 0 γ q such that v 2

´" v

´, and let s be the unique element of R such that

g ´sv P W `pv 1 q. The map v Þ Ñ ps, v 1 , v 2 q is a homeomorphism from a neighbourhood of v 0 γ in T 1 Ă M to a neighbourhood of p0, v 0 γ , v 0 γ q in R ˆW ´pv 0 γ q ˆW `pv 0 γ q. Note that if v " g r v 0 γ for some r P R close to 0, then w ´" v γ , w `" v γ , s " r, v 1 " v 2 " v 0 γ , s ´" λ γ ´t 2 `s, s `" λ γ ´t 2 ´s .
Up to increasing t 0 (which does not change Step 4, up to increasing c 4 ), we may assume that for every pt, vq P A η,γ pT q, the vector v belongs to the domain of this local product structure of T 1 Ă M at v 0 γ . The vectors v, v 1 , v 2 are close to v 0 γ if t is large and η small, as the following result shows. We denote (also) by d the Riemannian distance induced by Sasaki's metric on T 1 Ă M .

Lemma 11.5. For every pt, vq P A η,γ pT q, we have dpv, v 0 γ q, dpv 1 , v 0 γ q, dpv 2 , v 0 γ q " Opη `e´t{2 q. 168 19/12/2016

Proof. Consider the distance d 1 on T 1 Ă M , defined by

@ v 1 , v 2 P T 1 Ă M , d 1 pv 1 , v 2 q " max rPr´1,0s d `πpg r v 1 q, πpg r v 2 q ˘.
As seen in (iii) of Step 3M, we have dpπpw ˘q, πpv γ qq, dpπpvq, α γ q " Ope ´t{2 q, and furthermore, dpπpg t{2 w ´q, πpvqq, λγ 2 ´t 2 " Opη `e´t{2 q. Hence dpπpvq, πpv 0 γ qq " Opη `e´t{2 q. By [PauPS, Lem. 2.4], we have dpπpg ´t 2 ´s´v q, πpv γ qq ď dpπpg ´t 2 ´s´v q, πpw ´qq `dpπpw ´q, πpv γ qq ď R `c0 e ´t{2 .

By an exponential pinching argument, we hence have d 1 pv, v 0 γ q " Opη `e´λγ{2 q. Since d and d 1 are equivalent (see [START_REF] Ballmann | Lectures on spaces of nonpositive curvature[END_REF]page 70]), we therefore have dpv, v 0 γ q " Opη `e´λγ{2 q. For all w P T 1 Ă M and V P T w T 1 Ă M , we may uniquely write V " V ´`V 0 `V `with V ´P T w W ´pwq, V 0 P R d dt |t 0 g t w and V `P T w W `pwq. By [PauPS, §7.2] (building on [Brin] whose compactness assumption on M and torsion free assumption on Γ are not necessary for this, the pinched negative curvature assumption is sufficient), Sasaki's metric (with norm } ¨}) is equivalent to the Riemannian metric with (product) norm

}V } 1 " a } V ´}2 `} V 0 } 2 `} V `}2 .
By the dynamical local product structure of T 1 Ă M in the neighbourhood of v 0 γ and by the definition of v 1 , v 2 , the result follows, since the exponential map of T 1 Ă M at v 0 γ is almost isometric close to 0 and the projection to a factor of a product norm is Lipschitz. l

We now use the local product structure of the Gibbs measure to prove the following result.

Lemma 11.6. For every pt, vq P A η,γ pT q, we have dt d r m F pvq " e Oppη`e ´λγ {2 q c 2 q dt ds dµ W ´pv 0 γ q pv 1 q dµ W `pv 0 γ q pv 2 q .

Proof. By the definition of the measures (see Equations ( 4.3) and (7.9)), since the above parameter s differs, when v ´, v `are fixed, only up to a constant from the time parameter in Hopf's parametrisation, we have d r m F pvq " e C v´p x 0 , πpvqq `Cv `px 0 , πpvqq dµ x0 pv ´q dµ x0 pv `q ds dµ W ´pv 0 γ q pv 1 q " e C v1 `px 0 , πpv 1 qq

dµ x0 pv 1 `q , dµ W `pv 0 γ q pv 2 q " e C v2 ´px 0 , πpv 2 qq dµ x0 pv 2 ´q .
By Proposition 3.10 (2) since F is bounded, we have | C ξ pz, z 1 q | " Opdpz, z 1 q c 2 q for all ξ P B 8 Ă M and z, z 1 P Ă M with dpz, z 1 q bounded. Since the map π :

T 1 Ă M Ñ Ă M is Lipschitz, and since v `" v 1 `and v ´" v 2
´, the result follows from Lemma 11.5 and the cocycle property (3.7). l

When λ γ is large, the submanifold g λγ {2 Ω ´has a second order contact at v 0 γ with W ´pv 0 γ q and similarly, g ´λγ {2 Ω `has a second order contact at v 0 γ with W `pv 0 γ q. Let P γ be the plane domain of pt, sq P R 2 such that there exist s ˘P s ´η, ηr with s ¯" λγ ´t 2 ˘s `Ope ´λγ {2 q. 169 19/12/2016

Note that its area is p2η `Ope ´λγ {2 qq 2 . By the above, we have (with the obvious meaning of a double inclusion)

A η,γ pT q " P γ ˆB´p v 0 γ , r λγ e Opη`e ´λγ {2 q q ˆB`p v 0 γ , r λγ e Opη`e ´λγ {2 q q . By Lemma 11.6, we hence have ż Aη,γ pT q dt d r m F pvq " e Oppη`e ´λγ {2 q c 2 q p2η `Ope ´λγ {2 qq 2 μW ´pv 0 γ q pB ´pv 0 γ , r λγ e Opη`e ´λγ {2 q qq µ W `pv 0 γ q pB `pv 0 γ , r λγ e Opη`e ´λγ {2 q qq . (11.21)

The last ingredient of the proof of Step 4M is the following continuity property of strong stable and strong unstable ball volumes as their centre varies. See [START_REF] Roblin | Ergodicité et équidistribution en courbure négative[END_REF]Lem. 1.16], [START_REF] Paulin | Equilibrium states in negative curvature[END_REF]Prop. 10.16] for related properties, although we need a more precise control for the error term in Section 12.3.

Lemma 11.7. Assume that p Ă M , Γ, r F q has radius-continuous strong stable/unstable ball masses. There exists c 5 ą 0 such that for every ą 0, if η is small enough and λ γ large enough, then for every pt, vq P A η,γ pT q, we have µ W ´pw t q pB ´pw t , r t qq " e Op c 5 q µ W ´pv 0 γ q pB ´pv 0 γ , r λγ qq and µ W `pw t q pB `pw t , r t qq " e Op c 5 q µ W `pv 0 γ q pB `pv 0 γ , r λγ qq .

If we furthermore assume that the sectional curvature of Ă M has bounded derivatives and that p Ă M , Γ, r F q has radius-Hölder-continuous strong stable/unstable ball masses, then we may replace by pη `e´λγ{2 q c 6 for some constant c 6 ą 0.

Proof. We prove the (second) claim for W `, the (first) one for W ´follows similarly. The final statement is only used for the error estimates in Section 12.3.

v γ w ´t{2 γ {2 v 0 γ w t Opη `e´ γ {2q Ope ´ γ {2 q v γ w B`p w ´, Rq B `pv γ , R e Opη`e γ {2 q q
Using respectively Equation (2.13) since w t " g t{2 w ´and r t " e ´t{2 R, Equation (7.11) where pv, t, wq is replaced by pg t{2 v, t{2, g t{2 w ´q, and Equation (3.8), we have µ W `pw t q pB `pw t , r t qq " (11.23) Let h ´: B `pw ´, Rq Ñ W `pv γ q be the map such that ph ´pvqq ´" v ´, which is well defined and a homeomorphism onto its image if λ γ is large enough (since R is fixed). By Proposition 7.4 applied with D " HB `pw ´q and D 1 " HB `pv γ q, we have, for every v P B `pw ´, Rq, dµ W `pw ´qpvq " e ´Cv ´pπpvq, πph ´pvqqq dµ W `pv γ q ph ´pvqq .

Let us fix ą 0. The strong stable balls of radius R centred at w ´and v γ are very close (see the picture in the beginning of the proof). More precisely, recall that R is fixed, and that dpπpw ´q, πpv γ qq " Ope ´λγ {2 q and dpπpg t{2 w ´q, πpg λγ {2 v γ qq " Opη `e´λγ{2 q. Therefore we have dpπpvq, πph ´pvqqq ď for every v P B `pw ´, Rq if η is small enough and λ γ is large enough. If we furthermore assume that the sectional curvature has bounded derivatives, then by Anosov's arguments, the strong stable foliation is Hölder-continuous, see for instance [START_REF] Paulin | Equilibrium states in negative curvature[END_REF]Theo. 7.3]. Hence we have dpπpvq, πph ´pvqqq " Oppη `e´λγ{2 q c 5 q for every v P B `pw ´, Rq, for some constant c 5 ą 0, under the additional regularity asumption on the curvature. We also have h ´pB `pw ´, Rqq " B `pv γ , R e Op q q and, under the additional hypothesis on the curvature, h ´pB `pw ´, Rqq " B `pv γ , R e Oppη`e ´λγ {2 q c 5 q q.

In what follows, we assume that " pη `e´λγ{2 q c 5 under the additional assumption on the curvature. By Proposition 3.10 (2), we hence have, for every v P B `pw ´, Rq, dµ W `pw ´qpvq " e Op c 2 q dµ W `pv γ q ph ´pvqq and, using Proposition 3.10 (3),

ż πpg t{2 vq πpvq p r F ´δq ´ż πpg t{2 h ´pvqq πph ´pvqq p r F ´δq " Op c 2 q .
The result follows, by Equation (11.22) and (11.23) and the continuity property in the radius. l Now Lemma 11.7 (with as in its statement, and when its hypotheses are satisfied) implies that ij pt,vqPAη,γ pT q dt d r m F pvq µ Ẃ `pw t q pB `pw t , r t qq µ Ẁ ´pw t q pB ´pw t , r t qq " e Op c 5 q ť pt,vqPAη,γ pT q dt d r m F pvq µ Ẃ `pv 0 γ q pB `pv 0 γ , r t qq µ Ẁ ´pv 0 γ q pB ´pv 0 γ , r t qq .

By Equation (11.20) and Equation (11.21), we hence have j η, γ pT q " e Oppη`e ´λγ {2 q c 2 q e c 5 p2η `Ope ´λγ {2 qq 2 p2ηq 2

under the technical assumptions of Lemma 11.7. The assumption on radius-continuity of strong stable/unstable ball masses can be bypassed using bump functions, as explained in [START_REF] Roblin | Ergodicité et équidistribution en courbure négative[END_REF]page 81]. l 171 19/12/2016

Equidistribution of common perpendiculars in simplicial trees

In this Section, we prove a version of Theorem 11.1 for the discrete time geodesic flow on quotients of simplicial trees (and we leave to the reader the version without the assumption that the critical exponent of the system of conductances is positive). Let X be a locally finite simplicial tree without terminal vertices. Let Γ be a nonelementary discrete subgroup of AutpXq. Let r c : EX Ñ R be a Γ-invariant system of conductances on X, let r

F c be its associated potential (see Section 3.5), and let m c " m Fc . Let D ´" pD í q iPI ánd D `" pD j q jPI `be locally finite Γ-equivariant families of nonempty proper simplicial subtrees of X.

For every edge path α " pe 1 , . . . , e n q in X, we set cpαq "

n ÿ i"1
r cpe i q .

Theorem 11.8. Assume that the critical exponent δ c of r c is positive and that the Gibbs measure m c is finite and mixing for the discrete time geodesic flow on ΓzG X. Then

lim tÑ`8 e δc ´1 e δc }m c } e ´δc t ÿ iPI ´{", jPI `{", γPΓ D í XD γj "H, λ i, γj ďt e cpα i,γj q ∆ α í, γj b ∆ α γ´1 i, j " r σ D´b r σ Df
or the weak-star convergence of measures on the locally compact space p G X ˆp G X.

Proof. The proof is a modification of the continuous time proof for metric trees in Sections 11.1 and 11.2. Here, we indicate the changes to adapt the proof to the discrete time. We use the conventions for the discrete time geodesic flow described in Section 2.7.

Note that for all i P I ´, j P I `, γ P Γ, the common perpendicular α i,γj is now an edge path from D í to D γj , and that by Proposition 3.11, we have ş α i,γj r F c " cpα i,γj q. In the definition of the bump functions in Section 10.1, we assume (as we may) that η ă 1, so that for all η 1 P s0, 1r and w P B 1 ¯D˘s uch that w ¯P ΛΓ, we have

V w,η,η 1 " B ˘pw, η 1 q ,
see Equation (2.12) and recall that we are only considering discrete geodesic lines. As p0q " wp0q for every P B ˘pw, ηq since η ă 1, and as the time is now discrete, Equations (10.1) and (7.12) give

h η, η 1 pwq " 1 µ W ¯pwq pB ˘pw, η 1 qq . (11.24)
This is a considerable simplification compared with the inequalities of Equation ( 10.3).

In the whole proof, we restrict to t " n P N, T " N P N. In Steps 1 and 2, we define instead of Equation (11.3) φ ή pg ´tn{2u q φ ὴ pg rn{2s γ ´1 q d r m c p q .

I Ω ´, Ω `pN q " pe δc ´1q }m c } e ´δcpN `1q
Equation (11.6) is replaced by

i η pN q " N ÿ n"0
e δc n a η pnq , so that by a geometric sum argument, the pair of inequalities (11.7) becomes

e ´ e δc pN `1q r σ `pΩ ´q r σ ´pΩ `q pe δc ´1q }m c } ´c ď i η pN q ď e e δc pN `1q r σ `pΩ ´q r σ ´pΩ `q pe δc ´1q }m c } `c .

Step 3 is unchanged up to replacing ş T 0 by ř N n"0 , r F by r F c , δ by δ c and t{2 by either tn{2u or rn{2s, so that Equation (11.10) becomes, since tn{2u `rn{2s " n,

h ή, R ˝f D´p q h ὴ, R ˝f D`p γ ´1 q " e ´δc n e ş w ´ptn{2uq w ´p0q r Fc`ş w `p0q w `p´rn{2sq
r Fc h ή, e ´tn{2u R pg tn{2u w ´q h ὴ, e ´rn{2s R pg ´rn{2s w `q .

The proof then follows similarly as in Section 11.2, with the simplifications in the point (iii) that, taking η ă 1{2, we have λ γ equal to t " n, and the points w ´pt n 2 uq, w `p´r n 2 sq and p0q are equal. In particular, Equation (11.11) Fc j η, γ pN q ď c 4 .

The statement of Step 4T now simplifies as j η,γ pN q " 1 , if η ă 1 2 , and if γ P Γ is such that D ´and γD `do not intersect and λ γ is large enough. We introduce in its proof the slightly modified notation

r ń " e ´t n 2 u R, r ǹ " e ´r n 2 s R, w ń " g t n 2 u w ´and w ǹ " g ´r n 2 s w `.
and we now take as x γ the point at distance t n 2 u from its origin on the common perpendicular α γ . Equation (11.13) becomes (using Equation (11.24) instead of Equation ( 10.3)) j η, γ pN q " ij pn, qPAη,γ pN q dn d r m c p q µ W `pw ń q pB `pw ń , r ń qq µ W ´pw ǹ q pB ´pw ǹ , r ǹ qq .

173 19/12/2016 Since p0q " x γ if pn, q P A η, γ pN q, Equation (11.14) simplifies as d r m c p q " dµ xγ p ´q dµ xγ p `q ds , with ds the counting measure on the Hopf parameter s P Z of . Replacing P γ with its intersection with Z 2 reduces it to one point pλ γ , 0q, and now s " s ˘" 0. Lemma 11.4 becomes pB `pw ń , r ń qq ´" B dx γ pξ γ , R e ´t λγ 2 u q, pB ´pw ǹ , r ǹ qq `" B dx γ pξ γ , R e ´r λγ 2 s q , so that A η,γ pN q " tpλ γ , 0qu ˆBdx γ pξ γ , R e ´t λγ 2 u q ˆBdx γ pξ γ , R e ´r λγ 2 s q . Finally, since p0q " x γ if pn, q P A η, γ pN q, Equation (11.15) becomes µ W `pw ń q pB `pw ń , r ń qq " µ xγ pB dx γ pξ γ , R e ´t λγ 2 u qq, µ W ´pw t q pB ˘pw ǹ , r n qq " µ xγ pB dx γ pξ γ , R e ´r λγ 2 s qq .

The last centred equation in

Step 4T now reduces to j η, γ pT q " 1. l

For lattices in regular trees, we get more explicit expressions.

Corollary 11.9. Let X be a pq `1q-regular simplicial tree (with q ě 2) and let Γ be a lattice of X such that ΓzX is not bipartite. Assume that the Patterson density is normalised to be a family of probability measures. Let D ˘be nonempty proper simplicial subtrees of X with stabilisers Γ D˘i n Γ, such that D ˘" pγD ˘qγPΓ{Γ D ˘is locally finite. Then

lim tÑ`8 q ´1 q `1 VolpΓz zXq q ´t ÿ pα, β, γqPΓ{Γ D ´ˆΓ{Γ D `ˆΓ 0ădpαD ´, γβD `qďt ∆ α ά, γβ b ∆ α γ´1 α, β " r σ D´b r σ D`,
for the weak-star convergence of measures on the locally compact space p G X ˆp G X. If the measure σ D`i s nonzero and finite, then

lim tÑ`8 q ´1 q `1 VolpΓz zXq }σ D`} q ´t ÿ γPΓ{Γ D `, 0ădpD ´, γD `qďt ∆ α é, γ " r σ D´,
for the weak-star convergence of measures on the locally compact space p G X.

Proof. In order to prove the first claim, we apply Theorem 11.8 with r c " 0, so that by Propositions 4.14, 4.15, and 8.1 (3), we have δ c " ln q ą 0, m c " m BM is finite and mixing, and }m BM } " q q`1 VolpΓz zXq. The second claim follows by restricting to α " β " e and integrating on an appropriate fundamental domain (note that Equation 11.4 does not require Ω `to be relatively compact, just to have finite measure for r σ ´). l

The mixing assumption in Theorem 11.8 implies that the length spectrum L Γ of Γ is equal to Z. 5 The next result considers the other case, when only the square of the geodesic flow is mixing, when appropriately restricted. Note that the smallest nonempty Γ-invariant simplicial subtree of X is uniform, without vertices of degree 2, for instance in the case when X is pp `1, q `1q-biregular with p, q ě 2 and Γ is a lattice of X. 19/12/2016 Theorem 11.10. Assume that the smallest nonempty Γ-invariant simplicial subtree of X is uniform, without vertices of degree 2, and that the length spectrum L Γ of Γ is 2Z. Assume that the critical exponent δ c of r c is positive, that the Gibbs measure m c is finite and that its restriction to ΓzG even X is mixing for the square of the discrete time geodesic flow on ΓzG even X. Then

lim tÑ`8 e 2 δc ´1 2 e 2 δc }m c } e ´δc t ÿ iPI ´{", jPI `{", γPΓ D í XD γj "H, λ i, γj ďt e cpα i,γj q ∆ α í, γj b ∆ α γ´1 i, j " r σ D´b r σ Df
or the weak-star convergence of measures on the locally compact space p G X ˆp G X.

Proof. We denote by r σ D¯, even the restriction of r σ D¯t o p G even X, and by r σ D¯, odd the restriction

of r σ D¯t o p G odd X " p G X ´p G even X.
We denote by V even X the subset of V X consisting of the vertices at even distance from x 0 , and by V odd X " V X ´Veven X its complement. The subsets V even X and V odd X are Γ-invariant if L Γ " 2Z by Equation (4.13).

Let us first prove that lim tÑ`8 e 2 δc ´1 2 e 2 δc }m c } e ´δc t ÿ iPI ´{", jPI `{", γPΓ πpα í,γj q, πpα γ´1 i,j q P VevenX D í XD γj "H, λ i, γj ďt 11.25) for the weak-star convergence of measures on the locally compact space p G even X ˆp G even X.

e cpα i,γj q ∆ α í, γj b ∆ α γ´1 i, j " r σ D´, even b r σ D`, even (
The proof of this Equation (11.25) is a modification of the proof of the previous Theorem 11.8. We now restrict to t " 2n P N, T " 2N P N, and we replace r m c by p r m c q |GevenX and pg t q tPZ by pg 2t q tPZ . Note that since r m c is invariant under the time 1 of the geodesic flow, which maps ΓzG even X to ΓzG X ´ΓzG even X, we have (11.26) Note that for all i P I ´, j P I `and γ P Γ, if πpα í,γj q and πpα γ´1 i,j q belong to V even X, then the distance between D í and γD j is even (since for all x, y, z in a simplicial tree, if p is the closest point from x to ry, zs, then dpy, zq " dpy, xq `dpx, zq ´2 dpx, pq ).

}p r m c q |GevenX } " 1 2 } r m c } .
In Steps 1 and 2, we now consider Ω ˘two Borel subsets of B 1 ¯D˘X p G even X, and we define instead of Equation (11.3)

I Ω ´, Ω `p2N q " pe 2 δc ´1q }m c } 2 e ´2 δcpN `1q ÿ γPΓ : 0ăλγ ď2N, πpα γ q, πpα γ q P VevenX α γ | s0,λγ s PΩ ´|s0,λγ s , α γ | s´λγ ,0s PΩ `|s´λγ ,0s
a η p2nq " ÿ γPΓ ż PGevenX φ ή pg ´2tn{2u q φ ὴ pg 2rn{2s γ ´1 q d r m c p q .
Equation (11.6) is replaced by

i η p2N q " N ÿ n"0
e δc 2n a η p2nq .

The mixing property of the square of the geodesic flow on ΓzG even X for the restriction of the Gibbs measure m c gives that, for every ą 0, there exists T " T ,η ě 0 such that for all n ě T , we have

e ´ ş GevenX φ ή d r m c ş GevenX φ ὴ d r m c }p r m c q |GevenX } ď a η p2nq ď e ş GevenX φ ή d r m c ş GevenX φ ὴ d r m c }p r m c q |GevenX } .
Note that G even X is saturated by the strong stable and strong unstable leaves, since two points x, y on a given horosphere of centre ξ P B 8 X are at even distance one from another (equal to 2dpx, pq where rx, ξr X ry, ξr " rp, ξr ). By the disintegration proposition 7.6, when ranges over U D X G even X, we have

d r m F |U D X GevenX p q " ż ρPB 1 ˘DX p Geven X
dν ρ p q dr σ Dpρq .

Hence the proof of Lemma 10.1 extends to give (11.27) where in order to simplify notation r σ ȇven " r σ D¯, even . Therefore, by Equations (11.26) and (11.27), and by a geometric sum argument, the pair of inequalities (11.7) becomes 2 e ´ e 2δc pN `1q r σ èven pΩ ´q r σ éven pΩ `q pe 2δc ´1q }m c } ´c ď i η p2N q ď 2 e e 2δc pN `1q r σ èven pΩ ´q r σ éven pΩ `q pe 2δc ´1q }m c } `c .

ż GevenX φ η d r m c " r σ ȇven pΩ ¯q ,
Up to replacing the summations from n " 0 to N to summations on even numbers between 0 to 2N , and replacing tn{2u by 2tn{2u as well as rn{2s by 2tn{2u, the rest of the proof applies and gives the result, noting that in claim (iii) of Step 3T, we furthermore have that the origin and endpoint of the constructed common perpendicular α γ are in V even X. This concludes the proof of Equation (11.25) e 2 δc ´1 2 e 2 δc }m c } e ´δc t ÿ iPI ´{", jPI `{", γPΓ πpα í,γj q, πpα γ´1 i,j q P V odd X 11.28) for the weak-star convergence of measures on the locally compact space p G odd X ˆp G odd X.

D í XD γj "H, λ i, γj ďt e cpα i,γj q ∆ α í, γj b ∆ α γ´1 i, j " r σ D´, odd b r σ D`, odd (
Let us now apply Equation (11.25) by replacing D ´" pD í q iPI ´by N 1 D ´" pN 1 D í q iPI

´.

Let us consider the map ϕ `: p G X Ñ p G X, which maps a generalised geodesic line to the generalised geodesic line which coincides with g `1 on r0, `8r and is constant (with value p1q) on s ´8, 0r. Note that this map is continuous and Γ-equivariant, and that it maps

p G even X in p G odd X and p G odd X in p G even X. Furthermore, by convexity, ϕ `induces for every i P I ´an homeomorphism from B 1 `Dí to B 1 `N1 D í , which sends B 1 `Dí X p G odd X to B 1 `N1 D í X p G even X, such that, by Equation (7.8),
for all w P B 1 `Dí X p G odd X, if e w is the first (respectively the last) edge followed by w dr σ Dí , odd pwq " e cpewq´δc dr σ Ǹ1 D í , even pϕ `pwqq .

Note that for all ą 0, there is a one-to-one correspondance between the set of common perpendiculars of length , with origin and endpoint both in V even , between N 1 D í and γD j for all i P I ´, j P I `and γ P Γ, and the set of common perpendiculars of length `1, with origin in V odd and endpoint in V even , between D í and γD j for all i P I ´, j P I `and γ P Γ. In particular, ϕ `pα í,γj q is the common perpendicular between N 1 D í and γD j , starting at time t " 0 from N 1 D í .

Therefore Equation (11.25) applied by replacing

D ´" pD í q iPI ´by N 1 D ´" pN 1 D í q iPI ǵives lim tÑ`8
e 2 δc ´1 2 e 2 δc }m c } e ´δc t ÿ iPI ´{", jPI `{", γPΓ πpα í,γj q P V odd X, πpα γ´1 i,j q P VevenX D í XD γj "H, λ i, γj ďt`1 e cpe α í, γj q`cpϕ `pα í,γj qq

∆ α í, γj b ∆ α γ´1 i, j " e δc r σ D´, odd b r σ D`, even
for the weak-star convergence of measures on the locally compact space p G odd X ˆp G even X.

19/12/2016

Since cpe α í, γj q `cpϕ `pα í,γj qq " cpα í,γj q, replacing t by t ´1 and simplifying by e δc , we get lim tÑ`8 e 2 δc ´1 2 e 2 δc }m c } e ´δc t ÿ iPI ´{", jPI `{", γPΓ πpα í,γj q P V odd X, πpα γ´1 i,j q P VevenX D í XD γj "H, λ i, γj ďt e cpα í,γj q ∆ α í, γj b ∆ α γ´1 i, j

" r σ D´, odd b r σ D`, even (11.29)

for the weak-star convergence of measures on the locally compact space p G odd X ˆp G even X. Now Theorem 11.10 follows by summing Equation (11.25), Equation (11.28), Equation (11.29) and the formula, proven similarly, obtained from Equation (11.25) by replacing D `" pD j q jPI `by pN 1 D j q jPI `. l

The following result for bipartite graphs (of groups) is used in the arithmetic applications in Part III (see Section 15.4).

Corollary 11.11. Let X be a pp`1, q `1q-biregular simplicial tree (with p, q ě 2, possibly with p " q), with corresponding partition V X " V p X \ V q X. Let Γ be a lattice of X such that this partition is Γ-invariant. Assume that the Patterson density is normalised so that }µ x } " p`1 ? p for every x P V p X. Let D ˘be nonempty proper simplicial subtrees of X with stabilisers Γ Dȋ n Γ, such that the families D ˘" pγD ˘qγPΓ{Γ D ˘are locally finite. Then ´, γD `qďt

∆ α é, γ " r σ D´,
for the weak-star convergence of measures on the locally compact space p G X.

Proof. In order to prove the first result, we apply Theorem 11.10 with r c " 0, so that by Propositions 4.14, 4.15, and 8.1 (2), we have δ c " 1 2 lnppqq ą 0, m c " m BM is finite and its restriction to ΓzG even X is mixing under the square of the geodesic flow, and }m BM } " TVolpΓz zXq.

The second claim follows as in the proof of Corollary 11.9. l

Remark. In some special occasions, the measures involved in the statements of Theorem 11.10 and Corollary 11.11 (whether skinning measures or Dirac masses) are actually all supported on p G even X (up to choosing appropriately x 0 ). This is in particular the case if X " rX| 1 is a simplicial tree and if D ˘" pγD ˘qγPΓ{Γ D ˘with D ´, D `at even signed distance (see below), as the following proposition shows.

178 19/12/2016 `H H 1 H H 1
The signed distance between horoballs H and H 1 in an R-tree that are not centred at the same point at infinity is the distance between them (that is, the length of their common perpendicular) if they are disjoint, or the opposite of the diameter of their intersection otherwise. Note that if nonempty, the intersection of H and H 1 is a ball centred at the midpoint of the segment contained in the geodesic line between the two points at infinity of the horoballs, which lies in both horoballs.

Lemma 11.12. Let X be a simplicial tree, Γ a subgroup of AutpXq and H , H 1 two horoballs in X (whose boundaries are contained in V X), which either are equal or have distinct points at infinity. If ΛΓ Ă 2Z and H , H 1 are at even signed distance, then the signed distance between H and γH 1 is even for every γ P Γ such that H and γH 1 do not have the same point at infinity.

Proof. For every horoball H 2 and for all s P N, let H 2 rss be the horoball contained in H , whose boundary is at distance s from the boundary of H . Shrinking the horoballs H and H 1 , by replacing them by the horoballs H rss and H 1 rss for any s P N, only changes by ˘2s the considered signed distances. Hence, taking s big enough, we may assume that H and γH 1 are disjoint, and that H and H 1 are disjoint or equal. Let rx, x 1 s be the common perpendicular between H and H 1 with x P BH , x 1 P BH 1 , and let ry, y 1 s be the one between H and γH 1 , with y P BH , y 1 P BpγH 1 q. Note that γx 1 P BpγH 1 q.

x H

y 1 y γx 1 γH 1
The distance between two points x, y of a horosphere is always even (equal to twice the distance from x to the geodesic ray from y to the point at infinity of the horosphere). Since geodesic triangles in trees are tripod, for all a, b, c in a simplicial tree, since dpa, cq " dpa, bq `dpb, cq ´2dpb, ra, csq , if dpa, bq and dpb, cq are even, so is dpa, cq.

Since ΛΓ Ă 2Z, the distance between x 1 and γx 1 is even by Equation (4.13). Since dpx, x 1 q is even by assumption, we hence have that dpx, γx 1 q is even. Therefore dpy, y 1 q " dpx, γx 1 q ´dpx, yq ´dpy 1 , γx 1 q is even. l 179 19/12/2016

Chapter 12

Equidistribution and counting of common perpendiculars in quotient spaces

In this Chapter, we use the results of Chapter 11 to prove equidistribution and counting results in Riemannian manifolds (or good orbifolds) and in metric and simplicial graphs (of groups). Let X, x 0 , Γ and r F be as in the beginning of Chapter 11.

Multiplicities and counting functions in Riemannian orbifolds

In this Section, we assume that X " Ă M is a Riemannian manifold. We denote its quotient Riemannian orbifold under Γ by M " Γz Ă M , and the quotient Riemannian orbifold under Γ of its unit tangent bundle by T 1 M " ΓzT 1 Ă M . We use the identifications G X " G ˘, 0 X " T 1 X " T 1 Ă M explained in Chapter 2. Let D " pD i q iPI be a locally finite Γ-equivariant family of nonempty proper closed convex subsets of Ă M . Let Ω " pΩ i q iPI be a Γ-equivariant family of subsets of T 1 Ă M , where Ω i is a measurable subset of B 1 ˘Di for all i P I (the sign ˘being constant). The multiplicity of an element v P T 1 M with respect to Ω is

m Ω pvq " Card ti P I{ " : r v P Ω i u CardpStab Γ r vq , for any preimage r v of v in T 1 Ă M .
The numerator and the denominator are finite by the local finiteness of the family D and the discreteness of Γ, and they depend only on the orbit of r v under Γ.

The numerator takes into account the multiplicities of the images of the elements of Ω in T 1 M . The denominator of this multiplicity is also natural, as any counting problem of objects possibly having symmetries, the appropriate counting function consists in taking as the multiplicity of an object the inverse of the cardinality of its symmetry group.

Examples 12.1. The following examples illustrate the behaviour of the multiplicity when Γ is torsion-free and Ω " B 1 ˘D . 181 19/12/2016

(1) If for every i P I, the quotient Γ D i zD i of D i by its stabiliser Γ D i maps injectively in M by the map induced by the inclusion of D i in Ă M , and if for every i, j P I such that j R Γi, the intersection D i X D j is empty, then the nonzero multiplicities m Ω p q are all equal to 1.

(2) Here is a simple example of a multiplicity different from 0 or 1. Let c be a closed geodesic in the Riemannian manifold M , let r c be a geodesic line in Ă M mapping to c in M , let D " pγ r cq γPΓ , let x be a double point of c, let v P T 1

x M be orthogonal to the two tangent lines to c at x (this requires the dimension of Ă

M to be at least 3, if x is a transverse self-intersection point). Then m B 1 ˘D pvq " 2.

x v c

Given t ą 0 and two unit tangent vectors v, w P T 1 M , we define the number n t pv, wq of locally geodesic paths having v and w as initial and terminal tangent vectors respectively, weighted by the potential F , with length at most t, by n t pv, wq "

ÿ α CardpΓ α q e ş α F ,
where the sum ranges over the locally geodesic paths α : r0, ss Ñ M in the Riemannian orbifold M such that 9

αp0q " v, 9 αpsq " w and s P s0, ts, and Γ α is the stabiliser in Γ of any geodesic path r α in Ă M mapping to α by the quotient map Ă M Ñ M . If F " 0 and Γ is torsion free, then n t pv, wq is precisely the number of locally geodesic paths having v and w as initial and terminal tangent vectors respectively, with length at most t.

Let Ω ´" pΩ í q iPI ´and Ω `" pΩ j q jPI `be Γ-equivariant families of subsets of T 1 Ă M , where Ω k is a measurable subset of B 1 ˘Dk for all k P I ¯. We will denote by N Ω ´, Ω `, F : s0, `8r Ñ R the following counting function: for every t ą 0, let N Ω ´, Ω `, F ptq be the number of common perpendiculars whose initial vectors belong to the images in T 1 M of the elements of Ω ´and terminal vectors to the images in T 1 M of the elements of Ω `, counted with multiplicities and weighted by the potential F , that is:

N Ω ´, Ω `, F ptq " ÿ v, wPT 1 M
m Ω ´pvq m Ω `pwq n t pv, wq .

When Ω ˘" B 1 ¯D ˘, we denote N Ω ´, Ω `, F by N D ´, D `, F . Remark 12.2. Let Y be a negatively curved complete connected Riemannian manifold and let r Y Ñ Y be its Riemannian universal cover. Let D ˘be a locally convex 1 geodesic metric space endowed with a continuous map f ˘: Let us continue fixing the notation used in Sections 12.2 and 12.3. For every pi, jq in I ´ˆI `such that D í and D j have a common perpendicular 2 , we denote by α i, j this common perpendicular, by λ i, j its length, by v í, j P B 1 `Dí its initial tangent vector and by v ì, j P B 1 ´Dì its terminal tangent vector. Note that if i 1 " i, j 1 " j and γ P Γ, then γ α i 1 , j 1 " α γi, γj , λ i 1 , j 1 " λ γi, γj and γ v ȋ1 , j 1 " v γi, γj .

D ˘Ñ Y such that if r D ˘Ñ D ˘is a locally isometric universal cover and if r f ˘: r D ˘Ñ r Y is a lift of f ˘, then
(12.1)

When Γ has no torsion, we have, for the diagonal action of Γ on I ´ˆI `,

N D ´, D `, F ptq " ÿ pi, jqPΓzppI ´{"qˆpI `{"qq : D í XD j "H, λ i, j ďt e ş α i, j r F .
When the potential F is zero and Γ acts without torsion, N D ´, D `, F ptq is the number of common perpendiculars of length at most t, and the counting function t Þ Ñ N D ´, D `, 0 ptq has been studied in various special cases of negatively curved manifolds since the 1950's and in a number of recent works, see the Introduction. The asymptotics of N D ´, D `, 0 ptq as t Ñ `8 in the case when X is a Riemannian manifold with pinched negative curvature are described in general in [START_REF] Parkkonen | Counting common perpendicular arcs in negative curvature[END_REF]Thm. 1], where it is shown that if the skinning measures σ D´a nd σ Dà re finite and nonzero, then as s Ñ `8,

N D ´, D `, 0 psq " }σ D´} }σ D`} }m BM } e δ Γ s δ Γ . (12.2)

Common perpendiculars in Riemannian orbifolds

Corollary 12.3 below is the main result of this text on the counting with weights of common perpendiculars and on the equidistribution of their initial and terminal tangent vectors in negatively curved Riemannian manifolds endowed with a Hölder potential. We use the notation of Section 12.1.

The following observation on the behaviour of induced 3 measures under quotients by properly discontinuous group actions will be used in the proof of the following result and also those of its analogues in Section 12.4. Let G be a discrete group that acts properly on a Polish space r Y and let Y " Gz r Y . Let r µ k for k P N and r µ be G-invariant locally finite measures on r Y , with finite induced measures µ k for k P N and µ on Y . If for every Borel subset B of r Y with r µpBq finite and r µpBBq " 0 we have lim kÑ8 r µ k pBq " r µpBq, then the sequence pµ k q kPN narrowly converges to µ.

Corollary 12.3. Let Ă M be a complete simply connected Riemannian manifold with pinched negative sectional curvature at most ´1. Let Γ be a nonelementary discrete group of isometries of Ă M . Let r F : T 1 Ă M Ñ R be a bounded Γ-invariant Hölder-continuous function with positive critical exponent δ. Let D ´" pD í q iPI ´and D `" pD j q jPI `be locally finite Γ-equivariant families of nonempty proper closed convex subsets of Ă M . Assume that the Gibbs measure m F is finite and mixing for the geodesic flow on T 1 M . Then,

lim tÑ`8 δ }m F } e ´δ t ÿ v, wPT 1 M m B 1 `D ´pvq m B 1 ´D `pwq n t pv, wq ∆ v b ∆ w " σ D´b σ D`( 12.3)
2 that is, whose closures D í and D j in X Y B8X have empty intersection 3 see for instance [START_REF] Paulin | Equilibrium states in negative curvature[END_REF]§2.6] for a definition 183 19/12/2016 for the weak-star convergence of measures on the locally compact space T 1 M ˆT 1 M . If σ Dá nd σ D`a re finite, the result also holds for the narrow convergence.

Furthermore, for all Γ-equivariant families Ω ˘" pΩ k q kPI ˘of subsets of T 1 Ă M with Ω k a Borel subset of B 1 ˘Dk for all k P I ¯, with nonzero finite skinning measure and with boundary in B 1 ˘Dk of zero skinning measure, we have, as t Ñ `8,

N Ω ´, Ω `, F ptq " }σ Ὼ´} }σ Ώ`} δ }m F } e δ t .
Proof. Note that the sum in Equation ( 12.3) is locally finite, hence it defines a locally finite measure on T 1 M ˆT 1 M . We are going to rewrite the sum in the statement of Theorem 11.1 in a way which makes it easier to push it down from

T 1 Ă M ˆT 1 Ă M to T 1 M ˆT 1 M . For every r v P T 1 Ă M , let
m ¯pr vq " Card tk P I ¯{" : r v P B 1 ˘Dk u , so that for every v P T 1 M , the multiplicity of v with respect to the family B 1

˘D ¯is 4 m B 1 ˘D ¯pvq " m ¯pr vq CardpStab Γ r vq ,
for any preimage r v of v in T 1 Ă M . For all γ P Γ and r v, r w P T 1 Ă M , there exists pi, jq P pI ´{" q ˆpI `{" q such that r v " v í,γj and r

w " v γ´1 i,j " γ ´1v ì,γj if and only if γ r w P g R r v, there exists i 1 P I ´{" such that r v P B 1 `Dí 1 and there exists j 1 P I `{" such that γ r w P B 1 ´Dj 1 . Then the choice of such elements pi, jq, as well as i 1 and j 1 , is free. We hence have

ÿ iPI ´{", jPI `{", γPΓ 0ăλ i, γj ďt , v í, γj "r v , v γ´1 i, j " r w e ş α i, γj r F ∆ v í, γj b ∆ v γ´1 i, j " ÿ γPΓ, 0ăsďt γ r w"g s r v e ş γπp r wq πpr vq r F Card pi, jq P pI ´{" q ˆpI `{" q : v í, γj " r v , v γ´1 i, j " r w ( ∆ r v b ∆ r w " ÿ γPΓ, 0ăsďt γ r w"g s r v e ş γπp r wq πpr vq r F m ´pr vq m `pγ r wq ∆ r v b ∆ r w .
Therefore By definition, σ D¯i s the measure on T 1 M induced by the Γ-invariant measure r σ D¯. Thus Corollary 12.3 follows from Theorem 11.1 and Equation (11.4) after a similar reduction 5 as in Section 11.1, and since no compactness assumptions were made on Ω ˘to get this equation, by [START_REF] Paulin | Equilibrium states in negative curvature[END_REF]§2.6]. l

ÿ iPI ´{", jPI `{", γPΓ 0ăλ i, γj ďt e ş α i, γj r F ∆ v í, γj b ∆ v γ´1 i, j " ÿ r v, r w P T 1 Ă M ´ÿ γPΓ, 0ăsďt γ r w"g s r v
In particular, if the skinning measures σ D´a nd σ D`a re positive and finite, Corollary 12.3 gives, as t Ñ `8,

N D ´, D `, F ptq " }σ D´} }σ D`} δ }m F } e δ t .
Remark 12.4. Under the assumptions of Corollary 12.3 with the exception that δ may now be nonpositive, we have the following asymptotic result as t Ñ `8 for the growth of the weighted number of common perpendiculars with lengths in st ´τ, ts for every fixed τ ą 0:

N D ´, D `, F ptq ´ND ´, D `, F pt ´τ q " p1 ´e´δ τ q }σ D´} }σ D`} δ }m F } e δ t .
This result follows by considering a large enough constant σ such that δ Γ, F `σ " δ `σ ą 0, by applying Corollary 12.3 with the potential F `σ (see Remark 7.1 (2)), and by an easy subdivision and geometric series argument, see [START_REF] Paulin | Equilibrium states in negative curvature[END_REF]Ch. 9].

Using the continuity of the pushforwards of measures for the weak-star and the narrow topologies, applied to the basepoint maps π ˆπ from

T 1 Ă M ˆT 1 Ă M to Ă M ˆĂ M
, and from T 1 M ˆT 1 M to M ˆM , we have the following result of equidistribution of the ordered pairs of endpoints of common perpendiculars between two equivariant families of convex sets in Ă M or two families of locally convex sets in M . When M has constant curvature and finite volume, F " 0 and D ´is the Γ-orbit of a point and D `is the Γ-orbit of a totally geodesic cocompact submanifold, this result is due to Herrmann [Her]. When D ˘are Γ-orbits of points and F is a Hölder potential, see [START_REF] Paulin | Equilibrium states in negative curvature[END_REF]Thm. 9.1,9.3], and we refer for instance to [BoyM] for an application of this particular case.

Corollary 12.5. Let Ă M , Γ, r F , D ´, D `be as in Corollary 12.3. Then

lim tÑ`8 δ }m F } e ´δ t ÿ iPI ´{", jPI `{", γPΓ 0ăλ i, γj ďt e ş α i, γj r F ∆ πpv í, γj q b ∆ πpv γ´1 i, j q " π ˚r σ D´b π ˚r σ D`,
for the weak-star convergence of measures on the locally compact space Ă M ˆĂ M , and

lim tÑ`8 δ }m F } e ´δ t ÿ v, wPT 1 M m B 1 `D ´pvq m B 1 ´D `pwq n t pv, wq ∆ πpvq b ∆ πpwq " π ˚σD ´b π ˚σD `,
for the weak-star convergence of measures on M ˆM . If the measures σ D¯a re finite, then the above claim holds for the narrow convergence of measures on M ˆM . l 5 See Step 1 of the proof of Theorem 11.1.

19/12/2016

We will now prove Theorems 1.4 and 1.5 (1) in the Introduction for Riemannian manifolds. Recall from Remark 12.2 the definition of proper nonempty properly immersed closed locally convex subsets D ˘in a pinched negatively curved complete connected Riemannian manifold Y and the associated maps r f ˘: r D ˘Ñ r Y .

Proof of Theorems 1.4 and 1.5 (1) for Riemannian manifolds. Let Y, F, D ˘be as in these statements and assume that Y is a Riemannian manifold. Let Γ be the covering group of the universal Riemannian cover r Y Ñ Y . Let I ˘" Γ ˆπ0 p r D ˘q with the action of Γ defined by γ ¨pα, cq " pγα, cq for all γ, α P Γ and every component c of r D ˘. Consider the families D ˘" pD k q kPI ˘where D k " α r f ˘pcq if k " pα, cq. Then D ˘are Γ-equivariant families of nonempty proper closed convex subsets of r Y , which are locally finite since D ˘are properly immersed in Y . The conclusions in Theorems 1.4 and 1.5 (1) when Y is a manifold then follow from Corollary 12.3, applied with Ă M " r Y and with r F the lift of

F to T 1 Ă M . l Corollary 12.6. Let Ă M , Γ, r F , D
´, D `be as in Corollary 12.3. Assume that σ D¯a re finite and nonzero. Then

lim sÑ`8 lim tÑ`8 δ }m F } 2 e ´δt }σ D´} }σ D`} ÿ vPT 1 M m B 1 `D ´pvq n t, D `pvq ∆ g s v " m F , where n t,D `pvq " ÿ wPT 1 M m B 1 ´D `pwq n t pv, wq
is the number (counted with multiplicities) of locally geodesic paths in M of length at most t, with initial vector v, arriving perpendicularly to D `.

Proof. For every s P R, by Corollary 12.3, using the continuity of the pushforwards of measures by the first projection pv, wq Þ Ñ v from T 1 M ˆT 1 M to T 1 M , and by the geodesic flow on T 1 M at time s, since pg s q ˚∆v " ∆ g s v , we have

lim tÑ`8 δ }m F } e ´δt ÿ vPT 1 M m B 1 `D ´pvq n t,D `pvq ∆ g s v " pg s q ˚σD ´}σ D`} .
The result then follows from Theorem 10.2 with Ω " B 1 `D ´. l 12.3 Error terms for equidistribution and counting for Riemannian orbifolds

In Section 9.1, we discussed various results on the rate of mixing of the geodesic flow for Riemannian manifolds. In this Section, we apply these results to give error bounds to the statements of equidistribution and counting of common perpendicular arcs given in Section 12.2. We use again the notation of Section 12.1.

Theorem 12.7. Let Ă M be a complete simply connected Riemannian manifold with pinched negative sectional curvature at most ´1. Let Γ be a nonelementary discrete group of isometries of Ă M . Let r F : T 1 Ă M Ñ R be a bounded Γ-invariant Hölder-continuous function with positive critical exponent δ. Assume that p Ă M , Γ, r F q has radius-Hölder-continuous strong stable/unstable ball masses. Let D ´" pD í q iPI ´and D `" pD j q jPI `be locally finite Γequivariant families of nonempty proper closed convex subsets of Ă M , with finite nonzero skinning measure σ D ´and σ D `. Let M " Γz Ă M and let F : T 1 M Ñ R be the potential induced by r F .

186 19/12/2016

(1) Assume that M is compact and that the geodesic flow on T 1 M is mixing with exponential speed for the Hölder regularity for the potential F . Then there exist α P s0, 1s and κ 1 ą 0 such that for all nonnegative ψ ˘P C α c pT 1 M q, we have, as t Ñ `8,

δ }m F } e δ t ÿ v, wPT 1 M m B 1 `D ´pvq m B 1 ´D `pwq n t pv, wq ψ ´pvq ψ `pwq " ż T 1 M ψ ´dσ D´ż T 1 M ψ `dσ D``O pe ´κ1 t }ψ ´}α }ψ `}α q .
(2) Assume that Ă M is a symmetric space, that D k has smooth boundary for every

k P I ˘,
that m F is finite and smooth, and that the geodesic flow on T 1 M is mixing with exponential speed for the Sobolev regularity for the potential F . Then there exist P N and κ 1 ą 0 such that for all nonnegative maps ψ ˘P C c pT 1 M q, we have, as t Ñ `8,

δ }m F } e δ t ÿ v, wPT 1 M m B 1 `D ´pvq m B 1 ´D `pwq n t pv, wq ψ ´pvq ψ `pwq " ż T 1 M ψ ´dσ D´ż T 1 M ψ `dσ D``O pe ´κ1 t }ψ ´} }ψ `} q .
Furthermore, if D ´and D `respectively have nonzero finite outer and inner skinning measures, and if p Ă M , Γ, r F q satisfies the conditions of (1) or (2) above, then there exists κ 2 ą 0 such that, as t Ñ `8,

N D ´, D `, F ptq " }σ D´} }σ D`} δ }m F } e δ t `1 `Ope ´κ2 t q ˘.
The maps Op¨q depend on Ă M , Γ, F, D, and the speeds of mixing. The proof is a generalization to nonzero potential of [START_REF] Parkkonen | Counting common perpendicular arcs in negative curvature[END_REF]Thm. 15].

Proof. We will follow the proofs of Theorem 11.1 and Corollary 12.3 to prove generalizations of the assertions (1) and ( 2) by adding to these proofs a regularisation process of the test functions r φ η as for the deduction of [START_REF] Parkkonen | Skinning measure in negative curvature and equidistribution of equidistant submanifolds[END_REF]Theo. 20] from [START_REF] Parkkonen | Skinning measure in negative curvature and equidistribution of equidistant submanifolds[END_REF]Theo. 19]. We will then deduce the last statement of Theorem 12.7 from these generalisations, again using this regularisation process.

Let β be either α P s0, 1s small enough in the Hölder regularity case or P N big enough in the Sobolev regularity case. We fix i P I ´, j P I `, and we use the notation D ˘, α γ , λ γ , v γ and r σ ˘of Equation (11.2). Let r ψ ˘P C β c pB 1 ¯D˘q . Under the assumptions of Assertion (1) or (2), we first prove the following avatar of Equation (11.4), indicating only the required changes in its proof: there exists κ 0 ą 0 (independent of r ψ ˘) such that, as T Ñ `8,

δ }m F } e ´δ T ÿ γPΓ, 0ăλγ ďT e ş αγ r F r ψ ´pv γ q r ψ `pv γ q " ż B 1 `D´r ψ ´dr σ `żB 1 ´D`r ψ `dr σ ´`Ope ´κ0 T } r ψ ´}β } r ψ `}β q . (12.4)
By [START_REF] Parkkonen | Counting common perpendicular arcs in negative curvature[END_REF]Lem. 6] and the Hölder regularity of the strong stable and unstable foliations under the assumptions of Assertion (1), or by the smoothness of the boundary of D ˘under 187 19/12/2016 the assumptions of Assertion (2), the maps f D¯: V η, R pB 1 ˘D¯q Ñ B 1 ˘D¯a re respectively Hölder-continuous or smooth fibrations, whose fiber over w P B 1 ˘D¯i s exactly V w, η, R . By applying leafwise the regularisation process described in the proof of [START_REF] Parkkonen | Skinning measure in negative curvature and equidistribution of equidistant submanifolds[END_REF]Theo. 20] to characteristic functions, there exist a constant κ 1 ą 0 and χ η,

R P C β pT 1 Ă M q such that ' }χ η, R } β " Opη ´κ1 q, ' 1 V η e ´Opηq , R e ´Opηq pB 1 ¯D˘q ď χ η, R ď 1 V η, R pB 1 ¯D˘q , ' for every w P B 1 ¯D˘, we have ż V w, η, R
χ η, R dν w " ν w pV w, η, R q e ´Opηq " ν w pV w, η e ´Opηq , R e ´Opηq q e Opηq .

We now define the new test functions (compare with Section 10.1). For every

w P B 1 ¯D˘, let H η, R pwq " 1 ş V w, η, R χ η, R dν w .
Let Φ η : T 1 Ă M Ñ R be the map defined by

Φ η " pH η, R r ψ ˘q ˝f D˘χ η, R .
The support of this map is contained in V η, R pB 1 ¯D˘q . Since M is compact in Assertion (1) and by homogeneity in Assertion (2), if R is large enough, by the definitions of the measures ν w , the denominator of H η, R pwq is at least c η where c ą 0. The map H η, R is hence Höldercontinuous under the assumptions of Assertion (1), and it is smooth under the assumptions of Assertion (2). Therefore Φ η P C β pT 1 Ă M q and there exists a constant κ 2 ą 0 such that

}Φ η } β " Opη ´κ2 } r ψ ˘}β q .
As in Lemma 10.1, the functions Φ η are measurable, nonnegative and satisfy

ż T 1 Ă M Φ η d r m F " ż B 1 ¯D˘r ψ ˘dr σ ¯.
As in the proof of Theorem 11.1, we will estimate in two ways the quantity

I η pT q " ż T 0 e δ t ÿ γPΓ ż T 1 Ă M pΦ ή ˝g´t{2 q pΦ ὴ ˝gt{2 ˝γ´1 q d r m F dt . (12.5)
We first apply the mixing property, now with exponential decay of correlations, as in Step 2 of the proof of Theorem 11.1. For all t ě 0, let

A η ptq " ÿ γPΓ ż vPT 1 Ă M Φ ή pg ´t{2 vq Φ ὴ pg t{2 γ ´1vq d r m F pvq .
Then with κ ą 0 as in the definitions of the exponential mixing for the Hölder or Sobolev regularity, we have

A η ptq " 1 }m F } ż T 1 Ă M Φ ή d r m F ż T 1 Ă M Φ ὴ d r m F `O `e´κ t }Φ ή } β }Φ ὴ } β " 1 }m F } ż B 1 `D´r ψ ´dr σ `żB 1 ´D`r ψ `dr σ ´`O `e´κ t η ´2κ 2 } r ψ ´}β } r ψ `}β ˘. 188 19/12/2016
Hence by integrating,

I η pT q " e δ T δ }m F } ´żB 1 `D´r ψ ´dr σ `żB 1 ´D`r ψ `dr σ ´`O `e´κ T η ´2κ 2 } r ψ ´}β } r ψ `}β ˘¯. (12.6)
Now, as in Step 3 of the proof of Theorem 11.1, we exchange the integral over t and the summation over γ in the definition of I η pT q, and we estimate the integral term independently of γ:

I η pT q " ÿ γPΓ ż T 0 e δ t ż T 1 Ă M pΦ ή ˝g´t{2 q pΦ ὴ ˝gt{2 ˝γ´1 q d r m F dt . Let p Φ η " H η, R ˝f D˘χ η, R , so that Φ η " r ψ ˘˝f D˘p Φ η .
By the last two properties of the regularised maps χ η, R , we have, with φ η defined as in Equation (10.4),

φ η e ´Opηq , R e ´Opηq , B 1 ¯D˘e ´Opηq ď p Φ η ď φ η e Opηq .
(12.7)

If v P T 1 Ă
M belongs to the support of pΦ ή ˝g´t{2 q pΦ ὴ ˝gt{2 ˝γ´1 q, then we have v P

g t{2 V ὴ, R pB 1 `D´q X g ´t{2 V ή, R pγB 1 ´D`q .
Hence the properties (i), (ii) and (iii) of Step 3M of the proof of Theorem 11.1 still hold (with Ω ´" B 1 `D´a nd Ω `" B 1 ´pγD `q). In particular, if w ´" f D´p vq and w `" f γD `pvq, we have, by Assertion (iii) in Step 3M of the proof of Theorem 11.1, 6 that dpw ˘, v γ q " Opη `e´λγ{2 q .

Hence, with κ 3 " α in the Hölder case and κ 3 " 1 in the Sobolev case (we may assume that ě 1), we have | r ψ ˘pw ˘q ´r ψ ˘pv γ q | " Oppη `e´λγ{2 q κ 3 } r ψ ˘}β q .

Therefore there exists a constant κ 4 ą 0 such that

I η pT q " ÿ γPΓ `r ψ ´pv γ q r ψ `pv γ q `Oppη `e´λγ{2 q κ 4 } r ψ ´}β } r ψ `}β q ˘ż T 0 e δ t ż vPT 1 Ă M p Φ ή pg ´t{2 vq p Φ ὴ pγ ´1g t{2 vq d r m F pvq dt .
Now, using the inequalities (12.7), Equation (12.4) follows as in Steps 3M and 4M of the proof of Theorem 11.1, by taking η " e ´κ5 T for some κ 5 ą 0 and using the effective control given by Equation (11.19) in Step 4M.

In order to prove Assertions (1) and (2) of Theorem 12.7, we may assume that the supports of ψ ˘are small enough, say contained in Bpx ˘, q for some x ˘P T 1 M and small enough. Let

r x ˘be lifts of x ˘and let r ψ ˘P C β c pT 1 Ă M q with support in Bp r
x ˘, q be such that r ψ ˘" ψ ˘˝T p where p : Ă M Ñ M is the universal cover. By a finite summation argument and Equation (12.4), we have

δ }m F } e ´δ T ÿ iPI ´{", jPI `{", γPΓ 0ăλ i, γj ďT e ş αγ r F r ψ ´pv γ q r ψ `pv γ q " ż B 1 `D´r ψ ´dr σ `żB 1 ´D`r ψ `dr σ ´`Ope ´κ0 T } r ψ ´}β } r ψ `}β q .
(12.8)

6 See also the picture at the beginning of the proof of Lemma 11.7.

19/12/2016

Assertions (1) and ( 2) are deduced from this equation in the same way that Corollary 12.3 is deduced from Theorem 11.1. Taking the functions ψ k to be the constant functions 1 in Assertion (1) gives the last statement of Theorem 12.7 under the assumptions of Assertion (1). An approximation argument gives the result under the assumptions of Assertion (2). l 12.4 Equidistribution and counting for quotient simplicial and metric trees

In this Section, we assume that X is the geometric realisation of a locally finite metric tree without terminal vertices pX, λq, and that Γ is a (nonelementary discrete) subgroup of AutpX, λq. Let r c : EX Ñ R be a system of conductances for Γ, and let c : ΓzEX Ñ R be its quotient function. We assume in this Section that the potential r F is the potential r F c associated7 with c. Let δ c " δ Γ, Fc be the critical exponent of pΓ, F c q and let r m c " r m Fc and m c " m Fc be the Gibbs measures of F c for the continuous time geodesic flow on respectively G X and ΓzG X, as well as for the discrete time geodesic flow on respectively G X and ΓzG X when pX, λq is simplicial, that is, if λ is constant with value 1.

Let D ˘be simplicial subtrees of X, with the edge length map induced by λ,8 such that the Γ-equivariant families D ˘" pγD ˘qγPΓ{Γ D ˘are locally finite in X.9

For all γ, γ 1 in Γ such that γD ´and γ 1 D `are disjoint, we denote by α γ, γ 1 the common perpendicular from γD ´to γ 1 D `(which is an edge path in X), with length λ γ, γ 1 " dpγD ´, γ 1 D `q P N, and by α γ, γ 1 P p G X its parametrisations as in the beginning of Chapter 11: it is the unique map from R to X such that α γ, γ 1 ptq P γV D ´is the origin opα γ, γ 1 q of the edge path α γ, γ 1 if t ď 0, α γ, γ 1 ptq P γ 1 V D `is the endpoint tpα γ, γ 1 q of the edge path α γ, γ 1 if t ě λ γ, γ 1 , and α γ, γ 1 |r0, λ γ, γ 1 s is the shortest geodesic arc starting from a point of γD ´and ending at a point of γ 1 D `.

For all γ, γ 1 in Γ such that γD ´and γ 1 D `are disjoint, we define the multiplicity of the common perpendicular α γ, γ 1 from γD ´to γ 1 D `as

m γ, γ 1 " 1 CardpγΓ D ´γ ´1 X γ 1 Γ D `γ 1 ´1q
.

(12.9)

Note that m γ, γ 1 " 1 for all γ, γ 1 P Γ when Γ acts freely on EX (for instance when Γ is torsion-free). Generalising the definition for simplicial trees in Section 11.4, we set cpαq "

k ÿ i"1 cpe i q λpe i q ,
for any edge path α " pe 1 , . . . , e k q.

For n P N ´t0u, let

N D ´, D `pnq " ÿ rγsP Γ D ´zΓ{Γ D 0ădpD
´,γD `qďn m e, γ e cpαe, γ q , where Γ acts diagonally on pΓ{Γ D ´q ˆpΓ{Γ D `q and d is the the distance on X " |X| λ . When Γ is torsion-free, N D ´, D `pnq is the number of edge paths in the graph ΓzX of length at most n, starting by an outgoing edge from the image of D ´and ending by the opposite of an outgoing edge from the image of D `, with multiplicities coming from the fact that Γ D ˘zD ȋs not assumed to be embedded in ΓzX, and with weights coming from the conductances.

In the next results, we distinguish the continuous time case (Theorem 12.8) from the discrete time case (Theorem 12.9). We leave to the reader the versions without the assumption δ c ą 0, giving for every τ P N ´t0u an asymptotic on

N D ´, D `, τ pnq " ÿ rγsP Γ D ´zΓ{Γ D ǹ´τ ădpD
´,γD `qďn m e, γ e cpαe, γ q .

When ΓzX is compact, c " 0 and D ˘are reduced to points, the counting results in Theorems 12.8 and 12.9 are proved in [Gui]. When D ˘are singletons, Theorem 12.8 is due to [START_REF] Roblin | Ergodicité et équidistribution en courbure négative[END_REF] if c " 0. Otherwise, the result seems to be new.

Theorem 12.8. Let pX, λq, Γ, D ˘and c be as in the beginning of this Section. Assume that the critical exponent δ c is finite and positive, that the skinning measures σ D¯a re finite and nonzero, and that the Gibbs measure m c is finite and mixing for the continuous time geodesic flow. Then as t Ñ `8, the measures

δ c }m c } e ´δc t ÿ rγs PΓ D ´zΓ{Γ D 0ădpD ´,γ D `qďt m e, γ e cpαe, γ q ∆ Γα é,γ b ∆ Γα γ´1 ,e narrow converge to σ D´b σ D`i n Γz p G X ˆΓz p G X, and
N D ´, D `ptq " }σ D´} }σ D`} δ c }m c } e δc t .
Proof. By Theorem 11.1, we have

lim tÑ`8 δ c }m c } e ´δc t ÿ pa,b,γqPΓ{Γ D ´ˆΓ{Γ D `ˆΓ 0ădpaD ´, γbD `qďt e ş α a,γb r Fc ∆ α á,γb b ∆ α γ´1 a,b " r σ D´b r σ D`,
not only for the weak-star convergence on p G X ˆp G X, but also by Step 1 of the proof of Theorem 11.1, for the narrow convergence, as σ D´a nd σ D`a re finite. Recall that given a discrete group G acting properly (but not necessarily freely) on a locally compact space Z, the induced measure on GzZ of a (positive, Radon) measure µ on Z is a measure µ which depends linearly and continuously (both for the weak-star and narrow topologies) on µ, and satisfies ∆ z " 1 |Gz| ∆ Gz for every z P Z. See for instance Section [START_REF] Paulin | Equilibrium states in negative curvature[END_REF]§2.4] for more details.

The group Γ ˆΓ acts on Γ{Γ D ´ˆΓ{Γ D `ˆΓ by pa 1 , b 1 q ¨pa, b, γq " pa 1 a, b 1 b, a 1 γpb 1 q ´1q . and the map from the discrete set Γ{Γ D ´ˆΓ{Γ D `ˆΓ to p G X ˆp G X which sends pa, b, γq to pα á,γb , α γ´1 a,b q is pΓ ˆΓq-equivariant. In particular, the pushforward of measures by this map sends the unit Dirac mass at pa, b, γq to

∆ α á,γb b ∆ α γ´1 a,b . 191 19/12/2016
Every orbit of ΓˆΓ on Γ{Γ D ´ˆΓ{Γ D `ˆΓ has a representative of the form pΓ D ´, Γ D `, γq for some γ P Γ, since pa, bq¨pΓ D ´, Γ D `, a ´1γ bq " paΓ D ´, bΓ D ´, γq. Furthermore the double class in Γ D ´zΓ{Γ D `of such a γ is uniquely defined, and the stabiliser of pΓ D ´, Γ D `, γq has cardinality

|Γ D ´X γΓ D `γ ´1|, since pa, bq ¨pΓ D ´, Γ D `, γq " pΓ D ´, Γ D `, γ 1 q if and only if a P Γ D ´, b P Γ D ànd aγ b ´1 " γ 1 . When γ 1 " γ, this happens if and only if b " γ ´1aγ and a P Γ D ´X γΓ D `γ ´1.
Hence the measures

δ c }m c } e ´δc t ÿ rγs PΓ D ´zΓ{Γ D 0ădpD ´,γ D `qďt 1 |Γ D ´X γΓ D `γ ´1| e ş Γαe,γ Fc ∆ Γα é,γ b ∆ Γα γ´1 ,e narrow converge as t Ñ `8 to σ D´b σ D`i n Γz p G X ˆΓz p G X.
By applying this convergence to the constant function 1, and by the finiteness and nonvanishing of σ D´a nd σ D`, the result follows using the defining property of the potential F c , see Proposition 3.11. l

In the remainder of this Section, we consider simplicial trees with the discrete time geodesic flow.

Theorem 12.9. Let pX, λq, Γ, r c and D ˘be as in the beginning of this Section, with λ constant with value 1. Assume that the critical exponent δ c is finite and positive. If the Gibbs measure m c is finite and mixing for the discrete time geodesic flow and the skinning measures σ D¯a re finite and nonzero, then as n Ñ `8, the measures

e δc ´1 e δc }m c } e ´δc n ÿ rγs PΓ D ´zΓ{Γ D 0ădpD ´,γ D `qďn m e, γ e cpαe, γ q ∆ Γα é,γ b ∆ Γα γ´1 ,e narrow converge to σ D´b σ D`i n Γz p G X ˆΓz p G X and N D ´, D `pnq " e δc }σ D´} }σ D`} pe δc ´1q }m c } e δc n .
Proof. The claims follow as in Theorem 12.8, replacing Theorem 11.1 by Theorem 11.8. l

Examples 12.10. (1) Let X, Γ, c be as in Theorem 12.9, and let D ´" txu and D `" tyu for some x, y P V X. If the Gibbs measure m c is finite and mixing for the discrete time geodesic flow, then we have a version of Roblin's simultaneous equidistribution theorem with potential, see Corollary 11.2, and the number N x, y pnq of nonbacktracking edge paths of length at most n from x to y (counted with weights and multiplicities) satisfies

N x, y pnq " e δc }µ x } }µ ý } pe δc ´1q }m c } |Γ x | |Γ y | e δc n .
(2) If Y is a finite connected nonbipartite pq `1q-regular graph (with q ě 2) and Y ˘are points, then the number of nonbacktracking edge paths from Y ´to Y `of length at most n is equivalent as n Ñ `8 to q `1 q ´1 q n |V Y| `Opr n q (12.10) for some r ă q. Indeed, by Theorem 12.9 with X the universal cover of Y, Γ its covering group and c " 0, we have δ c " ln q and m c is the Bowen-Margulis measure, so that normalizing the 192 19/12/2016

Patterson measures to be probability measures, we have }m c } " q q`1 |V Y| by Equation (8.3). We refer to Section 12.6 (see Remark (i) following Theorem 12.17) for the error term.

Let Y the figure 8-graph with a single vertex and four directed edges, and let Y ˘be the the singleton consisting of its vertex. In this simple example, it is easy to count by hand that the number of loops of length exactly n without backtracking in Y is 4 3 n´1 . Thus the number N pnq of common perpendiculars of the vertex to itself of length at most n is by a simple geometric sum 2p3 n ´1q. This agrees with Equation (12.10) that gives N pnq " 2 3 n as n Ñ `8.

(3) Let Y be a finite connected nonbipartite pq `1q-regular graph (with q ě 2). Let Y ˘be regular connected subgraphs of degrees q ˘ě 0. Then the number N pnq of edge paths of length at most n starting transversally to Y ´and ending transversally to Y `satisfies

N pnq " pq `1 ´q´q pq `1 ´q`q |V Y ´| |V Y `|
pq 2 ´1q |V Y| q n `Opr n q for some r ă q. This is a direct consequence of Theorem 12.9, using Proposition 8.1 (3) and Proposition 8.4 (3), and refering to Section 12.6 (see Remark (i) following Theorem 12.17) for the error term.

We refer for instance to Section 15.2 for examples of counting results in graphs of groups where the underlying graph is infinite.

Remark 12.11. A common perpendicular in a simplicial tree is, in the language of graph theory, a non-backtracking walk. Among other applications (when restricting to groups Γ acting freely, which is never the case if Γ is a nonuniform lattice in the tree X, that is, when the quotient graph of groups ΓzX is infinite but has finite volume), Theorem 12.9 gives a complete asymptotic solution to the problem of counting non-backtracking walks from a given vertex to a given vertex of a (finite) nonbipartite graph. See Theorem 12.12 for the corresponding result in bipartite graphs and for example [START_REF] Alon | Non-backtracking random walks mix faster[END_REF]Th. 1.1], [START_REF] Angel | The non-backtracking spectrum of the universal cover of a graph[END_REF]p. 4290,4302], [START_REF] Friedman | A proof of Alon's second eigenvalue conjecture and related problems[END_REF]L. 2.3], [START_REF] Sodin | Random matrices, nonbacktracking walks, and orthogonal polynomials[END_REF]Prop. 6.4] for related results. Anticipating on the error terms that we will give in Section 12.6, note that the paper [START_REF] Alon | Non-backtracking random walks mix faster[END_REF]Th. 1.1] for instance gives a precise speed using spectral properties, more precise than the ones we obtain.

In some applications (see the examples at the end of this Section), we encounter bipartite simplicial graphs and, consequently, their discrete time geodesic flow is not mixing. The following result applies in this context.

Until the end of this section, we assume that the simplicial tree X has a Γ-invariant structure of a bipartite graph, and we denote by V X " V 1 X \ V 2 X the corresponding partition of its set of vertices. For every i P t1, 2u, we denote by p G i X the space of generalised discrete geodesic lines P p G X such that p0q P V i X, so that we have a partition

p G X " p G 1 X \ p G 2 X. Note that if the basepoint x 0 lies in V i X, then G even X is equal to p G i XXG X. For all i, j P t1, 2u, we define N D ´, D `, i, j pnq " ÿ rγsP Γ D ´zΓ{Γ D 0ădpD
´,γD `qďn opαe,γ qPV i X, tpαe,γ qPV j X m e, γ e cpαe, γ q . Theorem 12.12. Let pX, λq, Γ and c be as in the beginning of this Section, with λ constant with value 1. Assume that the critical exponent δ c is finite and positive. If X has a Γ-invariant

193 19/12/2016
structure of a bipartite graph as above, if the restriction to ΓzG even X of the Gibbs measure m c is finite and mixing for the square of the discrete time geodesic flow, then for all i, j P t1, 2u such that the measures σ D´, i and σ D`, j are finite and nonzero, as n tends to `8 with n " i´j mod 2, the measures

e 2 δc ´1 2 e 2 δc }m c } e ´δc n ÿ rγs PΓ D ´zΓ{Γ D 0ădpD ´,γ D `qďn opαe,γ qPV i X, tpαe,γ qPV j X m e, γ e cpαe, γ q ∆ Γα é,γ b ∆ Γα γ´1 ,e narrow converge to σ D´, i b σ D`, j in Γz p G X ˆΓz p G X and N D ´, D `, i, j pnq " 2 e 2 δc }σ D´, i } }σ D`, j } pe 2 δc ´1q }m c } e δc n .
Proof. This Theorem is proved in the same way as the above Theorem 12.9 using Theorem 11.10. Note that we have a pΓ ˆΓq-invariant partition

p G X ˆp G X " ğ pi, jqPt1, 2u 2 p G i X ˆp G j X , that α é, γ P p G i X if
and only if opα e, γ q P V i X, and that α γ´1 , e P p G j X if and only if tpα e, γ q P V j X, since α γ´1 , e p0q " γ ´1α è, γ p0q " γ ´1tpα e, γ q. l Examples 12.13. (1) Let X, Γ, c be as in Theorem 12.12, and let D ´" txu and D `" tyu for some vertices x, y in the same V i X for i P t1, 2u. If the restriction to ΓzG even X of the Gibbs measure m c is finite and mixing for the square of the discrete time geodesic flow, then as n Ñ `8 is even,

N D ´, D `pnq " 2 e 2 δc e 2 δc ´1 }µ x } }µ ý } }m c } |Γ x | |Γ y | e δc n .
Indeed, we have N D ´, D `pnq " N D ´, D `, i, i pnq and σ D˘, i " σ D˘.

(2) Let Y be the complete biregular graph with q `1 vertices of degree p `1 and p `1 vertices of degree q `1. Let Y ˘" tyu be a fixed vertex of degree p `1. Note that Y being bipartite, all common perpendiculars from y to y have even length, (the shortest one having length 4). Then as n is even and tends to `8, we have

N Y ´,Y `pnq " qpp `1q pq `1qppq ´1q ppqq n{2 .
Indeed, the biregular tree X p,q of degrees pp `1, q `1q is a universal cover of Y with covering group Γ acting freely and cocompactly, so that with c " 0 we have δ c " ln ? pq and the Gibbs measure m c is the Bowen-Margulis measure m BM . If we normalise the Patterson density such that }µ y } " p`1 ? p , then by Proposition 8.1 (2), we have }m BM } " 2pp `1qpq `1q. Thus the result follows from Example (1). Note that if p " q, then N Y ´,Y `pnq " q q 2 ´1 q n , and the constant in front of q n is indeed different from that in the nonbipartite case. (3) Let Y be a finite biregular graph with vertices of degrees p `1 and q `1, where p, q ě 2, and let V Y " V p Y \ V q Y be the corresponding partition. If Y ´" tvu where v P V p Y and Y ìs a cycle of length L ě 2, then as N Ñ `8, the number of common perpendiculars of even length at most 2N from Y ´to Y `is equivalent to

L q pp ´1q 2 ppq ´1q |V p Y| ppqq N ,
and the number of common perpendiculars of odd length at most

2N ´1 from v to Y `is equivalent to L pq ´1q 2 ppq ´1q |V p Y| ppqq N .
Proof. The cycle Y `has even length L and has L 2 vertices in both V p Y and V q Y. A common perpendicular from Y ´to Y `has even length if and only if it ends at a vertex in V p Y.

Let X Ñ Y be a universal cover of Y, whose covering group Γ acts freely and cocompactly on X. Let D ´" tr vu where r v P V p X is a lift of v, and let D ´be a geodesic line in X mapping to Y `. We use Theorem 12.12 with V 1 X the (full) preimage of V p Y in X, with V 2 X the premiage of V q Y in X and with c " 0, so that δ c " ln ? pq and m c " m BM . Let us normalise the Patterson density of Γ as in Proposition 8.1 (2), so that

}σ D´, 1 } " }µ r v } " p `1 ? p .
The mass for the skinning measure of the part of the inner unit normal bundle of Y `with basepoint in V p Y is (see Corollary 8.5)

L 2 p `1 ? p p ´1 p `1 " Lpp ´1q 2 ? p
and its complement has mass Lpq´1q 2 ?

q . Recall also that, by Proposition 8.1 (2), considering the graph Y as a graph of groups with trivial groups, }m BM } " TVolpYq " |EY| " 2pp `1q|V p Y| " 2pq `1q|V q Y| . The claim about the common perpendiculars of even length at most 2N follows from Theorem 12.12 with i " j " 1, since

2 e 2 δc }σ D´, i } }σ D`, j } pe 2 δc ´1q }m c } " 2 pq p`1 ? p Lpp´1q 2 ? p ppq ´1q 2 pp `1q |V p Y| " L q pp ´1q 2 ppq ´1q |V p Y| .
The claim about the common perpendiculars of odd length at most 2N ´1 follows similarly from Theorem 12.12 with i " 1 and j " 2. l

(4) Let Y be a finite biregular graph with vertices of degrees p `1 and q `1, where p, q ě 2, and let V Y " V p Y \ V q Y be the corresponding partition. If Y ´and Y `are cycles of length L ´ě 2 and L `ě 2 respectively, then as N Ñ `8, the number of common perpendiculars of length at most N from Y ´to Y `is equal to p ? q `?pq 2 L ´L2 ppq ´1q CardpEYq p ? pqq N `2 `Opr N q (12.11) 195 19/12/2016 for some r ă ? pq.

Proof. As in the above proof of Example (3), let X Ñ Y be a universal cover of Y, with covering group Γ and let D ˘be a geodesic line in X mapping to Y ˘. We normalise the Patterson density pµ x q xPV X of Γ so that }µ x } " deg X pxq ?

deg X pxq´1
. By Proposition 8.4 (3) with k " 1 and trivial vertex stabilisers, and since a simple cycle of length λ in a biregular graph of different degrees p `1 and q `1 has exactly λ 2 vertices of degree either p `1 or q `1, we have

}σ D¯} " ÿ ΓxPY ¯}µ x } pdeg X pxq ´kq deg X pxq " ÿ yPVpY ¯?p `ÿ yPVqY ¯?q " L ¯?p `?q 2 .
The result without the error term then follows from Theorem 12.12, using Proposition 8.1 (2) and Remark 2.11. We refer to Section 13.2 (see Remark (ii) following Theorem 12.17) for the error term. l

Remark 12.14. If Y is a finite bipartite pp `1q-regular graph, Y ´consists in a vertex and Y `is a cycle of length L, then Example (3) above gives

N Y ´, Y `pnq " L p pp ´1q 2 pp 2 ´1q |V Y| p n `L pp ´1q 2 pp 2 ´1q |V Y| p n " L |V Y| p n
for the number N Y ´, Y `pnq of common perpendiculars from Y ´to Y `with length at most n. This is the same result as for nonbipartite trees.

Counting for simplicial graphs of groups

In this Section, we give an intrinsic translation "a la Bass-Serre" of the counting result in Theorem 12.9 using graphs of groups (see [START_REF] Serre | Arbres, amalgames, SL 2 . 3ème éd. corr[END_REF] and Section 2.7 for background information).

Let pY, G ˚q be a locally finite, connected graph of finite groups, and let pY ˘, G ˘q be connected subgraphs of subgroups. 10 Let c : EY Ñ R be a system of conductances on Y.

Let X be the Bass-Serre tree of the graph of groups pY, G ˚q (with geometric realisation X " |X| 1 ) and Γ its fundamental group (for an indifferent choice of basepoint). Assume that Γ is nonelementary. We denote by G pY, G ˚q " ΓzG X and `gt : G pY, G ˚q Ñ G pY, G ˚q˘t PZ the quotient of the (discrete time) geodesic flow on G X, by r c : X Ñ R the (Γ-invariant) lift of c, with δ c its critical exponent and r F c : T 1 X Ñ R its associated potential, by m c the Gibbs measure on G pY, G ˚q associated with a choice of Patterson densities pµ x q xPX for the pairs pΓ, F c q, by D ˘two subtrees in X such that the quotient graphs of groups Γ D ˘z zD ˘identify with pY ˘, G ˘q (see below for precisions), and by σ pY ¯,G ¯q the associated skinning measures.

The fundamental groupoid πpY, G ˚q of pY, G ˚q11 is the quotient of the free product of the groups G v for v P V Y and of the free group on EY by the normal subgroup generated by the elements e e and e ρ e pgq e ρ e pgq ´1 for all e P EY and g P G e . We identify each G x for x P V Y with its image in πpY, G ˚q.

Let n P N ´t0u. A (locally) geodesic path of length n in the graph of groups pY, G ˚q is the image α in πpY, G ˚q of a word, called reduced in [START_REF] Bass | Covering theory for graphs of groups[END_REF]1.7

],

h 0 e 1 h 1 e 2 . . . h n´1 e n h n 10 See Section 2.7 for definitions and background. 11 denoted by F pY, G˚q in [Ser3, §5.1], called the path group in [START_REF] Bass | Covering theory for graphs of groups[END_REF]1.5], see also [Hig] 196

19/12/2016 with ' e i P EY and tpe i q " ope i`1 q for 1 ď i ď n ´1 (so that pe 1 , . . . , e n q is an edge path in the graph Y;

' h 0 P G ope 1 q and h i P G tpe i q for 1 ď i ď n;

' if e i`1 " e i then h i does not belong to ρ e i pG e i q, for 1 ď i ď n ´1.

Its origin is opαq " ope 1 q and its endpoint is tpαq " tpe n q. They do not depend on the chosen words with image α in πpY, G ˚q.

A common perpendicular of length n from pY ´, G ´q to pY `, G `q in the graph of groups pY, G ˚q is the double coset rαs " G ópαq α G tpαq of a geodesic path α of length n as above, such that:

' α starts transversally from pY ´, G ´q, that is, its origin opαq " ope 1 q belongs to V Y ánd h 0 R G ópe 1 q ρ e 1 pG e 1 q if e 1 P EY

´,

' α ends transversally in pY `, G `q, that is, its endpoint tpαq " tpe n q belongs to Y `and h n R ρ en pG en q G tpenq if e n P EY `.

Note that these two notions do not depend on the representative of the double coset G ópαq α G tpαq , and we also say that the double coset rαs starts transversally from pY ´, G ´q or ends transversally in pY ´, G ´q.

We denote by PerpppY ˘, G ˘q, nq the set of common perpendiculars in pY, G ˚q of length n from pY ´, G ´q to pY `, G `q. We denote by cpαq "

n ÿ i"1 cpe i q
the conductance of a geodesic path α as above, which depends only on the double class rαs. We define the multiplicity m α of a geodesic path α as above by

m α " 1 CardpG ópαq X α G tpαq α ´1q .
It depends only on the double class rαs of α. We define the counting function of the common perpendiculars in pY, G ˚q of length at most n from pY ´, G ´q to pY `, G `q (counted with multiplicities and with weights given by the system of conductances c) as

N pY ´,G ´q, pY `,G `qpnq " ÿ rαsPPerpppY ˘,G ˘q,nq m α e cpαq .
Theorem 12.15. Let pY, G ˚q, pY ˘, G ˘q and c be as in the beginning of this Section. Assume that the critical exponent δ c of c is positive and that the Gibbs measure m c on G pY, G ˚q is finite and mixing for the discrete time geodesic flow. Assume that the skinning measures σ pY ¯,G ¯q are finite and nonzero. Then as n P N tends to 8

N pY ´,G ´q, pY `,G `qpnq " e δc }σ pY ´,G ´q} }σ ṕY `,G `q} pe δc ´1q }m c } e δc n .
197 19/12/2016

Proof. Let X be the Bass-Serre tree of pY, G ˚q and Γ its fundamental group (for an indifferent choice of basepoint). As seen in Section 2.7, the Bass-Serre trees D ˘of pY ˘, G ˘q, with fundamental groups Γ ˘, identify with simplicial subtrees D ˘of X, such that Γ ˘are the stabilisers Γ D ˘of D ˘in Γ. In particular, the maps pΓ D ˘zD ˘q Ñ pΓzXq induced by the inclusion maps D ˘Ñ X by taking quotient, are injective:

@ γ P Γ, @ z P V D ˘Y ED ˘, if γz P V D ˘Y ED ˘, then D γ ˘P Γ D ˘, γ ˘z " γz . (12.12)
As in Definition 2.10, for all z P V Y Y EY and e P EY, we fix a lift r z P V X Y EX of z and g e P Γ, such that r e " r e, g e Ą tpeq " tpr eq, G z " Γ r z , and the monomorphism

ρ e : G e Ñ G tpeq is γ Þ Ñ g ´1
e γg e . We assume, as we may, that r

z P V D ˘Y ED ˘if z P V Y ˘Y EY ˘.
We assume, as we may using Equation (12.12), that if e P EY ˘, then g e P Γ D ˘. We denote by p : X Ñ Y " ΓzX the canonical projection.

f k γ k f 1 r e 1 γ 1 Ą e i`1 g ei Y ´g0 e 1 g i e i Ć tpe i q f i f i`1 γ i`1 g ei`1 r e i γ i Y èk g k γ 1 γ k`1 γ 1 ´1 γD γγ 0 γ ´1 γ Ć ope 1 q γ 1 Ć tpe k q γ 1 D tpf k q opf 1 q r e k γ 1 g ek Ć tpe k q γ g e1 Ć ope 1 q e i`1
For all γ, γ 1 P Γ such that γD ´and γ 1 D `are disjoint, the common perpendicular α γD ´, γ 1 D from γD ´to γ 1 D `is an edge path pf 1 , f 2 , . . . , f k q with opf 1 q P γD ´and tpf k q P γ 1 D `. Note that γ ´1 opf 1 q and ppopf 1 qq " are two vertices of D ´in the same Γ-orbit, and that γ 1 ´1 tpf k q and pptpf k qq " are two vertices of D `in the same Γ-orbit. Hence by Equation (12.12), we may choose

γ 0 P Γ D ´such that γ 0 γ ´1 opf 1 q " ppopf 1 qq " and γ k`1 P Γ D ´such that γ k`1 γ 1 ´1 tpf k q " pptpf k qq " . For 1 ď i ď k, choose γ i P Γ such that γ i f i " Ć ppf i q. We define ' e i " ppf i q for 1 ď i ď k, ' h i " g ´1 e i γ i γ i`1 ´1g e i`1 , which belongs to Γ Ć tpe i q " G tpe i q for 1 ď i ď k ´1, ' h 0 " γ 0 γ ´1γ ´1 1 g e 1 " γ ´1pγγ 0 γ ´1qγ ´1 1 g e 1 , which belongs to Γ Č ope 1 q " G ope 1 q , ' h k " g ´1 e k γ k γ 1 γ k`1 ´1 " g ´1 e k γ k pγ 1 γ k`1 γ 1 ´1q ´1γ 1 , which belongs to Γ Ć tpe k q " G tpe k q .
Lemma 12.16.

(1) The word h 0 e 1 h 1 . . . h k´1 e k h k is reduced. Its image α in the fundamental groupoid πpY, G ˚q does not depend on the choices of γ 1 , . . . , γ k , and starts transversally from pY ´, G ´q and ends transversally in pY `, G `q. The double class rαs of α is independent of the choices of γ 0 and γ k`1 .

198 19/12/2016

(2) The map r Θ from the set of common perpendiculars in X between disjoint images of D ánd D `under elements of Γ, into the set of common perpendiculars in pY, G ˚q from pY ´, G ´q to pY `, G `q, sending α γD ´, γ 1 D `to rαs, is constant under the action of Γ at the source, and preserves the lengths and the multiplicities.

(3) The map Θ induced by r Θ from the set of Γ-orbits of common perpendiculars in X between disjoint images of D ´and D `under elements of Γ into the set of common perpendiculars in pY, G ˚q from pY ´, G ´q to pY `, G `q is a bijection, preserving the lengths and the multiplicities.

Proof. (1) If e i`1 " e i , then by the definition of h i , we have

h i P ρ e i pG e i q " g ´1 e i Γ r e i g e i ðñ g e i h i g ´1 e i r e i " r e i ðñ g e i g ´1 e i γ i γ i`1 ´1g e i`1 g ´1 e i r e i " r e i ðñ γ i γ i`1 ´1 Ą e i`1 " r e i ðñ γ i`1 ´1 Ą e i`1 " γ ´1 i r e i ðñ f i`1 " f i .
Hence the word h 0 e 1 h 1 . . . h k´1 e k h k is reduced. The element γ i for i P t1, . . . , ku is uniquely determined up to multiplication on the left by an element of Γ r e i " G e i . If we fix 12 i P t1, . . . , ku and if we replace γ i by γ 1 i " α γ i for some α P G e i , then only the elements h i´1 and h i change, replaced by elements that we denote by h 1 i´1 and h 1 i respectively. We have (if 2 ď i ď k ´1, but otherwise the argument is similar by the definitions of h 0 and h k ) h 1 i´1 e i h 1 i " g ´1 e i´1 γ i´1 γ i ´1α ´1g e i e i g ´1 e i α γ i γ i`1 ´1g e i`1 " g ´1 e i´1 γ i´1 γ i ´1g e i ρ e i pαq ´1 e i ρ e i pαq g ´1 e i γ i γ i`1 ´1g e i`1 .

Since ρ e i pαq ´1 e i ρ e i pαq is equal to e i ´1 " e i in the fundamental groupoid, the words h 1 i´1 e i h 1 i and h i´1 e i h i have the same image in πpY, G ˚q. Therefore α is independent on the choices of γ 1 , . . . , γ k .

We have opαq " ope 1 q P V Y ´and tpαq " tpe k q P V Y `, hence α starts from Y ´and ends in Y `.

Assume that e 1 P EY ´. Let us prove that h 0 P G ópe 1 q ρ e 1 pG e 1 q if and only if

f 1 P γ ED ´. γ ´1f 1 Ć ope 1 q γ 0 g e1 opγ ´1f 1 q r e 1 D
By the definition of ρ e 1 , we have h 0 P G ópe 1 q ρ e 1 pG e 1 q if and only if there exists α P

Γ Č ope 1 q X Γ D ´such that α ´1 h 0 P g ´1 e 1 Γ Ă e 1 g e 1 .
By the definition of h 0 and since γ 1 maps f 1 to r e 1 , we have

α ´1 h 0 P g ´1 e 1 Γ Ă e 1 g e 1 ðñ g e 1 α ´1 `γ0 γ ´1γ ´1 1 g e 1 ˘g ´1 e 1 r e 1 " r e 1 ðñ f 1 " γ γ ´1 0 α g ´1 e 1 r e 1 .
12 We leave to the reader the verification that the changes induced by various i's do not overlap.

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Since r e 1 P ED ´and γ 0 , α, g e 1 all belong to Γ D ´, this last condition implies that f 1 P γ ED

´.

Conversely (for future use), if f 1 P γ ED ´, then (see the above picture) γ 0 γ ´1f 1 is an edge of D ´with origin Ć ope 1 q, in the same orbit that the edge g ´1 e 1 r e 1 of D ´, which also has origin Ć ope 1 q. By Equation (12.12), this implies that there exists α P Γ Č ope 1 q X Γ D ´such that f 1 " γ γ ´1 0 α g ´1 e 1 r e 1 . By the above equivalences, we hence have that h 0 P G ópe 1 q ρ e 1 pG e 1 q. Similarly, one proves that if e k P EY `, then h k P ρ e k pG e k q G tpe k q if and only if

f k P γ 1 ED `.
Since pf 1 , . . . , f n q is the common perpendicular edge path from γ D ´to γ 1 D `, this proves that α starts transversally from Y ´and ends transversally in Y

´.

Note that the element γ 0 P Γ D ´is uniquely defined up to multiplication on the left by an element of Γ Č ope 1 q X Γ D ´" G ópe 1 q , and appears only as the first letter in the expression of h 0 . Note that the element γ k`1 P Γ D `is uniquely defined up to multiplication on the left by an element of Γ Ć tpe k q X Γ D `" G tpe k q , hence γ ´1 k`1 is uniquely defined up to multiplication on the right by an element of G tpe k q , and appears only as the last letter in the expression of h k . Therefore α is uniquely defined in the fundamental groupoid πpY, G ˚q up to multiplication on the left by an element of G ópe 1 q and multiplication on the right by an element of G tpe k q , that is, the double class rαs Ă πpY, G ˚q is uniquely defined.

(2) Let β be an element in Γ and let x " α γD ´, γ 1 D `be a common perpendicular in X between disjoint images of D ´and D `under elements of Γ. Let us prove that r

Θpβ xq " r Θpxq. Since β x " α β γD ´, β γ 1 D `, in the construction of r
Θpβ xq, we may take, instead of the elements γ 0 , γ 1 , . . . , γ k , γ k`1 used to construct r Θpxq, the elements

γ 7 0 " γ 0 , γ 7 1 " γ 1 β ´1, . . . , γ 7 k " γ k β ´1, γ 7 k`1 " γ k`1 .
And instead of γ and γ 1 , we now may use γ 7 " β γ and γ 1 7 " β γ 1 . The only terms involving γ, γ 1 , γ 1 , . . . , γ k in the construction of r Θpxq come under the form γ ´1γ ´1

1 in h 0 , γ i γ i`1 ´1 in h i for 1 ď i ď k ´1, and γ k γ 1 in h k . Since pγ 7 q ´1pγ 7 1 q ´1 " γ ´1γ ´1 1 , pγ 7 i qpγ 7 i`1 q ´1 " γ i γ i`1 ´1 for 1 ď i ď k ´1, and pγ 7 k qpγ 1 7 q " γ k γ 1 , this proves that r Θpβ xq " r Θpxq, as wanted.

It is immediate that if the length of α γD ´, γ 1 D `is k, then the length of rαs is k.

Let us prove that the multiplicity, given in Equation (12.9),

m γ, γ 1 " 1 CardpγΓ D ´γ ´1 X γ 1 Γ D `γ 1 ´1q of the common perpendicular α γD ´, γ 1 D `in X between γ D ´and γ 1 D `is equal to the multi- plicity m α " 1 CardpG ópαq X α G tpαq α ´1q of the common perpendicular α in pY, G ˚q from pY ´, G ´q to pY `, G `q.
Since the multiplicity m γ, γ 1 is invariant under the diagonal action by left translations of γ ´1 0 γ ´1 P Γ on pγ, γ 1 q, we may assume that γ " γ 0 " id. Since the multiplicity m γ, γ 1 is invariant under right translation by γ k`1 ´1, which stabilises D `, on the element γ 1 , we may assume that γ k`1 " id. In particular, we have opf 1 q " Ć ope 1 q and tpf k q " γ 1 Ć tpe k q . 200 19/12/2016

We use the basepoint x 0 " ope 1 q in the construction of the fundamental group and the Bass-Serre tree of pY, G ˚q, so that (see in particular [START_REF] Bass | Covering theory for graphs of groups[END_REF]Eq. (1.3

)]) V X " ž βPπpY, G˚q : opβq"x 0 β G tpβq and
Γ " π 1 pY, G ˚q " tβ P πpY, G ˚q : opβq " tpβq " x 0 u .

Since an element in Γ which preserves D ´and γ 1 D `fixes pointwise its (unique) common perpendicular in X, we have

Γ D ´X γ 1 Γ D `γ 1 ´1 " Γ D ´X Γ γ 1 D `" pΓ opf 1 q X Γ D ´q X pΓ tpf k q X Γ γ 1 D `q " pΓ Č ope 1 q X Γ D ´q X pΓ γ 1 Ć tpe k q X Γ γ 1 D `q .
Note that Γ Č ope 1 q X Γ D ´" G ópe 1 q . By the construction of the edges in the Bass-Serre tree of a graph of groups (see [START_REF] Bass | Covering theory for graphs of groups[END_REF]page 11]), the vertex α G tpe k q is exactly the vertex tpf k q " γ 1 Ć tpe k q. By [Bass, Eq. (1.4)], we hence have

α G tpe k q α ´1 " Stab π 1 pY,G˚q pα G tpe k q q " Γ γ 1 Ć tpe k q .
Therefore m γ, γ 1 " m α .

(3) Let rαs " G opαq α G tpαq be a common perpendicular in pY, G ˚q from pY ´, G ´q to pY `, G `q, with representative α P πpY, G ˚q, and let h 0 e 1 h 1 . . . e k h k be a reduced word whose image in πpY, G ˚q is α.

We define

' γ 1 " g e 1 h ´1 0 , ' f 1 " γ ´1 1 r e 1 ,
' assuming that γ i and f i for some 1 ď i ď k ´1 are constructed, let

γ i`1 " g e i`1 h ´1 i g ´1 e i γ i and f i`1 " γ i`1 ´1 Ą e i`1 ,
' with γ k and f k constructed by induction, finally let γ 1 " γ ´1 k g e k h k . It is easy to check, using the equivalences in the proof of Lemma 12.16 (1) with γ " γ 0 " γ k`1 " id, that the sequence pf 1 , . . . , f k q is the edge path of a common perpendicular in X from D ´to γ 1 D `with origin Ć ope 1 q and endpoint γ 1 Ć tpe k q. If h 0 is replaced by α h 0 with α P G ópe 1 q , then by induction, f 1 , f 2 . . . , f k are replaced by αf 1 , αf 2 , . . . , αf k and γ 1 is replaced by αγ 1 . Note that pαf 1 , αf 2 , . . . , αf k q is then the common perpendicular edge path from D ´" αD ´to αγ 1 D `. If h k is replace by h k α with α P G tpe k q , then f 1 , f 2 . . . , f k are unchanged, and γ 1 is replaced by

γ 1 α. Note that γ 1 α D `" γ 1 D `.
Hence the map which associates to rαs the Γ-orbit of the common perpendicular in X from D ´to γ 1 D `with edge path pf 1 , . . . , f k q is well defined. It is easy to see by construction that this map is the inverse of Θ. l Theorem 12.15 now follows from Theorem 12.9. l 201 19/12/2016 12.6 Error terms for equidistribution and counting for metric and simplicial graphs of groups

In this Section, we give error terms to the equidistribution and counting results of Section 12.4, given by Theorem 12.8 for metric trees (and their continuous time geodesic flows) and by Theorem 12.9 for simplicial trees (and their discrete time geodesic flows), under appropriate bounded geometry and rate of mixing properties.

Let pX, λq, X, Γ, r c, c, r F c , F c , δ c , D ˘, D ˘, λ γ,γ 1 , α γ,γ 1 , α γ,γ 1 , m γ,γ 1 be as in Section 12.4. We first consider the simplicial case (when λ " 1), for the discrete time geodesic flow.

Theorem 12.17. Let X be a locally finite simplicial tree without terminal vertices, let Γ be a nonelementary discrete subgroup of AutpXq, let r c be a system of conductances on X for Γ and let D ˘be nonempty proper simplicial subtrees of X. Assume that the critical exponent δ c is finite and positive, that the Gibbs measure m c (for the discrete time geodesic flow) is finite and that the skinning measures σ D¯a re finite and nonzero. Assume furthermore that

(1) at least one of the following holds :

' Γ D ˘zBD ˘is compact ' C ΛΓ is uniform and Γ is a lattice of C ΛΓ,
(2) there exists β P s0, 1s such that the discrete time geodesic flow on pΓzG X, m c q is exponentially mixing for the β-Hölder regularity.

Then there exists κ 1 ą 0 such that for all ψ ˘P Proof. We follow the scheme of proof of Theorem 12.7, replacing aspects of Riemannian manifolds by aspects of simplicial trees as in the proof of Theorem 11.8. Let r ψ ˘P C β c p p G Xq. In order to simplify the notation, let λ γ " λ e,γ , α γ " α e,γ , α γ " α é,γ , α γ " α γ´1 ,e and r σ ˘" r σ D¯. Let us first prove the following avatar of Equation (12.4), indicating only the required changes in its proof: there exists κ 0 ą 0 (independent of r ψ ˘) such that, as n Ñ `8, e δc ´1 e δc }m c } e ´δc n ÿ γPΓ, 0ăλγ ďn e cpαγ q r ψ ´pα γ q r ψ `pα γ q " ż B 1 `D´r

ψ ´dr σ `żB 1 ´D`r ψ `dr σ ´`Ope ´κ0 n } r ψ ´}β } r ψ `}β q . (12.13) 202 19/12/2016
Most of the new work to be done in order to prove this formula concerns regularity properties of the test functions that will be introduced later on. We start with the regularity of the fibration of the dynamical neighbourhoods. 13 Let η, R ą 0 be such that η ă 1 ă ln R.

Lemma 12.18. Let Y be an R-tree and let D be a nonempty closed convex subset of Y .

Then the restriction to V η, R pB 1 ˘Dq of the fibration f D is (uniformly locally) Lipschitz, with constants independent of η.

Proof. We assume for instance that ˘" `. Let , 1 P V ὴ, R pB 1 `Dq and let w " f D p q, w 1 " f D p 1 q.

Since the fiber over ρ P B 1 `D of the restriction to V ὴ, R pB 1 `Dq of f D is V ρ, η, R (see the end of Section 2.5), there exist s, s 1 P s ´η, ηr such that g s P B `pw, Rq and g s 1 1 P B `pw 1 , Rq, so that g s ptq " wptq and g s 1 1 ptq " w 1 ptq for all t ě ln R by the definition of the Hamenstädt balls. Up to permuting and 1 , we assume that s 1 ě s.

w ď ln R 1 p0q s 1 ď η w 1 p0q 1 1
By (the proof of) Lemma 10.11 (1), there exists a constant c R ą 0 depending only on R such that if dp , 1 q ď c R and s 2 " dp p0q, 1 p0qq, then s 2 " s 1 ´s and the geodesic lines g s and g s 1 1 coincide at least on r´ln R ´1, ln R `1s . In particular, we have wpln Rq " ps `ln Rq " 1 ps 1 `ln Rq " w 1 pln Rq .

Since the origin of w is the closest point on D to any point of wpr0, `8rq, we hence have that wptq " w 1 ptq for all t P r0, ln Rs. Therefore (using Equation (2.5) for the last inequality),

dpw, w 1 q " ż `8 ln R dpwptq, w 1 ptqq e ´2 t dt " ż `8 ln R dpg s ptq, g s 1 1 ptqq e ´2 t dt " e 2s ż `8 ln R`s
dp puq, g s 2 1 puqq e ´2 u du ď e 2s dp , g s 2 1 q ď e 2s `dp , 1 q `dp 1 , g s 2 1 q ˘ď e 2s pdp , 1 q `s2 q " e 2s `dp , 1 q `dp p0q, 1 p0qq ˘, so that the result follows from Lemma 10.11 (2).

Note that when Y is (the geometric realisation of) a simplicial tree, then we have s " s 1 " s 2 " 0 and the above computations simplify to give dpw, w 1 q ď dp , 1 q. l 13 See Section 2.5 for notations.

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We fix R ą 0 big enough. With D ˘" |D ˘|1 , we introduce the following modification of the test functions φ η : 14

Φ η " ph η, R r ψ ˘q ˝f D˘1 V η, R pB 1 ¯D˘q .
As in Lemma 10.1, the functions Φ η are measurable and satisfy (12.15)

Proof. Since X is a simplicial tree and η ă 1, we have V w, η, R " B ˘pw, Rq for every w P B 1 ˘D¯.

As seen above, there exists c R ą 0 depending only on R such that if P B ˘pw, Rq and 1 P G X satisfy dp , 1 q ď c R , then 1 P B ˘pw, Rq. Hence (see Section 3.1) the characteristic function

1 V η, R pB 1
˘D¯q is c R -locally constant, thus β-Hölder-continuous by Remark 3.2. By Assumption (1) in the statement of Theorem 12.17, the denominator of

h η, R pwq " 1 µ W ˘pwq pB ˘pw, Rqq
is at least a positive constant depending only on R, hence h η, R is bounded by a constant depending only on R. Since the map 1 B ˘pw,Rq is c R -locally constant, so is the map h η, R . The result then follows from Lemma 12.18 and Equation (3.1). l

In order to prove Equation (12.13), as in the proofs of Theorems 12.7 and 11.8, for all N P N, we estimate in two ways the quantity

I η pN q " N ÿ n"0 e δc n ÿ γPΓ ż PG X Φ ή pg ´tn{2u q Φ ὴ pg rn{2s γ ´1 q d r m c p q . (12.16)
On the one hand, as in order to obtain Equation (12.6), using now Assumption (2) in the statement of Theorem 12.17 on the exponential mixing for the discrete time geodesic flow, a geometric sum argument and Equations (12.14) and (12.15), we have

I η pN q "
e δcpN `1q pe δc ´1q }m c } ´żB 1 `D´r

ψ ´dr σ `żB 1 ´D`r ψ `dr σ ´`Ope ´κ N } r ψ ´}β } r ψ `}β q ¯. (12.17)
On the other hand, exchanging the summations over γ and n in the definition of I η pN q, we have

I η pN q " ÿ γPΓ N ÿ n"0 e δc n ż G X Φ ή pg ´tn{2u q Φ ὴ pg rn{2s γ ´1 q d r m c p q .
14 See Equation (10.1) for the definition of h η, R , that simplifies as h η, R pwq " pµ W ˘pwq pB ˘pw, Rqqq ´1 since X is simplicial, as seen in Equation (11.24).

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With the simplifications in Step 3T of the proof of Theorem 11.1 given by the proof of Theorem 11.8, if η ă 1 2 , if P G X belongs to the support of Φ ή ˝g´tn{2u Φ ὴ ˝grn{2s ˝γ´1 , setting w ´" f D´p q and w `" f γD `p q, we then have λ γ " n, w ˘p0q " α γ p0q and w ´ptn{2uq " w `p´rn{2sq " p0q " α γ ptn{2uq " α γ p´rn{2sq , hence dpw ˘, α γ q " Ope ´λγ q .

Therefore, since r ψ ˘is β-Hölder-continuous, | r ψ ˘pw ˘q ´r ψ ˘pv γ q | " Ope ´βλγ } r ψ ˘}β q .

Note that now Φ η " r ψ ˘˝f D˘φ η , so that

I η pN q " ÿ γPΓ `r ψ ´pv γ q r ψ `pv γ q `Ope ´2βλγ } r ψ ´}β } r ψ `}β q ˘N ÿ n"0 e δc n ż G X
φ ή pg ´tn{2u q φ ὴ pg rn{2s γ ´1 q d r m c p q . Now if η ă 1 2 , Equation (12.13) with κ 0 " mint2β, κu follows as in Steps 3T and 4T of the proof of Theorem 11.1 with the simplifications given by the proof of Theorem 11.8.

The end of the proof of the equidistribution claim of Theorem 12.17 follows from Equation (12.13) as the one of Theorem 12.7 from Equation (12.4).

The counting claim follows from the equidistribution one by taking ψ ˘" 1 ΓV η,R pB 1 ¯D˘q, which has compact support since Γ D ˘zBD ˘is assumed to be compact, and is β-Höldercontinuous by previous arguments. l

Remarks. (i) Assume that r c " 0, that the simplicial tree X 1 with |X 1 | 1 " C ΛΓ is uniform without vertices of degree 2, that L Γ " Z and that Γ is a geometrically finite lattice of X 1 . Then all assumptions of Theorem 12.17 are satisfied by the results of Section 4.4 and by Corollary 9.6. Therefore we have an exponentially small error term in the (joint) equidistribution of the common perpendiculars, and in their counting if Γ D ˘zBD ˘is compact, see Example 12.10 (2).

(ii) Assume in this remark that Assumption (2) of the above theorem is replaced by the assumptions that C ΛΓ is uniform without vertices of degree 2, that L Γ " 2Z, and that there exists β P s0, 1s such that the square of the discrete time geodesic flow on pΓzG even X, m c q is exponentially mixing for the β-Hölder regularity, for instance if Γ is geometrically finite by Corollary 9.6 (2). Then a similar proof (replacing the references to Theorem 11.8 by references to Theorem 11.10) shows that there exists κ 1 ą 0 such that for all ψ ˘P C β c pΓz p G Xq, we have, as n Ñ `8, Let us now consider the metric tree case, for the continuous time geodesic flow, where the main change is to assume a superpolynomial decay of correlations and hence get a superpolynomial error term.

Theorem 12.20. Let pX, λq, Γ, r c and D ˘be as in the beginning of this Section, and let D ˘" |D ˘|λ . Assume that the critical exponent δ c is finite and positive, that the Gibbs measure m c (for the continuous time geodesic flow) is finite and that the skinning measures σ D¯a re finite and nonzero. Assume furthermore that

(1) at least one of the following holds :

' Γ D ˘zBD ˘is compact ' the metric subtree C ΛΓ is uniform and Γ is a lattice of C ΛΓ,
(2) there exists β P s0, 1s such that the continous time geodesic flow on pΓzG X, m c q has superpolynomial decay of β-Hölder correlations.

Then for every n P N there exists k P N such that for all ψ ˘P C k, β Remark. Assume that r c " 0, that the metric tree C ΛΓ is uniform, either that ΓzX is finite and the length spectrum L Γ of Γ is 2-Diophantine or that Γ is a geometrically finite lattice of C ΛΓ and that L Γ is 4-Diophantine. Then all assumptions of Theorem 12.20 are satisfied by the results of Section 4.4 and by Corollary 9.10. Therefore we have a superpolynomially small error term in the (joint) equidistribution of the common perpendiculars (and in their counting if Γ D ˘zBD ˘is compact).

Proof. The proof is similar to the one of Theorem 12.17, except that since the time is now continuous, we need to regularise our test functions in the time direction in order to obtain the regularity required for the application of the assumption on the mixing rate. We again use the simplifying notation λ γ " λ e,γ , α γ " α e,γ , α γ " α é,γ , α γ " α γ´1 ,e and r σ ˘" r σ D¯. We fix n P N ´t0u. Using the rapid mixing property, there exists a regularity k such that for all ψ, ψ 1 P C (12.19) In order to prove this formula, we introduce modified test functions, making them with bounded Hölder-continuous derivatives up to order k (by a standard construction) in the time direction (the stable leaf and unstable leaf directions remain discrete). We fix R ą 0 big enough.

For every η P s0, 1r , there exists a map x 1 η : R Ñ r0, 1s which has bounded β-Höldercontinuous derivatives up to order k, which is equal to 0 if t R r´η, ηs and to 1 if t P r´η e ´η, η e ´ηs (when k " 0, just take x 1 η to be continuous and linear on each remaining segment r´η, ´η e ´ηs and rη e ´η, ηs), such that, for some constant κ 1 ą 0,

} x 1 η } k, β " Opη ´κ1 q .
Using leafwise this regularisation process, there exists

χ η, R P C k, β b pG Xq such that ' }χ η, R } k, β " Opη ´κ1 q, ' 1 V η e ´η , R pB 1 ¯D˘q ď χ η, R ď 1 V η, R pB 1 ¯D˘q , ' for every w P B 1 ¯D˘, we have ż V w, η, R
χ η, R dν w " ν w pV w, η, R q e ´Opηq " ν w pV w, η e ´η , R q e Opηq .

As in the proof of Theorem 12.7 in the manifold case, the new test functions are defined, with

H η, R : w P B 1 ¯D˘Þ Ñ 1 ş V w, η, R χ η, R dν w , by Φ η " pH η, R r ψ ˘q ˝f D˘χ η, R : G X Ñ R . Let p Φ η " H η, R ˝f D˘χ η, R , so that Φ η " r ψ ˘˝f D˘p Φ η .
By the last two properties of the regularised maps χ η, R , we have, with φ η defined as in Equation (10.4), φ η e ´η e ´Opηq ď p Φ η ď φ η e Opηq . (12.20)

By Assumption (1), if R is large enough, by the definitions of the measures ν w , the denominator of H η, R pwq is at least c η where c ą 0. As in the proof of Theorem 12.7, there exists κ 2 ą 0 such that ż

G X Φ η d r m c " ż B 1 ¯D˘r ψ ˘dr σ ānd }Φ η } k, β " Opη ´κ2 } r ψ ˘}k, β q . 207 19/12/2016
We again estimate in two ways as T Ñ `8 the quantity (12.21) Note that as T Ñ `8,

I η pT q " ż T 0 e δc t ÿ γPΓ ż PG X Φ ή pg ´t{2 q Φ ὴ pg t{2 γ ´1 q d r m c p q dt .
e ´δc T ż T 1 e δc t t ´N n dt " e ´δc T ż T {2 1 e δc t t ´N n dt `e´δc T ż T T {2
e δc t t ´N n dt " Ope ´δc T {2 q `OpT ´N n`1 q " OpT ´pN ´1q n q .

Using Equation (12.18), an integration argument and the above two properties of the test functions, we hence have

I η pT q " e δc T δ c }m c } ´żB 1 `D´r ψ ´dr σ `żB 1 ´D`r ψ `dr σ ´`OpT ´pN ´1q n η ´2κ 2 } r ψ ´}k, β } r ψ `}k, β q ¯.
(12.22) As in Step 3T of the proof of Theorem 11.1, for all γ P Γ and t ą 0 big enough, if P G X belongs to the support of Φ ή ˝g´t{2 Φ ὴ ˝gt{2 ˝γ´1 (which is contained in the support of φ ή ˝g´t{2 φ ὴ ˝gt{2 ˝γ´1 ), then we may define w ´" f D´p q and w `" f γD `p q.

By the property (iii) in Step 3T of the proof of Theorem 11.1, the generalised geodesic lines w ´and α γ coincide, besides on s ´8, 0s, at least on r0, t 2 ´ηs, and similarly, w `and α γ coincide, besides on r0, `8r , at least on r´t 2 `η, 0s. Therefore, by an easy change of variable and since

| t 2 ´λγ 2 | ď η, dpw ´, α γ q ď ż `8 t 2
´η dpw ´psq, α γ psqq e ´2s ds ď e ´2p t 2 ´ηq ż `8 0 2s e ´2s ds " Ope ´tq " Ope ´λγ q .

Similarly, dpw `, α γ q " Ope ´λγ q. Hence since r ψ ˘is β-Hölder-continuous, we have | r ψ ˘pw ˘q ´r ψ ˘pα γ q | " Ope ´βλγ } r ψ ˘}β q .

Therefore, as in the proof of Theorem 12.17, we have

I η pT q " ÿ γPΓ `r ψ ´pα γ q r ψ `pα γ q `Ope ´2βλγ } r ψ ´}β } r ψ `}β q ˘ż T 0 e δ t ż PG X p Φ ή pg ´t{2 q p Φ ὴ pγ ´1g t{2 q d r m c p q dt .
Finally, Equation (12.19) follows as in the end of the proof of Equation (12.4), using Equations (12.20) and (11.16) instead of Equations (12.7) and (11.19), by taking η " T ´n and N " 2prκ 2 s `1q.

The end of the proof of the equidistribution claim of Theorem 12.20 follows from Equation (12.19) as the one of Theorem 12.7 from Equation (12.4).

The counting claim follows from the equidistribution one by taking ψ ˘to be β-Höldercontinuous plateau functions around ΓV η,R pB 1 ¯D˘q . l 208 19/12/2016

We are now in a position to prove one of the counting results in the introduction.

Proof of Theorem 1.9. Let X be the universal cover of Y, with fundamental group Γ for an indifferent choice of basepoint, and let D ˘be connected components of the preimages of Y ȋn X. Assertion (1) of Theorem 1.9 follows from Theorem 12.20 and its subsequent Remark. Assertion (2) of Theorem 1.9 follows from Theorem 12.17 and its subsequent Remarks (ii) and ( i

Geometric applications

In this final Chapter of Part II, we apply the equidistribution and counting results obtained in the previous Chapters in order to study geometric equidistribution and counting problems for metric and simplicial trees concerning conjugacy classes in discrete isometry groups and closed orbits of the geodesic flows.

Orbit counting in conjugacy classes for groups acting on trees

In this Section, we study the orbital counting problem for groups acting on metric or simplicial trees when we consider only the orbit points by elements in a given conjugacy class. We refer to the Introduction for motivations and previously known results for manifolds (see [START_REF] Huber | Über eine neue Klasse automorpher Funktionen und ein Gitterpunktproblem in der hyperbolischen Ebene[END_REF] and [START_REF] Parkkonen | On the hyperbolic orbital counting problem in conjugacy classes[END_REF]) and graphs (see [Dou] and [KeS]). The main tools we use are Theorem 12.8 for the metric tree case and Theorem 12.17 for the simplicial tree case, as well as their error terms.

We in particular obtain a much more general version of Theorem 1.12 in the Introduction.

Let pX, λq be a locally finite metric tree without terminal vertices, let X " |X| λ be its geometric realisation, let x 0 P V X and let Γ be a nonelementary discrete subgroup of AutpX, λq. 1 Let r c : EX Ñ R be a Γ-invariant system of conductances, let r F c and F c be its associated potentials on T 1 X and ΓzT 1 X respectively, and let δ c " δ Γ, F c be its critical exponent. 2 Let pµ x q xPX (respectively pµ x q xPV X ) be Patterson densities for the pairs pΓ, F c q, and let r m c " r m Fc and m c " m Fc be the associated Gibbs measures on G X and ΓzG X (respectively G X and ΓzG X) for the continuous time geodesic flow (respectively the discrete time geodesic flow, when λ " 1). 3 Recall that the virtual centre Z virt pΓq of Γ is the finite (normal) subgroup of Γ consisting of the elements γ P Γ acting by the identity on the limit set ΛΓ of Γ in B 8 X, see for instance [START_REF] Champetier | L'espace des groupes de type fini[END_REF]§5.1]. If ΛΓ " B 8 X (for instance if Γ is a lattice), then Z virt pΓq " teu.

For any nontrivial element γ in Γ with translation length λpγq in X, let C γ be 13.2 Equidistribution and counting of closed orbits on metric and simplicial graphs (of groups)

Classically, an important characterization of the Bowen-Margulis measure on compact negatively curved Riemannian manifolds is that it coincides with the weak-star limit of properly normalised sums of Lebesgue measures supported on periodic orbits, see [START_REF] Bowen | Periodic orbits for hyperbolic flows[END_REF]. Under much weaker assumptions than compactness, this result was extended to CATp´1q spaces with zero potential in [START_REF] Roblin | Ergodicité et équidistribution en courbure négative[END_REF] and to Gibbs measures in the manifold case in [START_REF] Paulin | Equilibrium states in negative curvature[END_REF]Thm. 9.11]. As a corollary of the simultaneous equidistribution results from Chapter 11, we prove in this Section the equidistribution towards the Gibbs measure of weighted closed orbits in quotients of metric and simplicial graphs of groups and as a corollary, in the standard manner, we obtain asymptotic counting results for weighted (primitive) closed orbits.

Let pX, λq be a locally finite metric tree without terminal vertices, and X " |X| λ its geometric realisation. Let Γ be a nonelementary discrete subgroup of AutpX, λq. Let r c : EX Ñ R be a Γ-invariant system of conductances, and c : ΓzEX Ñ R its induced function.

Given a periodic orbit g of the geodesic flow on ΓzG X, if pe 1 , . . . , e k q is the sequence of edges followed by g, we denote by L g the Lebesgue measure along g, by λpgq the length of g and by cpgq its period for the system of conductances c: λpgq "

k ÿ i"1 λpe i q and cpgq " k ÿ i"1 λpe i q cpe i q .
Let Perptq be the set of periodic orbits of the geodesic flow on ΓzG X and let Per 1 ptq be the subset of prime periodic orbits. We conjecture that if Γ is geometrically finite and if its length spectrum is 4-Diophantine6 , then for all n P N and β P s0, 1s, there exists k P N and an error term of the form Opt n }ψ} k, β q for these equidistribution claims evaluated on any ψ P C b, β c pΓzG Xq. But since we will not need this result and since the proof is likely to be very long, we do not address the problem here.

Proof. Let r F c and F c be the potentials on T 1 X and ΓzT 1 X respectively associated with 7 c, and note that the period 8 of a periodic orbit g for the geodesic flow on ΓzG X satisfies cpgq " L g pF 7 c q " Per Fc pγq , where F 7 c is the composition of the canonical map ΓzG X Ñ ΓzT 1 X with F c : ΓzT 1 X Ñ R, and γ P Γ is the loxodromic element of Γ whose conjugacy class corresponds to g.

Let H Γ,t be the subset of Γ that consists of loxodromic elements whose translation length is at most t, and let H 1 Γ,t be the subset of H Γ,t that consists of the primitive elements. The first claim is equivalent to the following assertion: we have

δ c e ´δc t ÿ γPH 1 Γ,t e Per Fc pγq L g á m c }m c } (13.2)
as t Ñ `8. We proceed with the proof of the convergence claimed in Equation (13.2) as in [START_REF] Paulin | Equilibrium states in negative curvature[END_REF]Thm. 9.11]. We first prove that

ν 2 t " δ c e ´δc t ÿ γPH Γ,t e Per Fc pγq L g á m c }m c } . (13.3)
We then refer to Step 2 of the proof of [START_REF] Paulin | Equilibrium states in negative curvature[END_REF]Thm. 9.11] for the fact that the contribution of the periods that are not primitive is negligible. Although the proof in loc. cit. is written for manifolds, the arguments are directly applicable for any CATp´1q space X and potential F satisfying the HC-property. 9 In particular, the use of Proposition 5.13 (i) and (ii) of loc. cit. in the proof of Step 2 in loc. cit. is replaced now by the use of Theorem 4.5 ( 1) and ( 4) respectively.

Let us fix x P X. Let The measures ν t weak-star converge to µ x b µ x as t Ñ `8 by Corollary 11.2 (taking in its statement y " x).

Let γ ˘be the attracting and repelling fixed points of any loxodromic element γ P Γ. Let

ν 3 t " δ c }m c } e ´δc t ÿ γPH Γ, t e Per Fc pγq ∆ γ ´b ∆ γ `.
Since X is an R-tree, every element γ P Γ such that x P sγ ´1x, γxr is loxodromic, and we have dpx, γxq " λpγq and

ż γx x r F c " Per Fc pγq .
If furthermore dpx, γxq is big, then γ ´1x and γx are respectively close to γ ´and γ `in X Y B 8 X.

Hence, for every continuous map ψ : pX Y B 8 Xq 2 Ñ r0, `8r with support contained in V pxq, and for every ą 0, if t is big enough, we have e ´ ν t pψq ď ν 3 t ψq ď e ν t pψq .

9 See Definition 3.4.

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Using Hopf's parametrisation with basepoint x, we have ν 3 t b ds " }m c } ν 2 t , and the support of any continuous function with compact support on G X may be covered by finitely many open sets V pxq ˆR where x P X. The proof of the claim (13.3) now follows as in Step 1 of the proof of Theorem [START_REF] Paulin | Equilibrium states in negative curvature[END_REF]Thm. 9.11].

The second claim of Theorem 13.3 follows from the first one in the same way as in [START_REF] Paulin | Equilibrium states in negative curvature[END_REF]Thm. 9.11] to which we refer for the proof. l

In a similar way, replacing in the above proof Corollary 11.2 of Theorem 11.1 by the similar corollary of Theorem 11.8 with D ´" pγxq γPΓ and D ´" pγyq γPΓ for any x, y P V X, we get the following analogous result for simplicial trees.

Theorem 13.4. Let X be a locally finite simplicial tree without terminal vertices, let Γ be a nonelementary discrete subgroup of AutpXq and let r c : EX Ñ R be a Γ-invariant system of conductances. Assume that the critical exponent δ c of c is finite and positive and that the Gibbs measure m c is finite and mixing for the discrete time geodesic flow. As n Ñ `8, the measures e δc ´1 e δc e ´δcn e cpgq L g pgq converge to mc }mc} for the weak-star convergence of measures. If Γ is geometrically finite, the convergence holds for narrow convergence. l

In the special case when ΓzX is a compact graph and F " 0, the following immediate corollary of Theorem 13.3 is proved in [Gui], and it follows from the results of [ParP]. 10 There are also some works on non-backtracking random walks with related results. For example, for regular finite graphs, [START_REF] Lubotzky | Ramanujan conjectures and explicit construction of expanders[END_REF] and [START_REF] Friedman | On the second eigenvalue and random walks in random d-regular graphs[END_REF] (see [START_REF] Friedman | A proof of Alon's second eigenvalue conjecture and related problems[END_REF]Lem. 2.3]) give an expression of the irreducible trace which is the number of closed walks of a given length.

Corollary 13.5. Let pX, λq be a locally finite metric tree without terminal vertices. Let Γ be a geometrically finite discrete subgroup of AutpX, λq. Let c : EX Ñ R be a Γ-invariant system of conductances, with finite and positive critical exponent δ c .

(1) If the Gibbs measure m c is finite and mixing for the continuous time geodesic flow, then Let F v be the completion of F with respect to this distance. The valuation v of F uniquely extends to a (normalised discrete) valuation on F v , again denoted by v.

Example 14.1. Let K " F q pY q be the field of rational functions in one variable Y with coefficients in a finite field F q of order a positive power q of a positive prime p in Z, let F q rY s be the ring of polynomials in one variable Y with coefficients in F q , and let v 8 : K ˆÑ Z be the valuation at infinity of K, defined on every P {Q P K with P P F q rY s and Q P F q rY s ´t0u by v 8 pP {Qq " deg Q ´deg P .

The absolute value associated with v 8 is

|P {Q| 8 " q deg P ´deg Q .
The completion of K for v 8 is the field K v8 " F q ppY ´1qq of formal Laurent series in one variable Y ´1 with coefficients in F q . The elements x in F q ppY ´1qq are of the form x " ÿ iPZ x i Y ´i where x i " 0 P F q for i P Z small enough. The valuation at infinity of F q ppY ´1qq extending the valuation at infinity of F q pY q is v 8 pxq " supti P Z : @ j ă i, x j " 0u , that is,

v 8 p 8 ÿ i"i 0 x i Y ´iq " i 0 if x i 0 ‰ 0.
The valuation ring of v 8 is the ring O v8 " F q rrY ´1ss of formal power series in one variable Y ´1 with coefficients in F q . The element π v8 " Y ´1 is a uniformizer of v 8 , the residual field

O v8 {π v8 O v8 of v 8 is k v8 " F q .
Example 14.2. Given a positive prime p P Z, the field of p-adic numbers Q p is the completion of Q with respect to the absolute value | ¨|p of the p-adic valuation v p defined by setting v p pp n a b q " n when n P Z, a, b P Z ´t0u are not divisible by p. Then the valuation ring O vp " Z p of Q p is the closure of Z for the absolute value | ¨|p , π vp " p is a uniformiser, and the residual field is k vp " Z p {pZ p " F p , a finite field of order p.

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A field endowed with a valuation is a non-Archimedean local field if it is complete with respect to its absolute value and if its residual field is finite. 1 Its valuation ring is then a compact open additive subgroup. Any non-Archimedean local field is isomorphic to a finite extension of the p-adic field Q p for some prime p, or to an extension of F p with transcendence degree 1 for some prime p.

The basic case of extensions of F p with transcendence degree 1 is described in Example 14.1 above, and the general case of the discussed in Section 14.2 below. The geometer reader may skip Section 14.2 and use only Example 14.1 in the remainder of Part III (using g " 0 when the constant g occurs).

Global function fields

In this Section, we fix a finite field F q with q elements, where q is a positive power of a positive prime p P Z, and we recall the definitions and basic properties of a function field K over F q , its genus g, its valuations v, its completion K v for the associated absolute value | ¨|v and the associated affine function ring R v . See for instance [Gos, Ros] for the content of this Section.

Let K be a (global) function field over F q , which can be defined in two equivalent ways as (1) the field of rational functions on a geometrically irreducible smooth projective curve C over F q , or (2) an extension of F q of transcendence degree 1, in which F q is algebraically closed.

There is a bijection between the set of closed points of C and the set of (normalised discrete) valuations of its function field K, the valuation of a given element f P K being the order of the zero or the opposite of the order of the pole of f at the given closed point. We fix such an element v from now on. We denote by g the genus of the curve C.

In the basic Example 14.1, C is the projective line P 1 over F q which is a curve of genus g " 0, and the closed point associated with the valuation at infinity is the point at infinity r1 : 0s.

We denote by K v the completion of K for v, and by O v " tx P K v : vpxq ě 0u the valuation ring of (the unique extension to K v ) of v. We choose a uniformizer π v of v. We denote by k v " O v {π v O v the residual field of v, which is a finite field of order

q v " |k v | .
The field k v is from now on identified with a fixed lift in O v (see for instance [START_REF] Colmez | Corps locaux[END_REF]Théo. 1.3]), and is an extension of the field of constants F q . The degree of this extension is denoted by deg v, so that q v " q deg v .

We denote by | ¨|v the (normalised) absolute value associated with v : for every x P K v , we have |x| v " pq v q ´vpxq " q ´vpxq deg v .

1 There are also two Archimedean local fields C and R, see for example [Cas].

223 19/12/2016 Every element x P K v is 2 a (converging) Laurent series x " ř iPZ x i pπ v q i in the variable π v over k v , where x i P k v is zero for i P Z small enough. We then have |x| v " pq v q ´suptjPZ : @ iăj, x i "0u , (14.1)

and O v consists of the (converging) power series x " ř iPN x i pπ v q i (where

x i P k v ) in the variable π v over k v .
We denote by R v the affine algebra of the affine curve C ´tvu, consisting of the elements of K whose only poles are at the closed point v of C. Its field of fractions is equal to K, hence we will often write elements of K as x{y with x, y P R v and y ‰ 0. In the basic Example 14.1, we have R v8 " F q rY s. Note that R v X O v " F q , (14.2)

since the only rational functions on C whose only poles are at v and whose valuation at v is nonnegative are the constant ones. We have (see for instance [START_REF] Serre | Arbres, amalgames, SL 2 . 3ème éd. corr[END_REF]II.2 Notation], [START_REF] Goss | Basic structures of function field arithmetic[END_REF]page 63]) pR v q ˆ" pF q q ˆ. (14.

3)

The following result is immediate when C " P 1 , since then R v `Ov " K v .

Lemma 14.3. The dimension of the quotient vector space K v {pR v `Ov q over F q is equal to the genus g of C.

Proof. (J.-B. Bost) We refer for instance to [START_REF] Serre | Faisceaux algébriques cohérents[END_REF] for background on sheaf cohomology. We denote in the same way the valuation v and the corresponding closed point on C.

Let O " K X O v be the discrete valuation ring of v restricted to K. Since K is dense in K v and O v is open and contains 0, we have K v " K `Ov . Therefore the canonical map K{pR v `Oq Ñ K v {pR v `Ov q is a linear isomorphism over F q . Let us hence prove that dim Fq K{pR v `Oq " g.

In what follows, V ranges over the affine Zariski-open neighbourhoods of v in C, ordered by inclusion. Let O C be the structural sheaf of C. Note that by the definition of R v , since the zeros of elements of K ˆare isolated and by the relation between valuations of K and closed points of C,

R v " H 0 pC ´tvu, O C q, K " lim Ý Ñ V H 0 pV ´tvu, O C q and O " lim Ý Ñ V H 0 pV , O C q .
Since V and C ´tvu are affine curves, we have H 1 pC ´tvu, O C q " H 1 pV , O C q " 0. By the Mayer-Vietoris exact sequence since tC ´tvu, V u covers C, we hence have an exact sequence

H 0 pC, O C q Ñ H 0 pC ´tvu, O C q ˆH0 pV , O C q Ñ H 0 pV ´tvu, O C q Ñ H 1 pC, O C q . Therefore K{pR v `Oq " lim Ý Ñ V H 0 pV ´tvu, O C q{ `H0 pC ´tvu, O C q `H0 pV , O C q » H 1 pC, O C q .
2 See for instance [START_REF] Colmez | Corps locaux[END_REF]Coro. 1.6] 224

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Since dim Fq H 1 pC, O C q " g by one definition of the genus of C, the result follows.

l

Recall that R v is a Dedekind ring. 3 In particular, every nonzero ideal (respectively fractional ideal) I of R v may be written uniquely as I " ś p p vppIq where p ranges over the prime ideals in R v and v p pIq P N (respectively v p pIq P Z), with only finitely many of them nonzero. By convention I " R v if v p pIq " 0 for all p. For every x, y P R v (respectively x, y P K), we denote by xx, yy " x R v `y R v the ideal (respectively fractional ideal) of R v generated by x, y. If I, J are nonzero fractional ideals of R v , we have We define the (absolute) norm of a nonzero ideal I "

I X J "
ś p p vppIq of R v by N pIq " rR v : Is " |R v {I| " ź p q vppIq deg p ,
where deg p is the degree of the field R v {p over F q , so that N pR v q " 1. By convention N p0q " 0. This norm is multiplicative:

N pIJq " N pIqN pJq ,
and the norm of a nonzero fractional ideal I " ś p p vppIq of R v is defined by the same formula. Note that if paq is the principal ideal in R v generated by a, we define N paq " N `paq ˘. We have (see for instance [START_REF] Goss | Basic structures of function field arithmetic[END_REF]page 63])

N paq " |a| v .
(14.4) Dedekind's zeta function of K is (see for instance [START_REF] Goss | Basic structures of function field arithmetic[END_REF]§7.8] or [START_REF] Rosen | Number theory in function fields[END_REF]§5])

ζ K psq " ÿ I 1 N pIq s
if Re s ą 1, where the summation is over the nonzero ideals I of R v . By for instance [START_REF] Rosen | Number theory in function fields[END_REF]§5], it has an analytic continuation on C ´t0, 1u with simple poles at s " 0, s " 1. It is actually a rational function of q ´s. In particular, if K " F q pY q, then (see [START_REF] Rosen | Number theory in function fields[END_REF]Theo. 5.9])

ζ FqpY q p´1q " 1 pq ´1qpq 2 ´1q . (14.5)

We denote by Haar Kv the Haar measure of the (abelian) locally compact topological group pK v , `q, normalised so that Haar Kv pO ν q " 1. 4 The Haar measure scales as follows under multiplication: for all λ, x P K v , we have d Haar Kv pλxq " |λ| v d Haar Kv pxq .

(14.6)

Note that any fractional ideal I of R v is a discrete subgroup of pK v , `q, and we will again denote by Haar Kv the Haar measure on the compact group K v {I which is induced by the above normalised Haar measure of K v .

Haar Kv pK v {Iq " q g´1 N pIq .

Proof. By the scaling properties of the Haar measure, we may assume that I is an ideal in R v . By Lemma 14.3, we have Card K v {pR v `Ov q " q g . By Equation (14.2) and by the normalisation of the Haar measure, we have

Haar Kv pR v `Ov q{R v " Haar Kv O v {pR v X O v q " Haar Kv O v {F q " 1 q .
Hence Haar Kv pK v {R v q " Card `Kv {pR v `Ov q ˘Haar Kv pR v `Ov q{R v " q g´1 .

Since Haar Kv pK v {Iq " N pIq Haar Kv pK v {R v q, the result follows. l

226 19/12/2016
Chapter 15

Bruhat-Tits trees and modular groups

In this Chapter, we give the background information and preliminary results on the main link between the geometry and the algebra used for our arithmetic applications: the (discrete time) geodesic flow on quotients of Bruhat-Tits trees by arithmetic lattices.

We denote the image in PGL 2 of an element

ˆa b c d ˙P GL 2 by " a b c d  P PGL 2 .

Bruhat-Tits trees

Let K v be a non-Archimedean local field, with valuation v, valuation ring O v , choice of uniformiser π v , and residual field k v of order q v (see Section 14.1 for definitions).

In this Section, we recall the construction and basic properties of the Bruhat-Tits tree X v of pPGL 2 , K v q, see for instance [Tit]. We use its description given in [START_REF] Serre | Arbres, amalgames, SL 2 . 3ème éd. corr[END_REF], to which we refer for proofs and further information.

An O v -lattice Λ in the K v -vector space K v ˆKv is a rank 2 free O v -submodule of K v ˆKv , generating K v ˆKv as a vector space. The Bruhat-Tits tree X v of pPGL 2 , K v q is the graph whose set of vertices V X v is the set of homothety classes (under pK v q ˆ) rΛs of O v -lattices Λ in K v ˆKv , and whose non-oriented edges are the pairs tx, x 1 u of vertices such that there exist representatives Λ of x and Λ 1 of x 1 for which Λ Ă Λ 1 and Λ 1 {Λ is isomorphic to O v {π v O v . If K is any field endowed with a valuation v whose completion is K v , then the similarly defined Bruhat-Tits tree of pPGL 2 , Kq coincides with X v , see [START_REF] Serre | Arbres, amalgames, SL 2 . 3ème éd. corr[END_REF]p. 71].

The graph X v is a regular tree of degree |P 1 pk v q| " q v `1. In particular, the Bruhat-Tits tree of pPGL 2 , Q p q is regular of degree p `1, and if K v " F q ppY ´1qq and v " v 8 , then the Bruhat-Tits tree X v of pPGL 2 , K v q is regular of degree q `1. More generally, if K v is the completion of a function field over F q endowed with a valuation v as in Section 14.2, then the Bruhat-Tits tree of pPGL 2 , K v q is regular of degree q v `1.

The standard base point ˚v of X is the homothety class rO v ˆOv s of the O v -lattice O v ˆOv , generated by the canonical basis of K v ˆKv . In particular, we have dp˚v, rO v ˆxO v sq " |vpxq| (15.1)

for every x P pK v q ˆ. The link lkp˚vq " ty P V X v : dpy, ˚vq " 1u of ˚v in X v identifies with the projective line P 1 pk v q.

227 19/12/2016

The left linear action of GL 2 pK v q on K v ˆKv induces a faithful, vertex-transitive left action by automorphisms of PGL 2 pK v q on X v . The stabiliser in PGL 2 pK v q of ˚v is PGL 2 pO v q. We will hence identify PGL 2 pK v q{ PGL 2 pO v q with V X v by the map g PGL 2 pO v q Þ Ñ g ˚v.

We identify the projective line P 1 pK v q with K v Y t8u using the map K v px, yq Þ Ñ x y , so that 8 " r1 : 0s .

The projective action of GL 2 pK v q or PGL 2 pK v q on P 1 pK v q is the action by homographies

1 on K v Y t8u, given by pg, zq Þ Ñ g ¨z " a z`b c z`d if g " ˆa b c d ˙P GL 2 pK v q, or g " " a b c d  P PGL 2 pK v q.
As usual we define 8 Þ Ñ a c and ´d c Þ Ñ 8. There exists a unique homeomorphism between the boundary at infinity B 8 X v of X v and P 1 pK v q such that the (continuous) extension to B 8 X v of the isometric action of PGL 2 pK v q on X v corresponds to the projective action of PGL 2 pK v q on P 1 pK v q. From now on, we identify B 8 X v and P 1 pK v q by this homeomorphism. Under this identification, O v consists of the positive endpoints `of the geodesic lines of X v with negative endpoint ´" 8 that pass through the vertex ˚v (see the picture below).

8 B 8 X v 0 O v 0 0 ˚v x 0 x 2 0 x 1 x i0 x P K v H 8
Let H 8 be the horoball centred at 8 P B 8 X v whose associated horosphere passes through ˚v. There is a unique labeling of the edges of X v by elements of P 1 pk v q " k v Y t8u such that • the label of any edge of X v pointing towards 8 P B 8 X v is 8, • for any x " ř iPZ x i pπ v q i P K v , the sequence px i q iPZ is the sequence of the labels of the (directed) edges that make up the geodesic line s8, xr oriented from 8 towards x • x 0 is the label of the edge of s8, xr exiting the horoball H 8 . We refer to [START_REF] Paulin | Groupe modulaire, fractions continues et approximation diophantienne en caractéristique p[END_REF]Sect. 5] when K v " F q ppY ´1qq and v " v 8 . For all η, η 1 P K v " B 8 X v ´t8u, we have In particular, the Hölder norms }ψ} β, | ¨´¨|v and }ψ} β 1 , d H8 of a function ψ : K v Ñ R, respectively for the distance px, yq Þ Ñ |x ´y| v and d H8 on K v , are related by the following formula: @ β P s0, 1 ln q v s, }ψ} β, | ¨´¨|v " }ψ} β ln qv, d H8 .

|η ´η1 | v " d H8 pη, η 1 q ln qv
(15.

3)

The group PGL 2 pK v q acts simply transitively on the set of ordered triples of distinct points in B 8 X v " P 1 pK v q. In particular, it acts transitively on the space G X v of discrete geodesic lines in X v . The stabiliser under this action of the geodesic line (from 8 " r1 : 0s to 0 " r0 : 1s)

˚: n Þ Ñ rO v ˆpπ v q ´nO v s is the maximal compact-open subgroup ApO v q " ! " a 0 0 d  : a, d P pO v q ˆ)
of the diagonal group

ApK v q " ! " a 0 0 d  : a, d P pK v q ˆ) .
We will hence identify PGL 2 pK v q{ApO v q with G X v by the mapping r Ξ :

gApO v q Þ Ñ g ˚. Define a v " " 1 0 0 π ´1 v  ,
which belongs to ApK v q and centralises ApO v q. The homeomorphism r Ξ is equivariant for the actions on the left of PGL 2 pK v q on PGL 2 pK v q{ApO v q and G X v . It is also equivariant for the actions on PGL 2 pK v q{ApO v q under translations on the right by pa v q Z and on G X v under the discrete geodesic flow pg n q nPZ : for all n P Z and x P PGL 2 pK v q{ApO v q, we have r Ξ px a v n q " g n r Ξ pxq .

(15.4) Furthermore, the stabiliser in PGL 2 pK v q of the ordered pair of endpoints p ˚" 8, ˚" 0q of ˚in B 8 X v " P 1 pK v q is ApK v q. Therefore any element γ P PGL 2 pK v q which is loxodromic on X v is diagonalisable over K v . By [START_REF] Serre | Arbres, amalgames, SL 2 . 3ème éd. corr[END_REF]page 108], the translation length on X v of γ 0 " " a 0 0 d  is λpγ 0 q " |vpaq ´vpdq| .

(15.5)

Note that if r γ 0 " ˆa 0 0 d ˙P GL 2 pK v q is a representative of γ 0 such that det r γ 0 P pO v q ˆ, then as 0 " vpdet r γ 0 q " vpadq " vpaq `vpdq and since vpaq ‰ vpdq if λpγ 0 q ‰ 0, we have vptr r γ 0 q " vpa `dq " maxtvpaq, vpdqu and vpaq ´vpdq " 2vpaq " ´2vpdq. Thus, λpγ 0 q " 2|vptr r γ 0 q| . (15.6) By conjugation, this formula is valid if γ 0 P PGL 2 pK v q is loxodromic on X v and represented by r γ 0 P GL 2 pK v q such that det r γ 0 P pO v q ˆ.

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Let H be a horoball in X v whose boundary is contained in V X v and whose point at infinity ξ is different from 8. The height of H is ht 8 pH q " maxtβ 8 px, ˚vq : x P BH u P Z , which is the signed distance between H 8 and H . 2 It is attained at the intersection point with BH of the geodesic line from 8 to ξ, which is then called the highest point of H . Note that the height of H is invariant under the action of the stabiliser of H 8 in PGL 2 pK v q on the set of such horoballs H .

The following lemma is a generalisation of [START_REF] Paulin | Groupe modulaire, fractions continues et approximation diophantienne en caractéristique p[END_REF]Prop. 6.1] that covers the particular case of K " F q pY q and v " v 8 .

Lemma 15.1. Assume that K v is the completion of a function field K over F q endowed with a valuation v, with associated affine function ring R v . For every γ "

" a b c d  P PGL 2 pKq
with a, b, c, d P K such that ad ´bc P pO v q ˆand c ‰ 0, the image of H 8 by γ is the horoball centred at a c P K Ă K v " B 8 X v ´t8u with height ht 8 pγH 8 q " ´2 vpcq .

Proof. It is immediate that γ8 " a c under the projective action. Up to multiplying γ on the left by "

1 ´a c 0 1  P PGL 2 pKq, which does not change c nor the height of γH 8 , we may assume that a " 0 and that b has the form c ´1u with u " bc ´ad P pO v q ˆ. Multiplying γ on the right by " 1 ´d c 0 1  P PGL 2 pKq preserves γH 8 and does not change a " 0, b " c ´1u or c. Hence we may assume that d " 0. Since γ then exchanges the points 8 and 0 in B 8 X v , the highest point of γH 8 is γ˚v. Assuming first that 0, γ˚v, ˚v, 8 are in this order on the geodesic line from 0 to 8, we have by Equation (15.1) ht 8 pγH 8 q " dp˚v, γ˚vq " dprO v ˆOv s, rc ´1uO v ˆcO v sq " dprO v ˆOv s, rO v ˆc2 O v sq " ´vpc 2 q " ´2 vpcq .

If 0, γ˚v, ˚v, 8 are in the opposite order, then the same computation holds, up to replacing the distance d by its opposite ´d. l

Modular graphs of groups

Let K be a function field over F q , let v be a (normalised discrete) valuation of K, let K v be the completion of K associated with v, and let R v be the affine function ring associated with v (see Section 14.2 for definitions).

The group Γ v " PGL 2 pR v q is a lattice in the locally compact group PGL 2 pK v q, and a lattice 3 of the Bruhat-Tits tree X v of pPGL 2 , K v q, called the modular group at v of K. The quotient graph Γ v zX v is called the modular graph at v of K, and the quotient graph of groups 4 Γ v z zX v is called the modular graph of groups at v of K. We refer to [START_REF] Serre | Arbres, amalgames, SL 2 . 3ème éd. corr[END_REF] for background 2 See the definition of signed distance just above Lemma 11.12. 3 See Section 2.7 for a definition. 4 See Section 2.7 for a definition.

230

19/12/2016 information on these objects, and for instance to [START_REF] Paulin | Groupe modulaire, fractions continues et approximation diophantienne en caractéristique p[END_REF] for a geometric treatment when K " F q pY q and v " v 8 . By for instance [START_REF] Serre | Arbres, amalgames, SL 2 . 3ème éd. corr[END_REF], the set of cusps Γ v zP 1 pKq is finite, and Γ v zX v is the disjoint union of a finite connected subgraph containing Γ v ˚v and of maximal open geodesic rays h z p s0, `8rq, for z " Γ v r z P Γ v zP 1 pKq, where h z (called a cuspidal ray) is the image by the canonical projection X v Ñ Γ v zX v of a geodesic ray whose point at infinity in P 1 pKq Ă B 8 X v is equal to r z. Conversely, any geodesic ray whose point at infinity lies in P 1 pKq Ă B 8 X v contains a subray that maps injectively by the canonical projection

X v Ñ Γ v zX v .
The group Γ v " PGL 2 pR v q is a geometrically finite lattice by for instance [START_REF] Paulin | Groupes géométriquement finis d'automorphismes d'arbres et approximation diophantienne dans les arbres[END_REF]. 5 The set of bounded parabolic fixed points of Γ v is exactly P 1 pKq Ă B 8 X v , and the set of conical limit points of Γ v is P 1 pK v q ´P1 pKq.

Let us denote by { Γ v zX v " pΓ v zX v q \ E v Freudenthal's compactification of Γ v zX v by its finite set of ends E v , see [Fre]. This set of ends is indeed finite, in bijection with Γ v zP 1 pKq by the map which associates to z P Γ v zP 1 pKq the end towards which the cuspidal ray h z converges. See for instance [START_REF] Serre | Arbres, amalgames, SL 2 . 3ème éd. corr[END_REF] for a geometric interpretation of E v in terms of the curve C.

Let I v be the set of classes of fractional ideals of R v . The map which associates to an element rx : ys P P 1 pKq the class of the fractional ideal xR v `yR v generated by x, y induces a bijection from the set of cusps ΓzP 1 pKq to I v .

The volume6 of the modular graph of groups can be computed using Equation ( 14.3) and Exercice 2 b) in [START_REF] Serre | Arbres, amalgames, SL 2 . 3ème éd. corr[END_REF]II.2.3]:

0 Γ 0 Γ 1 Γ 2
q `1 q 1 q 1 1 q 1 q where Γ ´1 " PGL 2 pF q q, Γ 1 0 " Γ 0 X Γ ´1 and, for every n P N,

Γ n "
" " a b 0 d  P PGL 2 pF q rY sq : a, d P F q , b P F q rY s, deg b ď n `1* .

Note that even though PGL 2 pK v q has inversions on X v , its subgroup Γ v " PGL 2 pR v q acts without inversion on X v (see for instance [START_REF] Serre | Arbres, amalgames, SL 2 . 3ème éd. corr[END_REF]II.1.3]). In particular, the quotient graph Γ v zX v is then well defined.

Computations of measures for Bruhat-Tits trees

In this Section, we compute explicit expressions for the skinning measures of horoballs and geodesic lines, and for the Bowen-Margulis measures, when considering lattices of Bruhat-Tits trees. See [START_REF] Parkkonen | Counting arcs in negative curvature[END_REF]Section 7] and [START_REF] Parkkonen | Counting and equidistribution in Heisenberg groups[END_REF]Section 4] for analogous computations in the real and complex hyperbolic spaces respectively, and [START_REF] Broise-Alamichel | Dynamique sur le rayon modulaire et fractions continues en caractéristique p[END_REF] for related computations in the tree case.

Let pK v , vq be as in the beginning of Section 15.1. Let Γ be a lattice of the Bruhat-Tits tree X v of pPGL 2 , K v q. Since X v is regular of degree q v `1, the critical exponent of Γ is

δ Γ " ln q v .
(15.8)

We normalise the Patterson density pµ x q xPV Xv of Γ as follows. Let H 8 be the horoball in X v centred at 8 whose associated horosphere passes through ˚v. Let t Þ Ñ x t be the geodesic ray in X v such that x 0 " ˚v and which converges to 8.

8 ˚v BH 8 K v " B 8 X v ´t8u π v O v H 8 x t O v
Hamenstädt's measure 8 associated with H 8 µ H8 " lim tÑ`8

e δ Γ t µ xt " lim tÑ`8 q v t µ xt
is a Radon measure on B 8 X v ´t8u " K v , invariant under all isometries of X v preserving H 8 , since Γ is a lattice. Hence it is invariant under the translations by the elements of K v . By the uniqueness property of Haar measures, µ H8 is a constant multiple of the chosen Haar measure 9 of K v , and we normalise the Patterson density pµ x q xPV Xv so that µ H8 " Haar Kv .

(15.9)

We summarise the various measure computations in the following result.

Proposition 15.2. Let Γ be a lattice of the Bruhat-Tits tree X v of pPGL 2 , K v q, with Patterson density normalised as above.

8 See Equation (7.5). 9 Recall that we normalise the Haar measure of pKv, `q such that HaarK v pOν q " 1.

}µ x } " q v `1 q v
for all x P V X v .

(3) The skinning measure of the horoball H 8 is the projection of the Haar measure of K v : For all ρ P B 1 ˘H8 , we have dr σ H8 pρq " dµ H8 pρ ˘q " d Haar Kv pρ ˘q .

(4) If 8 is a bounded parabolic fixed point of Γ, with Γ 8 its stabiliser in Γ, if D " pγH 8 q γPΓ{Γ8 , we have }σ D } " Haar Kv pΓ 8 zK v q " VolpΓ 8 z zBH 8 q .

(5) Let L be a geodesic line in X v with endpoints L ˘P K v " B 8 X v ´t8u. Then on the set of ρ P B 1 `L such that ρ `P K v " B 8 X v ´t8u and ρ `‰ L ˘, the outer skinning measure of L is (6) Let L be a geodesic line in X v , let Γ L be the stabiliser in Γ of L, and assume that Γ L zL has finite length. Then with D " pγLq γPΓ{Γ L , we have

}σ D } " q v ´1 q v
VolpΓ L z zLq .

Proof.

(1) For every ξ P K v , by the description of the geodesic lines in the Bruhat-Tits tree X v starting from 8 given in Section 15. (2) This Assertion follows from Assertion (1) by a geometric series argument, but we give a direct proof.

As Γ is a lattice, the family pµ x q xPV Xv is actually equivariant under AutpX v q,12 which acts transitively on the vertices of X v , and the stabiliser in AutpX v q of the standard base point ˚v acts transitively on the edges starting from ˚v.

Since X v is pq v `1q-regular, since the set of points at infinity of the geodesic rays starting from ˚v, whose initial edge has endpoint 0 P lkp˚vq " P 1 pk v q, is equal to π v O v , since all geodesic lines from 8 P B 8 X v to points of π v O v Ă B 8 X v pass through ˚v, and by the normalisation of the Patterson density and of the Haar measure, we have

}µ ˚v } " pq v `1q µ ˚v pπ v O v q " pq v `1q µ H8 pπ v O v q " pq v `1q Haar Kv pπ v O v q " q v `1 q v Haar Kv pO v q " q v `1 q v .
(3) This follows from Equation (7.4), and from the normalisation µ H8 " Haar Kv of the Patterson density.

(4) This follows from Assertion (3) and from Equation (8.10) (where the normalisation of the

ψ P C β b p p G Xq, | r ψpαq ´r ψpρq| ď e ´2 β T 4 β } r ψ } β .
With the notation of Theorem 15.4, we will use this result when X " |X v | 1 is the geometric realisation of the simplicial tree X v , α " α é, γ is 14 the common perpendicular between D ánd γD `for γ P Γ (when it exists), and ρ " ρ γ is any extension of α to a geodesic ray, or rather to a generalised geodesic line isometric exactly on r0, `8r . Under the assumptions of Theorem 15.4, we have

pq v 2 ´1qpq v `1q 2 q v 3 VolpΓz zX v q }σ D`} q ´n v ÿ γPΓ{Γ D 0ădpD
´, γD `qďn ∆ ργ á r σ D´, (15.13) with, if Γ is geometrically finite, an error term when evaluated on r ψ P C β c p p G Xq of the form Op } r ψ } β e ´κ n q for some κ ą 0 small enough (depending in particular on β P s0, 1s).

Proof. By Equation (2.4) defining the distance on p G X, we have, since dpαptq, ρptqq " 0 for all t P s ´8, T s and dpαptq, ρptqq " t ´T otherwise, dpα, ρq " ż `8 T pt ´T q e ´2t dt " e ´2 T ż `8 0 u e ´2u du " e ´2 T 4 .

The result follows. l

The second lemma is a metric estimate on the map which associates to a geodesic ray its point at infinity. We start by giving some definitions.

Let X be a geodesically complete proper CATp´1q space, and let D be a nonempty proper closed convex subset of X. The distance-like map d D : `B8 X ´B8 D ˘2 Ñ r0, `8r associated with D is defined in [HeP4, §2.2] as follows: For ξ, ξ 1 P B 8 X ´B8 D, let ξ t , ξ 1 t : r0, `8r Ñ X be the geodesic rays starting at the closest points P D pξq, P D pξ 1 q to ξ, ξ 1 on D and converging to ξ, ξ

d D pξ, ξ 1 q " $ ' &
' % e 1 2 dpP D pξq, P D pξ 1 qq ą 1 if P D pξq ‰ P D pξ 1 q d x pξ, ξ 1 q " e ´dpx, yq ď 1 if P D pξq " P D pξ 1 q " x and rx, ξr X rx, ξ 1 r " rx, ys .

14 the generalised geodesic line isometric exactly on r0, dpD ´, γD `qs parametrising 15 See Equation (2.1). 16 See Equation (2.8).

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In particular, although it is not an actual distance on its whole domain B 8 X ´B8 D, the map d D is locally a distance, and we can define with the standard formula the β-Hölder-continuity of maps with values in pB 8 X ´B8 D, d D q and the β-Hölder-norm of a function defined on pB 8 X ´B8 D, d D q. From now on, we endow B 8 X ´B8 D with the distance-like map d D .

Proposition 15.6. Let X be a locally finite simplicial tree without terminal vertices, and let D be a proper nonempty simplicial subtree of X. The homeomorphism B `: B 1 `D Ñ pB 8 X ´B8 Dq defined by ρ Þ Ñ ρ `is 1 2 -Hölder-continuous, and for all β P s0, 1s and ψ P C β b pB 8 X ´B8 Dq, the map ψ ˝B`: B 1 `D Ñ R is bounded and β 2 -Hölder-continuous, with

}ψ ˝B`} β 2 ď p1 `2 β 2 `1q }ψ} β .
With the notation of Theorem 15.4, using the claim following the statement of Lemma 15.5, we will use this result when X " X v and D " D ´. Under the assumptions of Theorem 15.4, with ρ γ any extension to a geodesic ray of the common perpendicular α é, γ between D ánd γD `for γ P Γ, since pushing forward measures on B 1 `D´b y the homeomorphism B `is continuous, we have by Equation (15.13)

pq v 2 ´1qpq v `1q 2 q v 3 VolpΓz zX v q }σ D`} q ´n v ÿ γPΓ{Γ D 0ădpD
´, γD `qďn ∆ pργ q `á pB `q˚r σ D´.

(15.15)

If Γ is geometrically finite, for all β P s0, 1s and ψ P C β c pB 8 X v ´B8 D ´q, using the error term in Equation (15.13) with regularity β 2 when evaluated on r ψ " ψ ˝B`P C β 2 c p p G X v q, we have by the last claim of Proposition 15.6 an error term in Equation (15.15) evaluated on ψ of the form Op }ψ } β e ´κ n q for some κ ą 0 small enough.

Proof. Let us prove that for every ρ, ρ 1 P B 1 `D, if dpρ, ρ 1 q ď 1, then ρp0q " ρ 1 p0q, and (15.16) This proves that the map B `is 1 2 -Hölder-continuous. We may assume that ρ ‰ ρ 1 . Let ρ, ρ 1 P B 1 `D. If ρp0q ‰ ρ 1 p0q, then the images of ρ and ρ 1 are disjoint and their connecting segment in the tree X joins ρp0q and ρ 1 p0q; hence for every t P r0, `8r , we have dpρptq, ρ 1 ptqq " dpρp0q, ρ 1 p0qq `dpρptq, ρp0qq `dpρ 1 ptq, ρ 1 p0qq ě 1 `2 t .

d D pρ `, ρ 1 `q " ? 2 dpρ, ρ 1 q 1 2 .

Thus

dpρ, ρ 1 q " ż 0 ´8 dpρp0q, ρ 1 p0qq e 2 t dt `ż `8 0 dpρptq, ρ 1 ptqq e ´2 t dt ě ż 0 ´8 e 2 t dt `ż `8 0 p1 `2tq e ´2 t dt ą 2 ż `8 0 e ´2 t dt " 1 .
Assume that x " ρp0q " ρ 1 p0q and let n be the length of the intersection of ρ and ρ 1 . Then

d D pρ `, ρ 1 `q " d x pρ `, ρ 1 `q " lim tÑ`
" sup ρ, ρ 1 PB 1 `D, 0ădpρ, ρ 1 qď1 |ψ ˝B`p ρq ´ψ ˝B`p ρ 1 q| dpρ, ρ 1 q β 2 ď sup ρ, ρ 1 PB 1 `D, 0ădpρ, ρ 1 qď 1 2 |ψ ˝B`p ρq ´ψ ˝B`p ρ 1 q| dpρ, ρ 1 q β 2 `2 }ψ ˝B`} 8 2 ´β 2 ď sup ξ, ξ 1 PB8X´B8D 0ăd D pξ, ξ 1 qď1 |ψpξq ´ψpξ 1 q| 2 ´β 2 d D pξ, ξ 1 q β `2 β 2 `1 }ψ} 8 ď 2 β 2 `1 }ψ} β . Since }ψ ˝B`} β 2 " }ψ ˝B`} 8 `}ψ ˝B`}1 β 2
, this proves the last claim of Proposition 15.6. l

We conclude Section 15.4 by giving a purely algebraic control of error terms, under the stronger regularity requirement on functions of being locally constant. We assume until the end of this Section that the lattice Γ is contained in G v " PGL 2 pK v q.

The group G v acts (on the left) on the complex vector space of maps ψ from ΓzG v to R, by right translation on the source: For every g P G v , we have gψ : x Þ Ñ ψpxgq. A function ψ : ΓzG v Ñ R is algebraically locally constant if there exists a compact-open subgroup U of PGL 2 pO v q which leaves ψ invariant: @ g P U, gψ " ψ , or equivalently, if ψ is constant on each orbit of U under the right action of G v on ΓzG v . Note that ψ is then continuous, since the orbits of U are compact-open subsets. We define d ψ " dim `Vect R pPGL 2 pO v qψq ȃs the dimension of the complex vector space generated by the images of ψ under the elements of PGL 2 pO v q, which is finite, and even satisfies d ψ ď rPGL 2 pO v q : U s .

We define the alc-norm of every bounded algebraically locally constant map ψ : ΓzG v Ñ R by }ψ} alc " a d ψ }ψ} 8 .

Though the alc-norm does not satisfy the triangle inequality, we have }λ ψ} alc " |λ| }ψ} alc for every λ P R. We denote by alcpΓzG v q the vector space of bounded algebraically locally constant maps ψ from ΓzG v to R.

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For every n P N, let U n be the compact-open subgroup of PGL 2 pO v q which is the kernel of the morphism PGL 2 pO v q Ñ PGL 2 pO v {π v n O v q of reduction modulo π v n . Note that any compact-open subgroup U of PGL 2 pO v q contains U n for some n P N. Hence ψ : ΓzG v Ñ R is algebraically locally constant if and only if there exists n P N such that ψ is constant on each right orbit of U n . For every n P N, since the order of PGL 2 pO v {π v n O v q is at most the order of pO v {π v n O v q 4 , which is q v 4n , if ψ : ΓzG v Ñ R is constant on each right orbit of U n , then }ψ} alc ď q v 2n }ψ} 8 .

(15.17)

The next result is an algebraic version of the error term statement in Theorem 15.4 (assuming for simplicity that L Γ " Z), which uses17 a stronger assumption on Γ, and obtains a weaker regularity (locally constant instead of Hölder-continuous, see Section 3.1 for definitions and notation). We will not use it in this book, but its version with L Γ " 2Z is used in the announcement [BrPP] which only considers the locally constant regularity.

Theorem 15.7. Let K v be the completion of a function field K over F q with respect to a valuation v of K and let Γ be a nonuniform lattice of G v with L Γ " Z. Then there exists κ ą 0 such that for every P s0, 1s and every -locally constant map r ψ :

p G X v Ñ R, we have, as n Ñ `8, pq v ´1q pq v `1q q v VolpΓz zX v q }σ D`} q ´n v ÿ γPΓ{Γ D 0ădpD ´, γD `qďn r ψpα é, γ q " ż p G Xv r ψ dr σ D´`O p } r ψ } lc, ln qv e ´κ n q .
Proof. This result follows by replacing in the proof of Theorem 12.17 (or rather Remark (ii) following its statement) the use of the exponential decay of β-Hölder correlations given by Corollary 9.6 by the following result of decay of correlations under locally constant regularity (which does follow from Corollary 9.6 by Remark 3.2).

Proposition 15.8. Let K v be the completion of a function field K over F q with respect to a valuation v of K and let Γ be a nonuniform lattice of G v with L Γ " Z. Then there exist C, κ ą 0 such that for every P s0, 1s, for all -locally constant maps φ, ψ : ΓzG X v Ñ R and n P Z, we have

ˇˇż ΓzG Xv φ ˝g´n ψ d m BM ´1 }m BM } ż ΓzG Xv φ d m BM ż ΓzG Xv ψ d m BM ˇď
C e ´κ|n| }φ} lc, ln qv }ψ} lc, ln qv .

Proof. Recall18 that we have a natural homeomorphism Ξ : ΓgApO v q Þ Ñ Γg ˚between ΓzG v {ApO v q and ΓzG X v . We denote by p G : ΓzG v Ñ ΓzG X v the composition map of the canonical projection `ΓzG v ˘Ñ `ΓzG v {ApO v q ˘and of Ξ. By Equation (15.4), for every ď e ´2p´1 2 ln q " , as wanted.

x P ΓzG v , we have p G px a v n q " g n p G pxq . ( 15
In order to prove that ψ ˝pG : ΓzG v Ñ R is U n -invariant, let x, x 1 P ΓzG v be such that x 1 P x U n . Since U n acts by the identity map on the ball of radius n in the Bruhat-Tits tree X v , the geodesic lines p G pxq and p G px 1 q in ΓzG X v coincide (at least) on r´n, ns. Hence, as we saw in the beginning of the proof, we have dpp G pxq, p G px 1 qq ď . Therefore ψpp G pxqq " ψpp G px 1 qq since ψ is -locally constant. Now, using Equation (15.17), we have }ψ ˝pG } alc ď q v 2 n }ψ ˝pG } 8 ď q v 2p1´1 2 ln q }ψ} 8 " q v 2 ´ln qv }ψ} 8 " q v 2 }ψ} lc, ln qv . l Now, in order to prove Proposition 15.8, we will use an algebraic result of exponential decay of correlations, Theorem 15.10 (see for instance [AtGP]). We first recall some definitions and notation, useful for its statement.

Recall that the left action of the locally compact unimodular group G v on the locally compact space G X v is continuous and transitive, and that its stabilisers are compact hence unimodular. Since Γ is a lattice, the (Borel, positive, regular) Bowen-Margulis measure r m BM on G X v is G v -invariant (see Proposition 4.14 (2)). Hence by [START_REF] Weil | L'intégration dans les groupes topologiques et ses applications[END_REF] (see also [START_REF] Gorodnik | Counting orbits of integral points in families of affine homogeneous varieties and diagonal flows[END_REF]Lem. 5]), there exists a unique Haar measure on G v , which disintegrates by the evaluation map

Ă p G : G v Ñ G X v defined by g Þ Ñ g ˚,
with conditional measure on the fiber over " g ˚P G X v the probability Haar measure on the stabiliser gApO v qg ´1 of under G v . Hence, taking the quotient under Γ and normalising in order to have probability measures, if µ v is the right G v -invariant probability measure on ΓzG v , we have cov m BM }m BM } , n pψ, ψ 1 q " cov µv, av n pψ ˝pG , ψ 1 ˝pG q .

pp G q ˚µv " m BM }m BM } . ( 15
(15.21)

Recall that the adjoint representation of G v " PGL 2 pK v q is the continuous morphism G v Ñ GLpM 2 pK v qq defined by rhs Þ Ñ tx Þ Ñ hxh ´1u, which is independent of the choice of the representative h P GL 2 pK v q of rhs P PGL 2 pK v q. For every g P G v , we denote by |g| v the operator norm of the adjoint representation of g. For instance, recalling that a v "

" 1 0 0 π ´1 v  ,
we have, for all n P Z,

|a v n | v " q v |n| . (15.22)
We refer for instance to [AtGP] for the following result of exponential decay of correlations.

Theorem 15.10. Let Γ be a nonuniform lattice of G v . There exist C 1 , κ 1 ą 0 such that, for all bounded locally constant functions r ψ, r

ψ 1 : ΓzG v Ñ R and g P G v , ˇˇcov µv, g p r ψ, r ψ 1 q ˇˇď C 1 } r ψ} alc } r ψ 1 } alc |g| v ´κ1
. l (15.23) Proposition 15.8 follows from this result applied to r ψ " ψ ˝pG , r ψ 1 " ψ 1 ˝pG and g " a v n by using Equations (15.21),(15.19) and (15.22) and by taking C " C 1 q 4 v and κ " κ 1 ln q v . l This concludes the proof of Theorem 15.7. l

Remark. There is a similar relationship between locally constant functions on K v in an algebraic sense and the ones in the metric sense.

The additive group pK v , `q acts on the complex vector space of functions from K v to R, by translations on the source: for all y P K v and ψ : K v Ñ R, the function y ¨ψ is equal to x Þ Ñ ψpx `yq. A map ψ : K v Ñ R is algebraically locally constant if there exists k P N such that ψ is invariant under the action of the compact-open subgroup pπ v q k O v of K v , that is, if for all x P K v and y P pπ v q k O v , we have ψpx `yq " ψpxq. Note that a locally constant map from K v to R is continuous.

For any locally constant function ψ : K v Ñ R, the complex vector space Vect R pO v ¨ψq generated by the images of ψ under the elements of O v is finite dimensional. Its dimension d ψ satisfies, with k as above,

d ψ ď rO v : pπ v q k O v s " q v k .
We define the alc-norm of every bounded algebraically locally constant map ψ :

K v Ñ R by }ψ} alc " a d ψ }ψ} 8 .
Though the alc-norm does not satisfy the triangle inequality, we have }λψ} alc " |λ| }ψ} alc for every λ P R, and the set of bounded algebraically locally constant maps from K v to R is a real vector space. Actually, a function ψ : K v Ñ R is algebraically locally constant if and only if it is locally constant. More precisely, for every P s0, 1s, since the closed balls of radius q ´k v in K v are the orbits by translations under pπ v q k O v , every -locally constant function ψ : K v Ñ R is constant under the additive action of pπ v q k O v for k " r ´ln ln qv s, hence 20 }ψ} alc ď }ψ} lc, 1 2 .

19 See Section 9.2 for a definition of covµ, n. 20 See Section 3.1 for the notation.

243

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Geometrically finite lattices with infinite Bowen-Margulis measure

This Section is a digression from the theme of arithmetic applications, in which we use the Nagao lattice defined in Section 15.2 in order to construct a geometrically finite discrete group of automorphisms of a simplicial tree which has infinite Bowen-Margulis measure. This example was promised towards the end of Section 4.4. We will equivariantly change the lengths of the edges of a simplicial tree X endowed with a geometrically finite (nonuniform) lattice Γ in order to turn it into a metric tree in which the group Γ remains a geometrically finite lattice, but now has a geometrically finite subgroup with infinite Bowen-Margulis measure. This example is an adaptation of the negatively curved manifold example of [START_REF] Dal'bo | Séries de Poincaré des groupes géométriquement finis[END_REF]§4]. The simplicial example is obtained as a modification of the metric tree example.

Theorem 15.11. There exists a geometrically finite discrete group of automorphisms of a regular metric tree with infinite Bowen-Margulis measure.

There exists a geometrically finite discrete group of automorphisms of a simplicial tree with uniformly bounded degrees whose Bowen-Margulis measure is infinite.

Proof. Let p

K " F q ppY ´1qq, O " F q rrY ´1ss and R " F q rY s. Let X be the Bruhat-Tits tree of pPGL 2 , p Kq with base point ˚" rO ˆOs. Let Γ " PGL 2 pRq, which is a lattice of X, with quotient the modular ray Γz zX described in Section 15.2. We denote by py i q i"´1,0,... the ordered vertices along Γz zX with vertex stabilisers pΓ i q i"´1,0,... , and by pe i q iPN the ordered edges along Γz zX (pointing away from the origin of the modular ray).

The subgroup

P " ď iě0 Γ i " " " a Q 0 d  : Q P F q rY s, a, d P F q *
is the stabiliser in Γ of 8 P B 8 X. Let P 0 be the finite index subgroup of P consisting of the elements

" 1 Q 0 1
 with Qp0q " 0. Observing that dpγ˚, ˚q " 2pi `1q for any γ P Γ i ´Γi´1 and that the cardinality of pΓ i ´Γi´1 q X P 0 is pq ´1qq i`1 , it is easy to see that the Poincaré series

Q P, 0, ˚, ˚psq "
ÿ γPP e ´s dp˚,γ˚q of the discrete (though elementary) subgroup P of IsompXq is (up to a multiplicative constant) equal to ř 8 i"0 q i e ´2si , which gives δ P " ln q 2 for the critical exponent of P on X. Let h be a loxodromic element of Γ whose fixed points belong to the open subset Y ´1O of p K " B 8 X ´t8u. Hence the horoball H 8 centred at 8 P B 8 X, whose horosphere contains ˚, is disjoint from the translation axis Ax h of h. Note that the stabiliser of H 8 in Γ is P and that P 0 acts freely on the edges exiting H 8 . Let x 0 P V X be the closest point on Ax h to H 8 , let e ˚be the edge with origin ˚pointing towards x 0 , and let e ´, e `be the two edges with origin x 0 on Ax h . Let U h be the set of points x in V X ´tx 0 u such that the geodesic segment from x 0 to x starts either by the edge e ´or by e `. Let U P 0 be the set of points y in V X ´ttpe ˚qu such that the geodesic segment from tpe ˚q to y starts by the edge e ˚. We have (1) U h X U P 0 " H and x 0 R U h Y U P 0 ,

(2) h k pV X´U h q Ă U h for every k P Z´t0u and wpV X´U P 0 q Ă U P 0 for every w P P 0 ´tidu, (3) dpx, yq " dpx, x 0 q `dpx 0 , yq for all x P U h and y P U P 0 .

Let Γ 1 be the subgroup of Γ generated by P 0 and h. By a ping-pong argument, Γ 1 is a free product of P 0 and of the infinite cyclic group generated by h. Hence every element γ in Γ 1 ´teu may be written uniquely w 0 h n 0 w 1 h n 1 . . . w k h n k with k P N, w i P P 0 , n i P Z with w i ‰ e if i ‰ 0 and n i ‰ 0 if i ‰ k. Using the above properties, we have by induction dpx 0 , w 0 h n 0 w 1 h n 1 . . . w k h n k x 0 q " ÿ 0ďiďk dpx 0 , h n i x 0 q `ÿ 0ďiďk dpx 0 , w i x 0 q . (15.24)

Let λ : EX Ñ R `be the Γ 1 -invariant length map on the set of edges of X such that for every i P N, the length of e P EX is 1 if e is not contained in Ť γPΓ 1 γH 8 , and otherwise, if e maps to e i or to e i under the canonical map X Ñ ΓzX, then λpeq " 1 `ln i`1 i if i ě 1 and λpeq " 1 if i " 0. Note that the distance in the metric graph |X| λ from ˚to the vertex on the geodesic ray from ˚to 8 originally at distance i from ˚is now i `ln i. The distances along the translation axis of h have not changed. Equation (15.24) remains valid with the new distance.

Let us now prove that the discrete subgroup Γ 1 of automorphisms of the regular metric tree pX, λq satisfies the first claim of Theorem 15.11.

By Γ 1 -invariance of λ, the group Γ 1 remains a subgroup of AutpX, λq. The elements of Γ 1 8, or equivalently, the points at infinity of the horoballs in the Γ 1 -equivariant family of horoballs pγH 8 q γPΓ 1 {Γ 1 8 with pairwise disjoint interiors in X, remain bounded parabolic fixed points of Γ 1 , and the other limit points remain conical limit points of Γ 1 . Hence Γ 1 remains a geometrically finite discrete subgroup of AutpX, λq.

The Poincaré series of the action of P 0 on pX, λq is (up to a multiplicative and additive constant) ř 8 i"1 q i e ´2si i ´2s , which has the same critical exponent δ P " ln q 2 as previously, but it is easy to see that the group P is now of convergence type if q ě 3.

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The computations of [START_REF] Dal'bo | Séries de Poincaré des groupes géométriquement finis[END_REF]§4] now apply to our situation (with C " 0 in their notation, and we sum over P 0 instead of over the infinite cyclic group generated by the parabolic element p in their notation). Their argument shows that Γ 1 is of convergence type with critical exponent δ P , up to replacing h by a big enough power. By Corollary 4.6, the Bowen-Margulis measure of Γ 1 is infinite.

In order to prove the second claim of Theorem 15.11, we first define a new length map λ : EX Ñ R `which coincides with the previous one on every edge e of X, unless e maps to e i or to e i under the canonical map X Ñ ΓzX, in which case we set λpeq " 1 `tlnpi `1qu ´tln iu if i ě 1 and λpeq " 1 if i " 0 (where t¨u is the largest previous integer map). This map λ now has values in t1, 2u, and we subdivide each edge of length 2 into two edges of length 1. The tree Y thus obtained has uniformly bounded degrees (although it is no longer a uniform tree), and the group Γ 1 defines a geometrically finite discrete subgroup of AutpYq with infinite Bowen-Margulis measure. l 246 19/12/2016

Chapter 16

Rational point equidistribution and counting in completed function fields

Let K be a function field over F q , let v be a (normalised discrete) valuation of K, and let R v be the affine function ring associated with v. In this Chapter, we prove analogues of the classical results on the counting and equidistribution towards the Lebesgue measure on R of the Farey fractions p q with pp, qq P Z ˆpZ ´t0uq relatively prime (see for instance [Nev], as well as [START_REF] Parkkonen | On the arithmetic of crossratios and generalised Mertens' formulas. Numéro Spécial "Aux croisements de la géométrie hyperbolique et de l'arithmétique[END_REF] for an approach using methods similar to the ones in this text). In particular, we prove various equidistribution results of locally finite families of elements of K towards the Haar measure on K v , using the geometrical work on equidistribution of common perpendiculars done in Section 11.4 and recalled in Section 15.4.

Equidistribution of non-Archimedian Farey fractions

The first result of this Section is an analog in function fields of the equidistribution of Farey fractions to the Lebesgue measure in R, see the Introduction, and for example [START_REF] Parkkonen | On the arithmetic of crossratios and generalised Mertens' formulas. Numéro Spécial "Aux croisements de la géométrie hyperbolique et de l'arithmétique[END_REF]p. 978] for the precise statement and a geometric proof. For every px 0 , y 0 q P R v ˆRv ´tp0, 0qu, let m v, x 0 , y 0 " Cardta P pR v q ˆ: D

b P x 0 R v X y 0 R v , pa ´1qx 0 y 0 ´bx 0 P y 2 0 R v u .
For future use, note that by Equation (14.3)

m v, 1, 0 " q ´1 . (16.1)
For every pa, bq P R v ˆRv and every subgroup H of GL 2 pR v q, let H pa,bq be the stabiliser of pa, bq for the linear action of H on R v ˆRv .

Theorem 16.1. Let G be a finite index subgroup of GL 2 pR v q, and let px 0 , y 0 q P R v ˆRv tp0, 0qu. Then as s Ñ `8, if c " pq v 2 ´1q pq v `1q ζ K p´1q m v, x 0 , y 0 pN xx 0 , y 0 yq 2 rGL 2 pR v q : Gs pq ´1q q g´1 q 3 v rGL 2 pR v q px 0 ,y 0 q : G px 0 ,y 0 q s , then c s ´2 ÿ px, yqPGpx 0 , y 0 q, |y|vďs ∆ x y á Haar Kv .

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For every β P s0, 1 ln qv s, there exists κ ą 0 such that for every β-Hölder-continuous function ψ : K v Ñ R with compact support, 1 as for instance if ψ : K v Ñ R is locally constant with compact support (see Remark 3.2), there is an error term in the equidistribution claim of Theorem 16.1 evaluated on ψ, of the form Ops ´κ}ψ} β q.

It is remarkable that due to the general nature of our geometrical tools, we are able to work with any finite index subgroup G of GL 2 pR v q, and not only with its congruence subgroups. In this generality, the usual techniques (for instance involving analysis of Eisenstein series) are not likely to apply. Also note that the Hölder regularity for the error term is a much weaker assumption than the locally constant one that is usually obtained by analytic number theory methods.

Theorem 1.13 in the Introduction follows from this result, by taking K " F q pY q (so that g " 0), v " v 8 and px 0 , y 0 q " p1, 0q, and by using Equations (14.5) and (16.1) in order to simplify the constant.

Before proving Theorem 16.1, let us give a counting result which follows from this equidistribution result by considering the locally constant characteristic function of a closed and open fundamental domain of K v modulo the action by translations of a finite index additive subgroup of R v , and by using Lemma 14.4 with I " R v .

The additive group R v acts on R v ˆRv by the horizontal shears (transvections):

@ k P R v , @ px, yq P R v ˆRv , k ¨px, yq " px `ky, yq ,
and this action preserves the absolute value |y| v of the vertical coordinate y. We may then define a counting function Ψ G, x 0 ,y 0 of elements in K in an orbit by homographies under a finite index subgroup G of GL 2 pR v q, as Ψ G, x 0 , y 0 psq " Card R v, G z px, yq P Gpx 0 , y 0 q, |y| v ď su ,

where R v, G is the finite index additive subgroup of R v consisting of the elements

x P R v such that ˆ1 x 0 1 ˙P G. Note that R v, G " R v if G " GL 2 pR v q.
Corollary 16.2. Let G be a finite index subgroup of GL 2 pR v q, and let px 0 , y 0 q P R v ˆRv tp0, 0qu. Then there exists κ ą 0 such that, as s Ñ `8, Ψ G, x 0 ,y 0 psq " pq ´1q q 2g´2 q 3 v rGL 2 pR v q px 0 ,y 0 q : G px 0 ,y 0 q s rR v : R v, G s pq v 2 ´1q pq v `1q ζ K p´1q m v, x 0 , y 0 pN xx 0 , y 0 yq 2 rGL 2 pR v q : Gs s 2 `Ops 2´κ q . l Let us fix some notation for this Section. For every subgroup H of GL 2 pR v q, we denote by H its image in Γ v " PGL 2 pR v q. Let X v be the Bruhat-Tits tree of pPGL 2 , K v q, which is regular of degree q v `1. Let r "

x 0 y 0 P K Y t8u .

If y 0 " 0, let g r " id P GL 2 pKq, and if y 0 ‰ 0, let g r " ˆr 1 1 0 ˙P GL 2 pKq . Proof of Theorem 16.1. Note that Γ has finite index in Γ v and, in particular, it is a lattice of X v . By [START_REF] Serre | Arbres, amalgames, SL 2 . 3ème éd. corr[END_REF]II.1.2,Coro.], for all x P V X v and γ P GL 2 pR v q, the distance dpx, γxq is even since vpdet γq " 0. Hence by the equivalence in Equation (4.13), the length spectrum

L Γv of Γ v is 2Z. The length spectrum of Γ is also 2Z, since it is contained in L Γv .
Note that D `is a horoball in X v centred at r " x 0 y 0 P B 8 X v , by Lemma 15.1. The stabiliser Γ D ´of D ´(respectively Γ D `of D `) coincides with the fixator Γ 8 of 8 P B 8 X v (respectively the fixator Γ r of r) in Γ. Note that the families D ˘" pγD ˘qγPΓ{Γ D ˘are locally finite, since Γ v , and hence its finite index subgroup Γ, is geometrically finite, 2 and since 8 and r P K are bounded parabolic limit points of Γ v , hence of its finite index subgroup Γ.

For every γ P Γ{Γ r such that D ´and γD `are disjoint, let ρ γ be the geodesic ray starting from α é, γ p0q and ending at the point at infinity γ ¨r of γD `. Note that ρ γ and α é, γ coincide on r0, dpD ´, γD `qs.

Since the Patterson densities of lattices of X v have total mass qv`1 qv by Proposition 15.2 (2), they are normalized as in Theorem 15.4. Then by Equation (15.15), we have

lim nÑ`8 pq v 2 ´1qpq v `1q 2 q 3 v VolpΓz zX v q }σ D`} q v ´n ÿ γPΓ{Γr 0ădpD ´, γD `qďn ∆ pργ q `" pB `q˚r σ D´.
(16.2) Furthermore, for every β P s0, 1 ln qv s, by the comment following Equation (15.15), we have an error term of the form Op}ψ} β ln qv e ´κ n q for some κ ą 0 in the above formula when evaluated on ψ P C β ln qv c pBX v ´t8uq, where BX v ´t8u is endowed with Hamenstädt's distance d H8 . Hence we have an error term Op}ψ} β e ´κ n q for some κ ą 0 in the above formula when evaluated on ψ P C β c pK v q, where K v " BX v ´t8u is endowed with the distance px, yq Þ Ñ |x ´y| v , see Equation (15.3).

By Proposition 15.2 (3), we have pB `q˚r σ D´" Haar Kv .

Hence Equation (16.2) gives, with the appropriate error term,

lim nÑ`8 pq v 2 ´1qpq v `1q 2 q 3 v VolpΓz zX v q }σ D`} q v ´n ÿ γPΓ{Γr 0ădpD ´, γD `qďn ∆ γ¨r " Haar Kv .
(16.3) Let g P GL 2 pKq be such that g8 ‰ 8. This condition is equivalent to asking that the p2, 1q-entry c of g is nonzero. By Lemma 15.1, the signed distance between the horospheres H 8 and gH 8 is dpH 8 , gH 8 q " ´2 vpcq " 2 ln |c| v ln q v . (16.4)

If y 0 ‰ 0, then px, yq " gpx 0 , y 0 q if and only if p x y 0 , y y 0 q " gg r p1, 0q, and the p2, 1q-entry of gg r is y y 0 . If y 0 " 0 (which implies that g r " id and x 0 ‰ 0), then px, yq " gpx 0 , y 0 q if and only if

2 See Section 15.2 249 19/12/2016 p x x 0 , y x 0 q " gp1, 0q, and the p2, 1q-entry of g " gg r is y x 0 . Let z 0 " " y 0 if y 0 ‰ 0 x 0 otherwise.

By Equation (16.4), the signed distance between D ´" H 8 and gD `" g g r H 8 is dpD ´, gD `q " 2 ln q v ln ˇˇy z 0 ˇˇv .

By discreteness, there are only finitely many double classes rgs P G p1,0q zG{G px 0 ,y 0 q such that D ´" H 8 and gD `" g g r H 8 are not disjoint. Let ZpGq be the centre of G, which is finite. Since ZpGq acts trivially on P 1 pK v q, the map G{G px 0 ,y 0 q Ñ Γ{Γ r induced by the canonical map GL 2 pR v q Þ Ñ PGL 2 pR v q is |ZpGq|-to-1. Using the change of variable

s " |z 0 | v q v n 2 , so that q v ´n " |z 0 | v 2 s ´2
pq v 2 ´1q pq v `1q |z 0 | v 2 2 q v 3 |ZpGq| VolpΓz zX v q }σ D`} s ´2 ÿ px, yqPGpx 0 , y 0 q, |y|vďs ∆ x y " Haar Kv . (16.5)
The order of the centre ZpGL 2 pR v qq " pR v q ˆid is q ´1 by Equation ( 14.3). The map GL 2 pR v q{G Ñ Γ v {Γ induced by the canonical map GL 2 pR v q Ñ PGL 2 pR v q is hence q´1 |ZpGq| -to-1. By Equation (15.7), we hence have VolpΓz zX v q " rΓ v : Γs VolpΓ v z zX v q " 2 ζ K p´1q rΓ v : Γs " 2 q ´1 ζ K p´1q |ZpGq| rGL 2 pR v q : Gs .

(16.6) Theorem 16.1 follows from Equations (16.5) and (16.6) and from Lemma 16.3 below. l Lemma 16.3. We have }σ D`} " q g´1 |z 0 | v 2 rGL 2 pR v q px 0 ,y 0 q : G px 0 ,y 0 q s m v, x 0 , y 0 pN xx 0 , y 0 yq 2 .

Proof. Let γ r be the image of g r in PGL 2 pKq. Let us define Γ 1 " γ r ´1Γγ r , which is a finite index subgroup in Γ 1 v " γ r ´1Γ v γ r and a lattice of X v . Since γ r maps 8 to r, the point 8 is a bounded parabolic limit point of Γ 1 , and we have pΓ 1 q 8 " γ r ´1Γ r γ r . Since the canonical map GL 2 pR v q Ñ PGL 2 pR v q is injective on the stabiliser GL 2 pR v q px 0 ,y 0 q , we have rpΓ 1 v q 8 : pΓ 1 q 8 s " rpΓ v q r : Γ r s " rGL 2 pR v q px 0 ,y 0 q : G px 0 ,y 0 q s .

Since the Patterson density of a lattice does not depend on the lattice (see Proposition 4.14 (1)), the skinning measures r σ H of a given horoball H do not depend on the lattice. Thus γ ˚r σ H " r σ γH 250 19/12/2016 for every γ P AutpX v q. Let D 1 " pγ 1 H 8 q γ 1 PΓ 1 {Γ 1 8 , which is a locally finite Γ 1 -equivariant family of horoballs. We hence have, using Proposition 15.2 (4) for the third equality, }σ D`} " }σ γrD 1 } " }σ D1 } " Haar Kv ppΓ 1 q 8 zK v q " rpΓ 1 v q 8 : pΓ 1 q 8 s Haar Kv ppΓ 1 v q 8 zK v q " rGL 2 pR v q px 0 ,y 0 q : G px 0 ,y 0 q s Haar Kv ppΓ 1 v q 8 zK v q .

(16.7)

Every element in the stabiliser of 8 in PGL 2 pK v q can be uniquely written in the form α "

" a b 0 1  with pa, bq P pK v q ˆˆK v . Note that ˆr 1 1 0 ˙ˆa b 0 1 ˙ˆ0 1 1 ´r˙" ˆbr `1 ar ´br 2 ´r b a ´br ˙.
When x 0 " 0 or y 0 " 0, we have α P Γ 1 v if and only if b P R v and a P pR v q ˆ.

When x 0 , y 0 ‰ 0, we have α P Γ 1 v if and only if γ r αγ r ´1 P Γ v , hence if and only if

b P R v X 1 r R v , a P pR v q ˆ, ar ´br 2 ´r P R v .
Let U 1 8 be the kernel of the map from pΓ 1 v q 8 to pK v q ˆsending " a b 0 1  to a, and let m v be its index in pΓ 1 v q 8 . If x 0 " 0 or y 0 " 0, then m v is equal to |pR v q ˆ|, so that, by Equation (14.3), m v " |pR v q ˆ| " |pF q q ˆ| " q ´1 .

If x 0 , y 0 ‰ 0, we have

m v " Card ta P pR v q ˆ: D b P R v X 1 r R v , ar ´br 2 ´r P R v u .
Note that the notation m v coincides with the constant m v, x 0 , y 0 defined before the statement of Theorem 16.1 in both cases.

If I px 0 ,y 0 q is the nonzero fractional ideal

I px 0 ,y 0 q " ! R v if x 0 " 0 or y 0 " 0 , R v X 1 r R v X 1 r 2 R v otherwise, then U 1 8 " ! " 1 b 0 1  : b P I px 0 ,y 0 q ) .
Therefore by Lemma 14.4,

Haar Kv ppΓ 1 v q 8 zK v q "
Haar Kv pI px 0 ,y 0 q zK v q rpΓ 1 v q 8 : U 1 8 s " q g´1 N pI px 0 ,y 0 q q m v .

(16.8) 251 19/12/2016

Let px 0 q " ś p p νppx 0 q and py 0 q " ś p p νppy 0 q be the prime decompositions of the principal ideals px 0 q and py 0 q. By the formulas of the prime decompositions of intersections, sums and products of ideals in Dedekind rings (see for instance [Nar, §1.1]), we have px 2 0 q X px 0 y 0 q X py 2 0 q " px 2 0 q X py 2 0 q " ź p p 2 maxtνppx 0 q, νppy 0 qu and xx 0 , y 0 y " ź p p mintνppx 0 q, νppy 0 qu .

By the definition of the ideal I px 0 ,y 0 q , by the multiplicativity of the norm, and by Equation (14.4), we hence have if x 0 ‰ 0 and y 0 ‰ 0 N pI px 0 ,y 0 q q pN xx 0 , y 0 yq 2

|y 0 | v 2 " N
´`px 2 0 q X px 0 y 0 q X py 2 0 q ˘xx 0 , y 0 y 2 px 0 q ´2py 0 q ´2¯" 1 . (16.9) If x 0 " 0 or y 0 " 0, then N pI px 0 ,y 0 q q " N pR v q " 1 .

(16.10) Lemma 16.3 follows from Equations (16.7), (16.8) and (16.9) if x 0 ‰ 0 and y 0 ‰ 0 or (16.10) if x 0 " 0 or y 0 " 0. l

Let us state one particular case of Theorem 16.1 in an arithmetic setting, using a congruence sugbroup.

Theorem 16.4. Let I be a nonzero ideal of R v . Then as t Ñ `8, we have

pq v 2 ´1q pq v `1q ζ K p´1q N pIq ś p|I p1 `1 N ppq q q g´1 q v pq v q ´2t ÿ px,yqPRvˆI xx, yy"Rv, vpyqě´t ∆ x y á Haar Kv ,
where the product ranges over the prime factors p of the ideal I. Furthermore, if

Ψptq " Card R v z px, yq P R v ˆI : xx, yy " R v , vpyq ě ´t( , then there exists κ ą 0 such that, as t Ñ `8,

Ψptq " q 2g´2 q v pq v 2 ´1q pq v `1q ζ K p´1q N pIq ś p|I p1 `1 N ppq q q v 2t `Opq v p2´κqt q .
For every β P s0, 1 ln qv s, there exists κ ą 0 such that for every ψ P C β c pK v q there is an error term in the above equidistribution claim evaluated on ψ, of the form Ops ´κ}ψ} β q.

Proof. The counting claim is deduced from the equidistribution claim in the same way that Corollary 16.2 is deduced from Theorem 16.1, noting that the action of R v by horizontal shears preserves R v ˆI.

In order to prove the equidistribution claim, we apply Theorem 16.1 with px 0 , y 0 q " p1, 0q and with G the Hecke congruence subgroup which is the preimage of the upper triangular subgroup of GL 2 pR v {Iq by reduction modulo I. In this case, the constant m v,x 0 ,y 0 appearing in the statement of Theorem 16.1 is equal to q ´1 by Equation (16.1). The group G I has finite index in GL 2 pR v q. The following result is well-known to arithmetic readers (see for instance [START_REF] Shimura | Introduction to the arithmetic theory of automorphic functions[END_REF]page 24] when R v is replaced by Z), we only give a sketch of proof (indicated to us by J.-B. Bost) for the sake of the geometer readers.

G I " " ˆa b c d ˙P GL 2 pR v q : c P I * , ( 16
Lemma 16.5. We have

rGL 2 pR v q : G I s " N pIq ź p|I p1 `1 N ppq q .
where the product ranges over the prime factors p of the ideal I.

Proof. In this proof, we denote by |E| the cardinality of a finite set E. For every commutative ring A with finite group of invertible elements A ˆ, we have

GL 2 pAq " ď aPA ˆˆa 0 0 1 ˙SL 2 pAq ,
Hence rGL 2 pAq : SL 2 pAqs " |A ˆ|. Since ˆa 0 0 1 ˙belongs to G I for all a P pR v q ˆ, we have

G I " ď aPpRvq ˆˆa 0 0 1 ˙GI X SL 2 pR v q , so that rGL 2 pR v q : G I s " rSL 2 pR v q : G I X SL 2 pR v qs.
The group morphism of reduction modulo I from SL 2 pR v q to SL 2 pR v {Iq is onto, by an argument of further reduction to the various prime power factors of I and of lifting elementary matrices. The order of the upper triangular subgroup of SL 2 pR v {Iq is |pR v {Iq ˆ| |R{I|, where pR v {Iq ˆis the group of invertible elements of the ring R v {I (that we will see again below). Hence

rGL 2 pR v q : G I s " rSL 2 pR v q : G I X SL 2 pR v qs " | SL 2 pR v {Iq| |pR v {Iq ˆ| |R{I| " | GL 2 pR v {Iq| |pR v {Iq ˆ|2 |R{I| . (16.12)
By the multiplicativity of the norm and by the Chinese remainder theorem, 3 one reduces the result to the case when I " p n is the n-th power of a fixed prime ideal p with norm N ppq " N , where n P N. Note that since R v {p is a field, we have

| GL 1 pR v {pq| " |pR v {pq ˆ| " |R v {p| ´1 " N ´1 and | GL 2 pR v {pq| " p|R v {p| 2 ´1qp|R v {p| 2 ´|R v {p|q " N 2 pN ´1q 2 `1 `1 N ˘.
3 saying that the rings Rv{I and ś p Rv{p vppIq are isomorphic, see for instance [START_REF] Narkiewicz | Elementary and analytic theory of algebraic numbers[END_REF]page 11] 253 19/12/2016

For k " 1 or k " 2, the kernel of the morphism of reduction modulo p from GL k pR v {Iq " GL k pR v {p n q to GL k pR v {pq has order N k 2 pn´1q . Hence

| GL 2 pR v {Iq| " N 4pn´1q N 2 pN ´1q 2 `1 `1 N ˘,
and |pR v {Iq ˆ| " N n´1 pN ´1q .

Therefore, by Equation (16.12), we have

rGL 2 pR v q : G I s " N 4pn´1q N 2 pN ´1q 2 `1 `1 N N 2pn´1q pN ´1q 2 N n " N n `1 `1 N ˘.
This proves the result. l

We can now conclude the proof of Theorem 16.4. Note that GL 2 pR v q p1,0q " pG I q p1,0q . The result then follows from Theorem 16.1 and its Corollary 16.2, using the change of variables s " pq v q t , since

G I p1, 0q " tpx, yq P R v ˆI : xx, yy " R v u . l
The following result is a particular case of Theorem 16.4.

Corollary 16.6. Let P 0 be a nonzero element of the polynomial ring R " F q rY s over F q , and let P 0 " a 0 ś k i"1 pP i q n i be the prime decomposition of P 0 . Then as t Ñ `8, pq `1q

ś k i"1 q n i deg P i p1 `q´deg P i q q ´1 q ´2t ÿ pP,QqPRˆpP 0 Rq P R`QR"R, deg Qďt ∆ P Q á Haar FqppY ´1qq .
For every β P s0, 1 ln qv s, there exists κ ą 0 such that for every ψ P C β c `Fq ppY ´1qq ˘there is an error term in the above equidistribution claim evaluated on ψ, of the form Ops ´κ}ψ} β q.

Proof. In this statement, we use the standard convention that k " 0 if P 0 is constant, a 0 P pF q q ˆand P i P R is monic.

We apply the first claim of Theorem 16.4 with K " F q pT q and v " v 8 so that g " 0, q v " q and R v " R, and with I " P 0 R, so that N pIq " ś k i"1 q n i deg P i . The result follows from Equation (14.5). l

Mertens's formula in function fields

In this Section, we recover the function field analogue of Mertens's classical formula on the average order of the Euler function. We begin with a more general counting and equidistribution result.

Let m be a (nonzero) fractional ideal of R v , with norm N pmq. Note that the action of the additive group R v on K v ˆKv by the horizontal shears k ¨px, yq " px `ky, yq preserves m ˆm. We consider the counting function ψ m : r0, `8r Ñ N defined by ψ m psq " Card `Rv z px, yq P m ˆm : 0 ă N pmq ´1N pyq ď s, xx, yy " m (˘.

Note that ψ m depends only on the ideal class of m and thus we can assume in the computations that m is integral.

19/12/2016

Corollary 16.7. There exists κ ą 0 such that, as s Ñ `8,

ψ m psq " pq ´1q q 2g´2 q v 3 pq v 2 ´1q pq v `1q ζ K p´1q m v, x 0 , y 0 s 2 `Ops 2´κ q ,
where m " xx 0 , y 0 y. Furthermore, as s Ñ `8,

pq v 2 ´1q pq v `1q ζ K p´1q m v, x 0 , y 0 pq ´1q q g´1 q v 3 s ´2 ÿ px,yqPmˆm N pmq ´1N pyqďs, xx,yy"m ∆ x y á Haar Kv .
For every β P s0, 1 ln qv s, there exists κ ą 0 such that for every ψ P C β c `Kv q ˘there is an error term in the above equidistribution claim evaluated on ψ, of the form Ops ´κ}ψ} β q.

Theorem 1.14 in the Introduction follows from this result, by taking K " F q pY q (so that g " 0) and v " v 8 . In order to simplify the constant, we use Equation (14.5) and the fact that the ideal class number of K, that equals the number of orbits of PGL 2 pF q rY sq on P 1 pF q pY qq, is 1. Thus, if m " xx 0 , y 0 y then the constant m v, x 0 , y 0 is equal to m v, 1, 0 , which is q ´1 by Equation (16.1).

Proof. Every nonzero ideal I in R v is of the form I " xR v `yR v for some px, yq P R v Rv ´tp0, 0qu, see for instance [START_REF] Narkiewicz | Elementary and analytic theory of algebraic numbers[END_REF]page 10]. For all px, yq and pz, wq in R v ˆRv , we have

x R v `y R v " z R v `w R v if
and only if pz, wq P GL 2 pR v qpx, yq. The ideal class group of K corresponds bijectively to the set PGL 2 pR v q zP 1 pKq of cusps of the quotient graph of groups PGL 2 pR v q z zX v (where X v is the Bruhat-Tits tree of pPGL 2 , K v q ), by the map induced by

I " x R v `y R v Þ Ñ rx : ys P P 1 pKq.
Given a fixed ideal m in R v , we apply Theorem 16.1 with G " GL 2 pR v q and px 0 , y 0 q P R v ˆRv ´tp0, 0qu a fixed pair such that x 0 R v `y0 R v " m. Using therein the change of variable s Þ Ñ N pmqs, the result follows from Theorem 16.1 and its Corollary 16.2. l

As already encountered in the proof of Lemma 16.5, the Euler function ϕ Rv of R v is defined on the set of (nonzero, integral) ideals I of R v by setting4 ϕ Rv pIq " CardppR v {Iq ˆq , and we denote ϕ Rv pyq " ϕ Rv py R v q for every y P R v . Thus, by the definition of the action of R v on R v ˆRv by shears, we have

ψ Rv psq " ÿ yPRv, 0ăN pyqďs Cardtx P R v {yR v : xx, yy " R v u " ÿ yPRv, 0ăN pyqďs ϕ Rv pyq . (16.13)
As a particular application of Corollary 16.7, we get a well-known asymptotic result on the number of relatively prime polynomials in F q rY s. The Euler function of the ring of polynomials R " F q rY s is then the map φ q : R ´t0u Ñ N defined by

φ q pQq " ˇˇ`R {QR ˘ˆˇˇ" Card P P R : xP, Qy " R, deg P ă deg Q ( .
Note that φ q pλQq " φ q pQq for every λ P pF q q ˆ.

Chapter 17

Equidistribution and counting of quadratic irrational points in non-Archimedean local fields

Let K v be a non-Archimedean local field, with valuation v, valuation ring O v , choice of uniformiser π v , and residual field k v of order q v . Let X v be the Bruhat-Tits tree of pPGL 2 , K v q.1 In this Chapter, we give counting and equidistribution results in K v " B 8 X v ´t8u of an orbit under a lattice of PGL 2 pK v q of a fixed point of a loxodromic element of this lattice. We use these results to deduce equidistribution and counting results of quadratic irrational elements in non-Archimedean local fields. When X v is replaced by a real hyperbolic space, or by a more general simply connected complete Riemannian manifold with negative sectional curvature, there are numerous quantitative results on the density of such an orbit, see the works of Patterson, Sullivan, Hill, Velani, Stratmann, Hersonsky-Paulin, Parkkonen-Paulin. See for instance [START_REF] Parkkonen | Counting arcs in negative curvature[END_REF] for references. The arithmetic applications when X v is replaced by the upper halfspace model of the real hyperbolic space of dimension 2, 3 or 5 are counting and equidistribution results of quadratic irrational elements in R, C and the Hamiltonian quaternions. See for instance [START_REF] Parkkonen | Équidistribution, comptage et approximation par irrationnels quadratiques[END_REF]Coro. 3.10] and [START_REF] Parkkonen | On the arithmetic of crossratios and generalised Mertens' formulas. Numéro Spécial "Aux croisements de la géométrie hyperbolique et de l'arithmétique[END_REF].

Counting and equidistribution of loxodromic fixed points

An element γ P PGL 2 pK v q is said to be loxodromic if it is loxodromic on the (geometric realisation of the) simplicial tree X v . 2 Its translation length is λpγq " min xPV Xv dpx, γxq ą 0 , and the subset Ax γ " tx P V X v : dpx, γxq " λpγqu is the image of a (discrete) geodesic line in X v , which we call the (discrete) translation axis of γ. The points at infinity of Ax γ are denoted by γ ´and γ `chosen so that γ translates away from γ ´and towards γ `on Ax γ . Note that for every γ 1 P PGL 2 pK v q, we have

γ 1 Ax γ " Ax γ 1 γ pγ 1 q ´1 and γ 1 γ ˘" pγ 1 γ pγ 1 q ´1q ˘.
If Γ is a discrete subgroup of PGL 2 pK v q and if α is one of the two fixed points of a loxodromic element of Γ, we denote the other fixed point of this element by α σ . Since Γ is discrete, the translation axes of two loxodromic elements of Γ coincide if they have a common point at infinity. Hence α σ is uniquely defined. We define the complexity hpαq of the loxodromic fixed point α by hpαq "

1 |α ´ασ | v (17.1)
if α, α σ ‰ 8, and by hpαq " 0 if α or α σ is equal to 8. We define ι α " 2 if there exists an element γ P Γ such that γ ¨α " α σ , and ι α " 1 otherwise.3 Following [Ser3, II.1.2], we denote by PGL 2 pK v q `the kernel of the group morphism PGL 2 pK v q Ñ Z{2Z defined by γ " rgs Þ Ñ vpdet gq mod 2. The definition does not depend on the choice of a representative g P GL 2 pK v q of an element γ P PGL 2 pK v q, since v `det ˆλ 0 0 λ ˙˘" 2 vpλq is even for every λ P pK v q ˆ. Note that when K v is as in Section 14.2 the completion of a function field over F q endowed with a valuation v, with associated affine function ring R v , the group Γ v " PGL 2 pR v q is contained in PGL 2 pK v q `, see [Ser3, II.1.2]: For every g P GL 2 pR v q, since det g P pR v q ˆ" pF q q ˆ,4 we have vpdet gq " 0.

The following result proves the equidistribution in K v of the loxodromic fixed points with complexity at most s in a given orbit by homographies under a lattice in PGL 2 pK v q as s Ñ `8, and its associated counting result. If ξ P B 8 X v " P 1 pK v q and Γ is a subgroup of PGL 2 pK v q, we denote by Γ ξ the stabiliser in Γ of ξ.

Theorem 17.1. Let Γ be a lattice in PGL 2 pK v q `, and let γ 0 P Γ be a loxodromic element of Γ. Then as s Ñ `8,

pq v `1q 2 VolpΓz zX v q 2 q 2 v VolpΓ γ 0 z z Ax γ 0 q s ´1 ÿ α P Γ¨γ 0 , hpαqďs ∆ α á Haar Kv and Cardtα P pΓ ¨γ0 q X O v : hpαq ď su " 2 q 2 v VolpΓ γ 0 z z Ax γ 0 q pq v `1q 2 VolpΓz zX v q s .
When Γ is geometrically finite, there is an error term of the form Ops 1´κ q for some κ " κ Γ ą 0 in the counting claim and, for every β P s0, 1 ln qv s, an error term of the form Ops ´κ}ψ} β q for some κ ą 0 in the equidistribution claim evaluated on any β-Hölder-continuous function In order to prove the equidistribution result, we apply Theorem 15.4 with D ´:" t˚vu and D `:" Ax γ 0 . The families D ˘" pγD ˘qγPΓ{Γ D ˘are locally finite, since Γ is discrete and the stabiliser Γ D `of D `acts cocompactly on D `. Furthermore, }σ D`} is finite and nonzero by Equation (8.11). Since Γ is contained in PGL 2 pK v q `, the length spectrum L Γ of Γ is contained in 2Z by [START_REF] Serre | Arbres, amalgames, SL 2 . 3ème éd. corr[END_REF]II.1.2,Coro.]. Hence, it is equal to 2Z by the equivalence given by Equation (4.13).

ψ : O v Ñ C.
For every γ P Γ such that dpD ´, γD `q ą 0,5 let ρ γ be the geodesic ray starting at time 0 from the origin of α é, γ (which is ˚v) with point at infinity γ ¨γ0 . Since X v is a tree and γ ¨γ0 is one of the two endpoints of γD `, the geodesic segment α é, γ | r0,dpD ´,γD `qs is an initial subsegment of ρ γ . 6 Therefore, by Equation (15.15), for the weak-star convergence of measures on B 1

`D´, we have

lim nÑ`8 pq v 2 ´1qpq v `1q 2 q 3 v VolpΓz zX v q }σ D`} q v ´n ÿ γPΓ{Γ D 0ădpD
´, γD `qďn ∆ γ¨γ 0 " pB `q˚r σ D´.

(17.2) Furthermore, for every β P s0, 1 ln qv s, by the comment following Equation (15.15), we have an error term of the form Op}ψ} β ln qv e ´κ n q for some κ ą 0 in the above formula when evaluated on ψ P C β ln qv c pB 8 X v q, where B 8 X v is endowed with the visual distance d ˚v . Note that on x, y P O v , the visual distance d ˚v and the distance px, yq Þ Ñ |x ´y| v are related by |x ´y| v " d H8 px, yq ln qv " d ˚v px, yq ln qv , see Equation (15.3). Hence we have an error term Op}ψ} β e ´κ n q for some κ ą 0 in the above formula when evaluated on ψ P C β c pO v q, where O v is endowed with the distance px, yq Þ Ñ |x ´y| v . The complexity of a quadratic irrational α

˚v

H 8 K v " B 8 X v ´t8u k k BH 8 O v π ´k v `Ov p H8 pπ ´k v q γ ¨γ0 γ ¨γ0 γD Let us fix for the moment k P N. For every ξ P π ´k v `Ov , we have |ξ| v " q ´vpξq v " q k v if k ě 1 and |ξ| v ď 1 if k " 0.
P K v over K is hpαq " 1 |α ´ασ | v ,
see for instance [START_REF] Hersonsky | On the almost sure spiraling of geodesics in negatively curved manifolds[END_REF]§6] for motivations and results when K " F q pY q and v " v 8 . Note that this complexity is invariant under the action of the stabiliser GL 2 pR v q 8 of 8 in GL 2 pR v q, which is its upper triangular subgroup. In particular, it is invariant under the action of R v by translations. 8 In [START_REF] Parkkonen | Équidistribution, comptage et approximation par irrationnels quadratiques[END_REF], where K and | ¨|v are replaced by Q and its Archimedean absolute value, there was, for convenience, an extra factor 2 in the numerator of the complexity, which is not needed here. We refer for instance to [START_REF] Parkkonen | Équidistribution, comptage et approximation par irrationnels quadratiques[END_REF]Lem. 4.2] for the connection of this complexity to the standard height, and to [START_REF] Parkkonen | Équidistribution, comptage et approximation par irrationnels quadratiques[END_REF]§4.2,4.4] and [PaP12, §6.1] for studies using this complexity.

The complexity hp¨q satisfies the following elementary properties, giving in particular its behaviour under the action of PGL 2 pR v q by homographies on the quadratic irrationals in K v over K. We also give the well-known computation of the Jacobian of the Haar measure for the change of variables given by homographies, and prove (using the properties of the complexity hp¨q) the invariance of a measure which will be useful in Section 18.1. For all g " ˆa b c d ˙P GL 2 pK v q and z P K v such that g ¨z ‰ 8, let

jgpzq " | det g| v |c z `d| 2 v .
Proposition 17.3. Let α P K v be a quadratic irrational over K.

(1) We have hpαq " 1 a | trpαq 2 ´4 npαq| v .

(2) For every g " ˆa b c d ˙P GL 2 pKq with | det g| v " 1, we have hpg ¨αq " | npd `c αq| v hpαq .

(3) If Q α : R v ˆRv Ñ r0, `8r is the map px, yq Þ Ñ | npx´y αq| v , then for every g P GL 2 pR v q, we have

Q g¨α " hpαq hpg ¨αq Q α ˝g´1 .
In particular, if g P GL 2 pR v q fixes α, then

Q α ˝g " Q α .
(4) For all x, y, z P K v and g P GL 2 pK v q such that g ¨x, g ¨y, g ¨z ‰ 8, we have on K v ´tα, α σ u is invariant under the stabiliser of α in PGL 2 pR v q Proof. (1) This follows from the formula pα ´ασ q 2 " pα `ασ q 2 ´4 α α σ .

|g
(2) Since g has rational coefficients (that is, coefficients in K), we have g ¨α ´pg ¨αq σ " g ¨α ´g ¨ασ " aα `b cα `d ´aα σ `b cα σ `d " pad ´bcqpα ´ασ q pcα `dqpcα σ `dq " pdet gqpα ´ασ q npd `cαq .

Taking absolute values and inverses, this gives Assertion (2).

(3) Let g " ˆa b c d ˙P GL 2 pR v q. Note that g ´1 ¨α " dα´b a´cα . For all x, y P R v , we hence have n `pax `byq ´pcx `dyqα ˘" n `xpa ´cαq ´ypdα ´bq " n `xpa ´cαq ´ypa ´cαq g ´1 ¨α"

npa ´cαq npx ´y g ´1 ¨αq .

Taking absolute values and using Assertion (2), we have

Q α ˝g " hpg ´1 ¨αq hpαq Q g ´1¨α .
Assertion (3) follows by replacing g by its inverse.

(4) Let g " ˆa b c d ˙P GL 2 pK v q. As seen in the proof of Assertion (2), we have g ¨x ´g ¨y " pdet gqpx ´yq pc x `dqpc y `dq .

Taking absolute values and squares, this gives the first claim of Assertion (4).

Recall that a homography z Þ Ñ a z`b c z`d is holomorphic 9 on K v ´t´d c u, with derivative z Þ Ñ ad´bc pcz`dq 2 . Hence infinitesimally close to z, the homography acts (up to translations which leave the Haar measure invariant) by a homothety of ratio We refer for instance to [START_REF] Serre | Lie algebras and Lie groups[END_REF] for background on holomorphic functions on non-Archimedean local fields.

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(5) Let g " ˆa b c d ˙P GL 2 pR v q fixing α. Note that an element of GL 2 pR v q which fixes α also fixes α σ . By Assertion (4), we have Again by Assertion (4), since | det g| v " 1 as det g P pR v q ˆ" pF q q ˆby Equation ( 14.3), by Assertion (2) and since g fixes α, we have 1 a jgpαq jgpα σ q " |c α `d| v |c α σ `d| v " | npc α `dq| v " hpg ¨αq hpαq " 1 .

The result follows. l

Let G be a finite index subgroup of GL 2 pR v q. We say that a quadratic irrational β P K v over K is G-reciprocal (simply reciprocal if G " GL 2 pR v q) if some element of G maps β to β σ . We define the G-reciprocity index ι G pβq as 2 if β is G-reciprocal and 1 otherwise. Similarly, we say that a loxodromic element γ of G is G-reciprocal (simply reciprocal if G " GL 2 pR v q) if there exists an element in G that switches the two fixed points of γ.

Proposition 17.4. Let G be a finite index subgroup of GL 2 pR v q, and let γ be a loxodromic element of G. The following assertions are equivalent:

(1) γ is conjugate in G to γ 1 γ ´1 for some γ 1 P G pointwise fixing Ax γ ,

(2) the loxodromic element γ is G-reciprocal,

(3) the quadratic irrational γ ´is G-reciprocal.

When G " GL 2 pR v q, Assertions (1), ( 2) and (3) are also equivalent to (4) the image of γ 2 γ in PGL 2 pR v q, for some γ 2 P G pointwise fixing Ax γ , is conjugate to the image in PGL 2 pR v q of t γ.

Proof. Most of the proofs are similar to the ones when R v , K and v are replaced by Z, Q and its Archimedean absolute value, see for instance [START_REF] Parkkonen | Équidistribution, comptage et approximation par irrationnels quadratiques[END_REF]. We only give hints for the sake of completeness. Let α " γ

´.

If α is G-reciprocal, then let β P G be such that β ¨α " α σ . Since R v Ă K, we have β ¨ασ " α. Hence βγβ ´1 is a loxodromic element of G fixing α and α σ , having the same translation length as γ, but translating in the opposite direction on Ax γ . Hence γ 1 " βγβ ´1γ fixes pointwise Ax γ . Therefore (3) implies (1).

If β P G conjugates γ to γ 1 γ ´1 for some γ 1 P G pointwise fixing Ax γ , then β preserves the set tα, α σ u. Hence, it preserves the translation axis of γ but it switches α and α σ since γ and γ 1 γ ´1 translate in opposite directions on Ax γ . Therefore (1) implies (2).

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The fact that (2) implies (3) is immediate, since α σ " γ `.

The equivalence between (1) and (4) when G " GL 2 pR v q follows from the fact that the stabiliser of Ax γ normalises the pointwise stabiliser of Ax γ , and from the formula The following result says that any orbit of a given quadratic irrational in K v over K, by homographies under a given finite index subgroup of the modular group PGL 2 pR v q, equidistributes to the Haar measure on K v . Again, note that we are not assuming the finite index subgroup to be a congruence subgroup.

Theorem 17.5. Let G be a finite index subgroup of GL 2 pR v q. Let α 0 P K v be a quadratic irrational over K. Then, as s Ñ `8, pq v `1q 2 ζ K p´1q m 0 rGL 2 pR v q : Gs 2 q 2 v pq ´1q |vptr g 0 q| s ´1 ÿ αPG¨α 0 : hpαqďs ∆ α á Haar Kv , where g 0 P G fixes α 0 with vptr g 0 q ‰ 0, and where m 0 is the index of g Z 0 in the stabiliser of α 0 in G. Furthermore, there exists κ ą 0 such that, as s Ñ `8, Cardtα P pG ¨α0 q X O v : hpαq ď su " 2 q 2 v pq ´1q |vptr g 0 q| pq v `1q 2 ζ K p´1q m 0 rGL 2 pR v q : Gs s `Ops 1´κ q .

For every β P s0, 1 ln qv s, there exists κ ą 0 such that for every ψ P C β c `Kv q ˘there is an error term in the above equidistribution claim evaluated on ψ, of the form Ops ´κ}ψ} β q.

Proof. We apply Theorem 17.1 with Γ the image of G in PGL 2 pR v q and with γ 0 the image in PGL 2 pR v q of the element g 0 introduced in the statement. Note that Γ, which is contained in Γ v , is indeed contained in PGL 2 pK v q `. By Equation (15.6), the translation length of g in X v is 2 |vptr gq|, and g P GL 2 pR v q is loxodromic if and only if vptr gq ‰ 0. This implies that g 0 exists, since G has finite index in GL 2 pR v q, and such an element exists in GL 2 pR v q by Proposition 17.2. Furthermore λpγ 0 q " 2 |vptr g 0 q| .

Since the centre of GL 2 pK v q acts trivially by homographies, we have G ¨α0 " Γ ¨α0 .

For every α P G ¨α0 , the complexities hpαq, when α is considered as a quadratic irrational or when α is considered as a loxodromic fixed point, coincide.

Since the centre ZpGq of G acts trivially by homographies, by the definition of m 0 in the statement, we have VolpΓ γ 0 z z Ax γ 0 q " 1 rΓ γ 0 : γ Z 0 s

Volpγ Z 0 z z Ax γ 0 q " λpγ 0 q rΓ γ 0 : γ Z 0 s " 2 |vptr g 0 q| |ZpGq| m 0 .

(17.7)

Theorem 17.5 now follows from Theorem 17.1 using Equations (16.6) and (17.7). l

Example 17.6. (1) Theorem 1.13 in the Introduction follows from this result, by taking K " F q pY q and v " v 8 , and by using Equation (14.5) in order to simplify the constant.

(2) Let G I be the Hecke congruence subgroup associated with a nonzero ideal I of R v , see Equation (16.11). By Lemma 16.5, we have, as s Ñ `8, pq v `1q 2 ζ K p´1q m 0 N pIq ś p|I p1 `1 N ppq q q 2 v pq ´1q |vptr g 0 q| s ´1 ÿ αPG I ¨α0 : hpαqďs ∆ α á Haar Kv .

We conclude this Section by a characterisation of quadratic irrationals and reciprocal quadratic irrationals in the field of formal Laurent series F q ppY ´1qq in terms of continued fractions. When F q rY s, F q pY q and v 8 are replaced by Z, Q and its Archimedean absolute value, we refer for instance to [Sarn] and [START_REF] Parkkonen | Équidistribution, comptage et approximation par irrationnels quadratiques[END_REF]Prop. 4.3] for characterisations of reciprocal quadratic irrationals.

Recall that Artin's continued fraction expansion of f P F q ppY ´1qq ´Fq pY q is the sequence pa i " a i pf qq iPN in F q rY s with deg a i ą 0 if i ą 0 such that f " a 0 `1 a 1 `1 a 2 `1 a 3 `1 . . .

.

See for instance the surveys [START_REF] Lasjaunias | A survey of Diophantine approximation in fields of power series[END_REF][START_REF] Schmidt | On continued fractions and Diophantine approximation in power series fields[END_REF], and [START_REF] Paulin | Groupe modulaire, fractions continues et approximation diophantienne en caractéristique p[END_REF] for a geometric interpretation. We say that the continued fraction expansion of f is eventually periodic if there exist n P N and N P N ´t0u such that a n`i " a n`N `i for every i P N, and we write f " ra 0 , . . . , a n´1 , a n , . . . , a n`N ´1s .

Such a sequence a n , . . . , a n`N ´1 is called a period of f , and if of minimal length, it is well defined up to cyclic permutation. Two elements β, β 1 P F q ppY ´1qq are in the same PGL 2 pF q rY sq-orbit if and only if their continued fraction expansions have equal tails up to an invertible element of F q rY s by [START_REF] Berthé | On continued fraction expansions in positive characteristic: equivalence relations and some metric properties[END_REF]Theo. 1]. More precisely, β, β 1 P K v are in the same PGL 2 pF q rY sq-orbit if and only if there exist m, n P N and x P F q such that for every k P N, we have a n`k pβ 1 q " x p´1q k a m`k pβq.

Proposition 17.7. Let K " F q pY q and v " v 8 .

(1) An element α P K v ´K is quadratic irrational over K if and only if its continued fraction expansion of β is eventually periodic, and if and only if it is a fixed point of a loxodromic element of PGL 2 pF q rY sq. (2) A quadratic irrational α P K v is reciprocal if and only if the period a 0 , . . . , a N ´1 of the continued fraction expansion of α is palindromic up to cyclic permutation and invertible elements, in the sense that there exist x P F q and p P N such that for k " 0, . . . , N ´1, we have a k`p " x p´1q k a N ´k´1 (with indices modulo N ).

Proof.

(1) The equivalence of being quadratic irrational and having an eventually periodic continued fraction expansion is well-known, see for instance the survey [START_REF] Lasjaunias | A survey of Diophantine approximation in fields of power series[END_REF]Theo. 3.1]. The rest of the claim follows from Proposition 17.2.

(2) The proof is similar to the Archimedean case in [START_REF] Perron | Die Lehre von den Kettenbrüchen[END_REF]§23]. 10 For every quadratic irrational f P F q ppY ´1qq, up to the action of GL 2 pF q rY sq, we may assume that f, pf σ q ´1 P Y ´1F q rrY ´1ss and f " r0, a 1 , a 2 , . . . , a n s. Then we may define by induction quadratic irrationals f 2 , . . . , f n P F q ppY ´1qq over F q pY q such that 1 f " a 1 `f2 , 1 f 2 " a 2 `f3 , . . . ,

1 f n`1 " a n`1 `fn , 1 f n " a n `f .
Passing to the Galois conjugates, we have

1 f σ " a 1 `f σ 2 , 1 f σ 2 " a 2 `f σ 3 , . . . , 1 f σ n " a n `f σ .
Taking these equations in the reverse order, we have

1 ´1 f σ " a n ´1 f σ n , 1
´1 f σ n " a n´1 ´1 f σ n´1 , . . . , 1
´1 f σ 2 " a 1 ´1 f σ ,
so that, since ´1 f σ P Y ´1F q rrY ´1ss, we have ´1 f σ " r0, a n , . . . , a 2 , a 1 s .

Therefore f σ " r´a n , . . . , ´a2 , ´a1 s. Thus, if f and f σ are in the same orbit, the periods are palindromic by [START_REF] Berthé | On continued fraction expansions in positive characteristic: equivalence relations and some metric properties[END_REF]Theo. 1]. l

Counting and equidistribution of quadratic irrationals in Q p

There are interesting arithmetic (uniform) lattices of PGL 2 pQ p q constructed using quaternion algebras. In this Section, we study equidistribution properties of loxodromic fixed points elements of these lattices. We use [Vig] as our standard reference on quaternion algebras.

Let F be a field and let a, b P F ˆ. Let D " p a,b F q be the quaternion algebra over F with basis 1, i, j, k as a F -vector space such that i 2 " a, j 2 " b and ij " ji " ´k. If x " x 0 `x1 i `x2 j `x3 k P D, then its conjugate is x " x 0 ´x1 i ´x2 j ´x3 k , 10 See also [START_REF] Berthé | On continued fraction expansions in positive characteristic: equivalence relations and some metric properties[END_REF]Coro. 1] by relating, using twice the period, what the authors call the ´continued fraction expansion to the standard expansion.

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19/12/2016 its (reduced) norm is

Npxq " x x " x 2 0 ´a x 2 1 ´b x 2 2 `ab x 2 3 and its (reduced) trace is Trpxq " x `x " 2 x 0 .

Let us fix two negative rational integers a, b and let D " p a,b Q q . For every field extension E of Q, we denote by D E the quaternion algebra D b Q E over E, and we say that D splits over E if the E-algebra D b Q E is isomorphic to M 2 pEq. The assumption that a, b are negative implies that D does not split over R. Furthermore, when p P N is an odd prime, D splits over Q p if and only if the equation a x 2 `b y 2 " 1 has a solution in Q p , see [START_REF] Vignéras | Arithmétique des algèbres de quaternions[END_REF]page 32].

The reduced discriminant of D is

Disc D " ź qPRampDq q .
where RampDq is the finite set of primes p such that D does not split over Q p . For instance, the quaternion algebra D " p ´1,´1 Q q splits over Q p if and only if p ‰ 2, hence it has reduced discriminant 2.

Assume from now on that p P N is a positive rational prime such that D splits over Q p and, for simplicity, that Q p contains square roots ? a and ? b of a and b. For example, if a " b " ´1, this is satisfied if p " 1 mod 4. We then have an isomorphism of Q p -algebras θ " θ a, b : D Qp Ñ M 2 pQ p q defined by θpx 0 `x1 i `x2 j `x3 kq " ˜x0 `x1 ? a ? b px 2 `?a x 3 q ? b px 2 ´?a x 3 q x 0 ´x1 ? a ¸, (17.8) so that detpθpxqq " Npxq and trpθpxqq " Trpxq .

If the assumption on the existence of the square roots in Q p is not satisfied, we can replace Q p by an appropriate finite extension, and prove equidistribution results in this extension.

Let O be a Z " 1 p ‰ -order in D Qp , that is, a finitely generated Z " 1 p ‰ -submodule of D Qp generating D Qp as a Q p -vector space, which is a subring of D Qp . Let O 1 be the group of elements of norm 1 in O. Then the image Γ 1 O of θpO 1 q in PGL 2 pQ p q is a cocompact lattice, see for instance [START_REF] Vignéras | Arithmétique des algèbres de quaternions[END_REF]Sect. IV.1]. In fact, this lattice is contained in PSL 2 pQ p q, hence in PGL 2 pQ p q `.

We denote by X p the Bruhat-Tits tree of pPSL 2 , Q p q, which is pp `1q-regular.

The next result computes the covolume of this lattice. Proof. We refer to [START_REF] Vignéras | Arithmétique des algèbres de quaternions[END_REF]page 53] for the (common) definition of the discriminant DiscpQ p q of the local field Q p and DiscpD Qp q of the quaternion algebra D Qp over the local field Q p . We will only use the facts that DiscpQ p q " 1 as it easily follows from the definition, and that DiscpD Qp q " DiscpQ p q 4 N ppZ p q 2 " p 2 (17.9)

11 The index q ranges over the primes dividing DiscD, that is, over the elements of RampDq .

268 19/12/2016 which follows by [START_REF] Vignéras | Arithmétique des algèbres de quaternions[END_REF]Lem. 4.7,page 53] and [START_REF] Vignéras | Arithmétique des algèbres de quaternions[END_REF]Cor. 1.7,page 35] for the first equality and N ppZ p q " CardpZ p {pZ p q " CardpZ{pZq " p for the second one. We refer to [START_REF] Vignéras | Arithmétique des algèbres de quaternions[END_REF]Sect. II.4,page 52] for the definition of the Tamagawa measure µ T on X ˆwhen X " D Qp or X " Q p . It is a Haar measure of the multiplicative locally compact group X ˆ, and understanding its explicit normalisation is the main point of this proposition. By [START_REF] Vignéras | Arithmétique des algèbres de quaternions[END_REF]Lem. 4.6, page 52], 12 with dx the Haar measure on the additive group X, 13 with }x} the module of the left multiplication by x P X ˆon the additive group X, 14 we have dµ T pxq " 1 a DiscpXq }x} dx .

By [START_REF] Vignéras | Arithmétique des algèbres de quaternions[END_REF]Lem. 4.3,page 49], identifying D Qp to M 2 pQ p q by θ, the measure of GL 2 pZ p q for the measure 1 p1´p ´1q }x} dx is 1 ´p´2 . Hence, by scaling and by Equation (17.9), we have µ T pGL 2 pZ p qq " p1 ´p´2 qp1 ´p´1 q a DiscpD Qp q " pp 2 ´1qpp ´1q p 4 .

Again by [START_REF] Vignéras | Arithmétique des algèbres de quaternions[END_REF]Lem. 4.3,page 49], the mass of Z p for the measure 1 p1´p ´1q }x} dx on Q p is 1, hence by scaling µ T pZ p q " 1 ´p´1 a DiscpQ p q " p ´1 p .

By [START_REF] Vignéras | Arithmétique des algèbres de quaternions[END_REF]page 54], since we have an exact sequence 1 ÝÑ SL 2 pQ p q ÝÑ GL 2 pQ p q det ÝÑ Q p ÝÑ 1 , the Tamagawa measure of GL 2 pQ p q disintegrates by the determinant over the Tamagawa measure of Q p with conditional measures the translates of a measure on SL 2 pQ p q, called the Tamagawa measure of SL 2 pQ p q and again denoted by µ T . Thus, µ T pSL 2 pZ p qq " µ T pGL 2 pZ p qq µ T pZ p q " p 2 ´1 p 3

By Example 3 on page 108 of [Vig], since the Z " 1 p ‰ -order O max is maximal, we have, with G " θpO 1 max q, µ T pGz SL 2 pQ p qq " 1 24 p1 ´p´2 q ź q| Disc D pq ´1q .

Since GL 2 pQ p q acts transitively on V X p with stabiliser of the base point ˚" rZ p ˆZp s the maximal compact subgroup GL 2 pZ p q, 15 and by the centred equation mid-page 116 of [START_REF] Serre | Arbres, amalgames, SL 2 . 3ème éd. corr[END_REF] 12 See also the top of page 55 in loc. cit.

13 with a normalisation that does not need to be made precise 14 so that pMxq˚dx " }x} dx where Mx : y Þ Ñ xy is the left multiplication by x on X 15 See Section 15.1.

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The natural homomorphism G " θpO 1 max q Ñ Γ 1 Omax is 2-to-1 and rΓ 1 Omax : Γ 1 O s " rO 1 max : O 1 s, so that

VolpΓ 1 Omax z zX p q " 2 VolpGz zX p q .

Proposition 17.8 follows. l

Note that the fixed points z for the action on P 1 pQ p q " Q p Y t8u Cardtα P pΓ ¨γ0 q X Z p : hpαq ď su " 24 p VolpΓ γ 0 z z Ax γ 0 q pp `1q 2 ś q| Disc D pq ´1q rO 1 max : O 1 s rΓ 1 O : Γs s `Ops 1´κ q . l Assume furthermore that the positive rational prime p P N is such that p " 1 mod 4 and that the integer p 2 ´1 4 is not of the form 4 a p8b `7q for a, b P N (for instance p " 5). By Legendre's three squares theorem (see for instance [Gros]), there exist x 1 1 , x 1 2 , x 1 3 P Z such that

p 2 ´1 4 " x 1 1 2 `x1 2 2 `x1 3
2 . Hence there are x 1 , x 2 , x 3 P 2Z such that p 2 ´1 " x 1 2 `x2 2 `x3 2 . A standard consequence of Hensel's theorem says that when p is odd, a number n P Z has a square root in Z p if n is relatively prime to p and has a square root modulo p, see for instance [START_REF] Knapp | Advanced algebra[END_REF]page 351]. Thus, 1 ´p2 has a square root in Z p , that we denote by a 1 ´p2 . As noticed above, since p " 1 mod 4, the element ´1 has a square root in Q p , that we denote by ε. The element α 0 " ε x 1 `a1 ´p2 x 3 `ε x 2 is a quadratic irrational in Q p over Qpεq.

The following result is a counting and equidistribution result of quadratic irrationals over Qpεq in Q p . We denote by α σ the Galois conjugate of a quadratic irrational α in Q p over Qpεq, and by where k Γ is the smallest positive integer such that " 1 `ε x 1 ´x3 `ε x 2 x 3 `ε x 2 1 ´ε x 1  k Γ P Γ. Furthermore, there exists κ ą 0 such that as s Ñ `8

Cardtα P pΓ ¨α0 q X Z p : hpαq ď su " 2 p 2 k Γ pp `1q 2 rΓ 1 O : Γs s `Ops 1´κ q .

Proof. The group O ˆof invertible elements of O is O ˆ" x P O : Npxq P p Z ( .

The centre of O ˆis ZpO ˆq " t˘p n : n P Zu and the centre of O 1 is ZpO 1 q " t˘1u. We identify O 1 {ZpO 1 q with its image in O ˆ{ZpO ˆq. The quotient group O ˆ{ZpO ˆq is a free group on s " p`1 2 generators γ 1 , γ 2 , . . . , γ s , which are the images modulo ZpO ˆq of some elements of O of norm p, see for instance [START_REF] Lubotzky | Discrete groups, expanding graphs and invariant measures[END_REF]Coro. 2.1.11].17

Since Nppq " p 2 , any reduced word of even length in S " tγ 1 , γ 2 , . . . , γ s u belongs to O 1 {ZpO 1 q. Two distinct elements in S differ by a reduced word of length 2, and γ 1 does not belong to O 1 {ZpO 1 q. Hence t1, γ 1 u is a system of left coset representatives of O 1 {ZpO 1 q in O ˆ{ZpO ˆq, and the index of O 1 {ZpO 1 q in O ˆ{ZpO ˆq is rO ˆ{ZpO ˆq : O 1 {ZpO 1 qs " 2 .

(17.10) Let g 0 "

˜1`ε x 1 p ´x3 `ε x 2 p x 3 `ε x 2 p 1´ε x 1 p ¸.
By the definition of the isomorphism θ in Equation (17.8) (with ? a " ? b " ε) and of the integers x 1 , x 2 , x 3 , the element g 0 belongs to θpOq since x 1 , x 2 , x 3 are even (and p is odd), and det g 0 " 1. Hence g 0 P θpO 1 q. Its fixed points for its action by homography on P 1 pQ p q are, by an easy computation, ε x 1 ˘a1 ´p2 x 3 `ε x 2 .

In particular, α 0 is one of these two fixed points. Note that tr g 0 " 2 p , hence |v p ptr g 0 q| " 1, and the image rg 0 s of g 0 in PGL 2 pQ p q is a primitive loxodromic element of Γ 1 O . Let us define γ 0 " rg 0 s k Γ where P t˘1u is chosen so that γ 0 " α 0 and where k Γ is defined in the statement of Theorem 17.10. Since Γ has finite index in Γ 1 O , some power of rg 0 s does belong to Γ, hence k Γ exists (and note that k Γ " 1 if Γ " Γ 1 O ). By the minimality of k Γ , the element γ 0 is a primitive loxodromic element of Γ. We will apply Theorem 17.1 to this γ 0 .

The algebra isomorphism θ induces a group isomorphism from O ˆ{ZpO ˆq onto its image in PGL 2 pQ p q, that we denote by Γ Ô . 18 By [START_REF] Lubotzky | Discrete groups, expanding graphs and invariant measures[END_REF]Lem. 7.4.1], the group Γ Ô acts simply transitively on the vertices of the Bruhat-Tits tree X p .

In particular, Γ 1 O acts freely on X p , and by Equation (17.10), we have

VolpΓ 1 O z zX p q " rΓ Ô : Γ 1 O s VolpΓ Ô z zX p q " rO ˆ{ZpO ˆq : O 1 {ZpO 1 qs Card `ΓÔ zV X p ˘" 2 .

(17.11)

Again since Γ 1 O (hence Γ) acts freely on X p , and since γ 0 is primitive loxodromic in Γ, we have VolpΓ γ 0 z z Ax γ 0 q " Card `Γγ 0 zV Ax γ 0 ˘" λpγ 0 q " k Γ λprg 0 sq " 2 k Γ |v p ptr g 0 q| " 2 k Γ .

(17.12) Chapter 18

Counting and equidistribution of crossratios

We use the same notation as in Chapter 17: K v is a non-Archimedean local field, with valuation v, valuation ring O v , choice of uniformiser π v , residual field k v of order q v , and X v is the Bruhat-Tits tree of pPGL 2 , K v q. Let Γ be a lattice in PGL 2 pK v q.

In this Chapter, we give counting and equidistribution results in K v " B 8 X v ´t8u of orbit points under Γ, using a complexity defined using crossratios, which is different from the one in Chapter 17. We refer to [START_REF] Parkkonen | On the arithmetic of crossratios and generalised Mertens' formulas. Numéro Spécial "Aux croisements de la géométrie hyperbolique et de l'arithmétique[END_REF] for the development when K v is R or C with its standard absolute value.

Recall that the crossratio of four pairwise distinct points a, b, c, d in P 1 pK v q " K v Y t8u is ra, b, c, ds " pc ´aq pd ´bq pc ´bq pd ´aq P pK v q ˆ, with the standard conventions when one of the points is 8. Adopting Ahlfors's terminology in the complex case, the absolute crossratio of four pairwise distinct points a, b, c, d P P 1 pK v q " K v Y t8u is |a, b, c, d| v " |ra, b, c, ds| v " |c ´a| v |d ´b| v |c ´b| v |d ´a| v , with conventions analogous to the previous ones when one of the points is 8. As in the classical case, the crossratio and the absolute crossratio are invariant under the diagonal projective action of GL 2 pK v q on the set of quadruples of pairwise distinct points in P 1 pK v q.

18.1 Counting and equidistribution of crossratios of loxodromic fixed points

Let α, β P K v be loxodromic fixed points of Γ. Recall that α σ , β σ is the other fixed point of a loxodromic element of Γ fixing α, β, respectively. The relative height of β with respect to α is1

h α pβq " 1 |α ´ασ | v |β ´βσ | v max |β ´α| v |β σ ´ασ | v , |β ´ασ | v |β σ ´α| v ( .
is not of the form 4 a p8b `7q for a, b P N (for instance p " 5). Let ε be a square root of ´1 in Q p . Let x 1 , x 2 , x 3 P 2Z be such that p 2 ´1 " x 1 2 `x2 2 `x3 2 . We again consider α 0 " ε x 1 `a1 ´p2 x 3 `ε x 2 , which is a quadratic irrational in Q p over Qpεq. We denote by α σ the Galois conjugate of a quadratic irrational α in Q p over Qpεq, and by h α pβq " 1 mint|α, β, α σ , β σ | p , |α, β σ , α σ , β| p u (18.5)

the relative height of a quadratic irrational β in Q p over Qpεq with respect to α, such that β R tα, α σ u. We again consider Hamilton's quaternion algebra D " p ´1,´1 Q q over Q and its Z "

1 p ‰ -order O " x P Z " 1 p ‰ `Z" 1 p ‰ i `Z" 1 p ‰ j `Z" 1 p ‰ k : x " 1 mod 2 ( .
The following result says that the orbit of α 0 in Q p by homographies under a given finite index subgroup of the arithmetic group Γ 1 O equidistributes, when its complexity is given by the relative height with respect to α 0 , to a measure absolutely continuous with respect to the Haar measure on Q p . where k Γ is the smallest positive integer such that " 1 `ε x 1 ´x3 `ε x 2 x 3 `ε x 2 1 ´ε x 1  k Γ P Γ. Furthermore, there exists κ ą 0 such that, as s Ñ `8, Card Γ α 0 ztα P Γ ¨α0 : h α 0 pαq ď su " 4 p pk Γ q 2 pp `1q rΓ 1 O : Γs s `Ops 1´κ q .

Proof. This follows, as in the proof of Theorem 17.10, from Theorem 18.4 using Equations (17.11) and (17.12), as well as Equation ( 18.3) for the counting claim. l 280 19/12/2016 19/12/2016

The function φ is continuous with summable variations. Following [START_REF] Bowen | Equilibrium states and the ergodic theory of Anosov diffeomorphisms[END_REF], let us prove that φ " φ ˝r. We have φ " φ `ÿ kě0 pφ ˝σk`1 ´φ ˝σk ˝r ˝σq ´ÿ kě0 pφ ˝σk ´φ ˝σk ˝rq " φ ´φ ´ÿ kě0 ´φ ˝σk ˝r ˝σ ´φ ˝σk ˝r" ÿ kě0 ´φ ˝σk ˝r ´φ ˝σk ˝r ˝σ¯.

Now, r 2 " r and r ˝σ ˝r " r ˝σ. Hence φ ˝r " φ as claimed. Thus, φ induces on the one-sided shift a function r φ : Σ `Ñ R defined by φ : px n q nPN Þ Ñ φp. . . z x 0 ´2z x 0 ´1x 0 x 1 . . . q , satisfying φ " r φ ˝π. To conclude, observe that φ ´φ is bounded and that cylinders defined by the same finite words have the same measure for an invariant probability measure m on the two-sided shift pΣ, σq and for its image π ˚m on the one-sided shift pΣ `, σ `q. Therefore m is a weak Gibbs measure for φ if and only if π ˚m is a weak Gibbs measure for r φ. By construction π ˚mp r φq " mpφq " mpφq since m is invariant. As it is well-known, the natural extension π preserves the entropy. Thus, the measure m is an equilibrium measure with respect to φ if and only if π ˚m is an equilibrium measure with respect to r φ. The reduction to one-sided topological Markov shifts is thus complete. l

A.2 Proof of the main result Theorem A.4

The uniqueness of the equilibrium state is given by Theorem A.3. We need to prove that weak Gibbs measures and equilibrium measures coincide under the integrability assumption on φ ´and that the number cpmq is equal to the pressure.

Step 1. If m is an equilibrium measure, then it is a weak Gibbs measure. This is a routine consequence of Theorem A.3. Our definition of an equilibrium measure m enforces ş φ ´dm ă `8 (hences excludes the concomitance of h m pσq " `8 and ş φ dm " ´8).

Recall from Theorem A.3 that dm " h dν. For v P V G and x P Fix n pΣq X rvs, we have mpC n pxqq " ż h 1 Cnpxq dν " e ´n P G pφq ż L n φ phq1 Cnpxq dν .

By definition, L n φ phq1 Cnpxq pzq " exp S n φpx 0 . . . x n´1 zq hpx 0 . . . x n´1 zq for all z P σ n pC n pxqq " rvs (and L n φ phq1 Cnpxq pzq " 0 otherwise). Hence mpC n pxqq " e ´n P G pφq exp ´Sn φpxq ˘n ÿ

k"1 var k pφq ¯żrvs h dν .

As 0 ă ş rvs h dν ă `8 and ř `8 k"1 var k pφq ă `8, the measure m is a weak Gibbs measure for φ with Gibbs constant cpmq " P G pφq. We now turn to the converse implication. Let m be a weak Gibbs measure for φ such that ş φ ´dm ă `8.

The weak Gibbs condition only controls the cylinders that start and end with the same symbol. Passing to an induced system (that is, considering a first return map on a 1-cylinder) will remove this restriction. More precisely, let a P V G be a vertex of G and let µ be an invariant probability measure on pΣ, σq with µprasq ą 0. The induced system on the 1-cylinder ras " tx P Σ : x 0 " au is the map σ : ras Ñ ras defined as follows:

' let τ pxq " inftn ě 1 : σ n x P rasu be the first-return time in ras, that we also denote by τ ras pxq when we want to emphasize ras;

' let σpxq " σ τ pxq pxq if τ pxq ă 8;

' let µpBq " µpB X rasq{µprasq for every Borel subset B of Σ be the restriction of µ to ras normalized to be a probability measure.

We also define τ 0 pxq " 0 and by induction τ n`1 pxq " τ pxq `τ n pσxq for every n P N. Note that σ can only be iterated on the subset tx P ras : @ n ě 1, τ n pxq ă 8u .

By Poincaré's recurrence theorem, this is a full measure subset of ras, hence the distinction will be irrelevant for our purposes. The induced partition is β " traξ 1 . . . ξ n´1 as ‰ H : n ě 1, ξ i ‰ au .

We note that σ : ras Ñ ras is topologically Bernoulli with respect to the partition β (that is, σ : b Ñ ras is a homeomorphism for each b P β). For every integer N ě 1, we define the N -th iterated partition β N of β by β N " tb 0 X σ ´1b 1 X ¨¨¨X σ ´N `1b N ´1 ‰ H : b 0 , . . . , b N ´1 P βu and we write β N pxq for the element of the partition β N that contains x.

Step 2. The topological Markov shift may be assumed to be topologically mixing.

This follows from the spectral decomposition for topological Markov shifts, see for instance [START_REF] Buzzi | Uniqueness of equilibrium measures for countable Markov shifts and multidimensional piecewise expanding maps[END_REF]Lem. 2.2].

Step 3. The Gibbs property implies full support and ergodicity.

Let A be an invariant (σ ´1pAq " A) measurable subset of Σ with mpAq ą 0 and let us prove that mpAq " 1.

Observe that the Gibbs property, together with the transitivity of Σ, implies that any cylinder has positive measure for m, hence that m has full support. Let a P V G be such that mpA X rasq ą 0.

As mprasq ą 0, we may consider the induced system on ras. Let N ě 1. When f is a homeomorphism between topological spaces, let f ˚denote the pushforwards of measures dpσ N q ˚m dm pyq " lim nÑ8 mpσ τ N pyq ry 0 . . . y n sq mpry 0 . . . y n sq " C ˘2e ´Sτ N pyq φpyq`τ N pyq cpmq .

Hence, mpA X rasq mprasq " C ˘4 mpA X β N pxqq mpβ N pxqq .

By Doob's increasing martingale convergence theorem (see for instance [Pet]), for m-almost every x P ras ´A, the ratio on the right hand side converges to 0 as N Ñ 8. Thus ras is contained in A modulo m. Therefore A " Ť aPW ras modulo m for some subset W of V G . Since Σ is topologically mixing, for any vertex b, the intersection ras X σ ´irbs X σ ´j ras is not empty for some integers 0 ă i ă j. Pick some point x in that set. By invariance, mprbsq ě mpσ i pC j pxqqq ě mpC j pxqq. But this last number is positive by the weak Gibbs property. Thus rbs is contained in A modulo m. Hence mpAq " 1, proving the ergodicity of m.

Step 4. The Gurevič pressure P G pφq is equal to cpmq, hence is finite. Furthermore h m pσq ă 8 and φ P L1 pmq.

Fix v P V G and let K " tx P Σ : x 0 " vu. Note that mpKq ą 0. The ergodicity of m gives a Cesaro convergence: as n Ñ 8, we have If we write Z n for the term between the parenthesis, we have by the definition of the Gurevič pressure:

P G pφq " lim sup nÑ8 1 n ln Z n .
As the value of the left hand side of Equation (A.2) is less than one, we see that cpmq ě P G pφq.

If this was a strict inequality, then the left hand side of Equation (A.2) would converge to zero, contradicting its Cesaro convergence to mpKq 2 ą 0. Therefore P G pφq " cpmq.

Since cpmq is finite, so is P G pφq. Hence Theorem A.2 implies that, for any ν P PpΣq with ş φ ´dν ă `8, we have h ν pσq ă 8 and φ is ν-integrable. In particular, this holds for ν " m, which finishes the proof of Step 4. standard point at infinity r1 : 0s of a projective plane ˘D with fibers the stable/unstable leaves r F potential on T 1 X F potential on ΓzT 1 X F q finite field of order a prime power q g genus of the smooth projective curve C p G X space of generalised geodesic lines in X p G X space of generalised discrete geodesic lines in a simplicial tree X p G even X space of generalised discrete geodesic lines in X with dp p0q, x 0 q even 298 19/12/2016 G even X space of discrete geodesic lines in X with p0q at even distance from x 0 G ˘X space of generalised positive/negative geodesic rays in X G ˘, 0 X space of generalised geodesic lines in X isometric exactly on ˘r0, `8r pg t q tPR (continuous time) geodesic flow on space of generalised geodesics p G X pg t q tPZ (discrete time) geodesic flow on space of generalised geodesics p G X h m pT q metric entropy of a transformation T with respect to a probability measure m h m pφ 1 q metric entropy of a flow pφ t q tPR with respect to a probability measure m hpαq complexity of loxodromic fixed point α hpαq , 116, 117 cohomologous, 45, 50, 87 common perpendicular, 34, 197 ending transversally, 197 endpoint, 197 multiplicity, 190 origin, 197 starting transversally, 197 complexity, 258, 262, 270 conductance, 50, 197

complexity

  p G Y ˆp G Y , we have lim tÑ`8 δ F }m F } e ´δF t ÿ αPPerppD ´, D `, tq e ş α F ∆ v ά b ∆ v ὰ " σ D´b σ D`.

  Define f D : U D Ñ B 1 ˘D as the composition of the continuous endpoint map Þ Ñ ˘from U D onto B 8 X ´B8 D and the homeomorphism P D from B 8 X ´B8 D to B 1 ˘D.

Proposition 4. 1 .

 1 There exists at least one Patterson density for the pair pΓ, r F ˘q.

F

  ˘´δq ď c .

rc

  ˘pxq is constant on the Γ-orbit of x. Hence, it defines a map deg c ˘: ΓzV X Ñ s0, `8r . 91 19/12/2016

  94 19/12/2016 of variable e Þ Ñ e in half the value of the second line, we have x∆ c f, gy vol " ÿ xPV Y 1 |Γ x | ÿ ePEY, opeq"x ipeq ppeq `f pxq ´f ptpeqq ˘gpxq " ÿ ePEY ipeq |Γ opeq | ppeq `f popeqq gpopeqq ´f ptpeqq gpopeqq " ppeq `f ptpeqq ´f popeqq ˘`gptpeqq ´gpopeqq " xd c f, d c gy Tvol .

D

  on B 1 `D and the inner skinning measure r σ D on B 1 ´D associated with the Patterson densities pµ x q xPX for pΓ, r F ˘q are the measures r σ D " r σ D, F ˘defined by dr σ Dpρq " e C ρ˘p x 0 , ρp0qq dµ x0 pρ ˘q , (7.1)

  Remark 7.1. (1) If D " txu is a singleton, then dr σ Dpρq " dµ x pρ ˘q (7.2) 103 19/12/2016

  Proposition 7.3. Let D be a nonempty proper closed convex subset of X, and let r σ D be the skinning measures on B 1 ˘D for the potential r F . (i) The skinning measures r σ D are independent of x 0 . (ii) For all γ P Γ, we have γ ˚r σ D " r σ γD . In particular, the measures r σ D are invariant under the stabiliser of D in Γ. (iii) For all s ě 0 and w P B 1 ˘D, denoting by pg ˘swq |˘r0,`8r the element of G ˘, 0 which coincides with g ˘sw on ˘r0, `8r, we have d r σ NsD ppg ˘swq |˘r0,`8r q " e C w˘p πpwq, πpg ˘swqq dr σ Dpwq " e ´şπpg ˘swq πpwq p r F ˘´δq dr σ Dpwq .

  109 19/12/2016 The measures r σ D and r σ D are respectively called the outer and inner skinning measures of D on p G X, and their induced 3 measures σ D and σ D on Γz p G X are the outer and inner skinning measures of D on Γz p G X.

  There are two cases to consider.

ÿ

  ePEX : opeqPV D, tpeqRV D µ opeq pB e Xq ∆ opeq " ÿ xPBV D ´ÿ ePEX : opeq"x, tpeqRV D µ x pB e Xq ¯∆x . and similarly π ˚r σ D " ÿ ePEX : tpeqPV D, opeqRV D µ tpeq pB e Xq ∆ tpeq " ÿ xPBV D ´ÿ ePEX : opeq"x, tpeqRV D µ x pB e Xq ¯∆x .As in the proof of Proposition 8.1 (2), since X is spherically homogeneous around each point and since Γ is a lattice (so that the Patterson density is AutpXq-equivariant, see Proposition 4.14), we have µ x pB e Xq " }µx} deg X pxq for all x P V X and e P EX with opeq " x. Assertion (1) of Proposition 8.4 follows, since ř ePEX : opeq"x, tpeqRV D 1 " codeg D pxq if x P BV D and codeg D pxq " 0 otherwise. 118 19/12/2016

  13

  either ΓzX is compact and the length spectrum of ΓzX is 2-Diophantine, (b) or there exists a finite subset E of vertices of ΓzX such that

d

  BW pry, s 1 s, ry, s 1 sq ď dpy, yq `|s 1 ´s1 | ď e N dpx, xq `|s ´s| ď e N d BW prx, ss, rx, s sq ď e T inf λ `1 d BW prx, ss, rx, s sq .

  B 1 `X1 by the canonical projection p G X Ñ Γz p G X, such that Θ ˚P is the normalised skinning measure σ D }σ D } and that the following diagram commutes for all n P N :

F

  ∆ γy b ∆ γ ´1x " µ x b µ ý for the weak-star convergence of measures on the compact space pX Y B 8 Xq ˆpX Y B 8 Xq. l

żF

  vPB `pw ´, Rq dµ W `pg t{2 w ´qpg t{2 vq " ż vPB `pw ´, Rq e C v´p πpvq, πpg t{2 vqq dµ W `pw ´qpvq " ż vPB `pw ´, Rq e ş πpg t{2 vq πpvq p r F ´δq dµ W `pw ´qpvq . (11.22) 170 19/12/2016Similarly, for every a ą 0, we have µ W `pv 0 γ q pB `pv 0 γ , ar t qq " ´δq dµ W `pv 0 γ q pvq .

ÿ

  γPΓ : 0ăλγ ďN α γ | s0,λγ s PΩ ´|s0,λγ s , α γ | s´λγ ,0s PΩ `|s´λγ ,0se

  , γqPΓ{Γ D ´ˆΓ{Γ D `ˆΓ 0ădpαD ´, γβD `qďt ∆ α ά, γβ b ∆ α γ´1 α, β " r σ D´b r σ Dfor the weak-star convergence of measures on the locally compact space p G X ˆp G X. If the measure σ D`i s nonzero and finite, then lim

  r f ˘is on each connected component of r D ˘an isometric embedding whose image is a proper nonempty closed locally convex subset of r Y , and the family of images under the covering group of r Y Ñ Y of the images by r f ˘of the connected components of r D ˘is locally finite. Then D ˘(or the pair pD ˘, f ˘q) is a proper nonempty properly immersed closed locally convex subset of Y . If Γ is a discrete subgroup without torsion of isometries of a CAT(´1) Riemannian manifold X, if D ˘" pγ r D ˘qγPΓ where r D ˘is a nonempty proper closed convex subset of X such that the family D ˘is locally finite, and if D ˘is the image of r D ˘by the covering map X Ñ ΓzX, then D ˘is a proper nonempty properly immersed closed convex subset of ΓzX. Under these assumptions, N D ´, D `, F is the counting function N D ´, D `, F given in the introduction. 1 not necessarily connected 182 19/12/2016

  we have, as n Ñ `8, e δc ´1 e δc }m c } e ´δc n ÿ rγs PΓ D ´zΓ{Γ D 0ădpD ´,γ D `qďn m e, γ e cpαe, γ q ψ ´pΓα é,γ q ψ `pΓα γ´1 ,e q " ż ψ ´dσ D´ż ψ `dσ D``O `e´κ 1 n }ψ ´}β }ψ ´}β ȃnd if Γ D ˘zBD ˘is compact, then N D ´, D `pnq " e δc }σ D´} }σ D`} pe δc ´1q }m c } e δc n `O `epδc´κ 1 qn ˘.

  The maps Φ η are β-Hölder-continuous with}Φ η } β " Op} r ψ ˘}β q .

  e 2δc ´1 2 e 2δc }m c } e ´δc n ÿ rγs PΓ D ´zΓ{Γ D 0ădpD´,γ D `qďn m e, γ e cpαe, γ q ψ ´pΓα é,γ q ψ `pΓα γ´1 ,e q "żψ ´dσ D´ż ψ `dσ D``O `e´κ 1 n }ψ ´}β }ψ ´}β 205 19/12/2016 and if Γ D ˘zBD ˘is compact, then N D ´, D `pnq " 2 e 2δc }σ D´} }σ D`} pe 2δc ´1q }m c } e δc n `O `epδc´κ 1 qn ˘.

  we have, as T Ñ `8,δ c }m c } e ´δc T ÿ rγs PΓ D ´zΓ{Γ D 0ădpD´,γ D `qďT m e, γ e cpαe, γ q ψ ´pΓα é,γ q ψ `pΓα γ´1 ,e q " }ψ ´}k, β }ψ ´}k, β ȃnd if Γ D ˘zBD ˘is compact, then for every n P N N D ´, D `pT q " }σ D´} }σ D`} δ c }m c } e δc T `O `eδc T T ´n˘.

  Theorem 13.3. Assume that the critical exponent δ c of c is finite and positive and that the Gibbs measure m c of c is finite and mixing for the continuous time geodesic flow. As t Ñ `8, the measures δ c e ´δc t ÿ gPPer 1 ptq e cpgq L g and δ c t e ´δc t ÿ gPPer 1 ptq e cpgq L g pgq converge to mc }mc} for the weak-star convergence of measures. If Γ is geometrically finite, the convergence holds for the narrow convergence.

VFc

  pxq " pξ, ηq P pX Y B 8 Xq 2 : ξ ‰ η, x P sξ, ηr ( , which is an open subset of X Y B 8 X. Note that the family pV pyqq yPX covers the set of pairs of distinct points of B 8 X. For every t ą 0, let ν t be the measure on pX Y B 8 Xq 2 defined by ν t " δ c }m c } e ´δct ÿ ∆ γ ´1x b ∆ γx .

  of the absolute value | ¨|v and of Hamenstädt's distance, see Equation (14.1) and the above geometric interpretation, and Equation (2.8). Note that in [Pau1], Hamenstädt's distance in a regular tree is defined in a different way: In that reference, the distance |η ´η1 | v equals Hamenstädt's distance between η and η 1 . 1 or linear fractional transformations 228 19/12/2016

  ´|v |ρ `´L ´|v |ρ `´L `|v d Haar Kv pρ `q .

Proof.

  The second result follows from the first one by integrating on the characteristic function of the compact-open subset O v , whose Haar measure is 1.

  By restricting the measures to the compact-open subset π ´k v `Ov 261 19/12/2016

  9

  dµpg ¨zq " d Haar Kv pg ¨zq |g ¨z ´α| v |g ¨z ´ασ | v " d Haar Kv pg ¨zq |g ¨z ´g ¨α| v |g ¨z ´g ¨ασ | v "

  rΓ γ 0 : γ Z 0 s " rG α 0 : g Z 0

  Theorem 18.7. With the above notation, let Γ be a finite index subgroup of Γ 1 O . Then, as s Ñ

  29119/12/2016 by f ´1. First note that, for almost every x P ras and every N P N ´t0u, since σ N is an homeomorphism from β N pxq onto ras, we have mpA X rasq mprasq " mpσ N pA X β N pxqqq mpσ N pβ N pxqqq " 1 that for m-almost every y P β N pxq:

  X σ ´kKq ÝÑ mpKq 2 ą 0 .The Gibbs property implies that, for all n ě 1, mpK X σ ´nKq " C

  a subset A " " " D equivalence relation on index set of an equivariant family D ||f || α α-Hölder norm of f P C α c pZq ||ψ|| Sobolev W ,2 -norm of ψ P C c pN q | ¨|v (normalised) absolute value associated to a valuation v ˚v base point ˚v " rO v ˆOv s of the Bruhat-Tits tree X vApO v q maximal compact-open diagonal subgroup of PGL 2 pK v q AutpX, λq automorphism group (edge-preserving, without inversion) of a metric tree pX, λq Aut X automorphism group (without inversion) of a simplicial tree X Bpx, rq closed ball of center x and radius r in a metric space B ˘pw, η 1 q Hamenstädt's ball of radius η 1 ą 0 with center any geodesic line extension of w P G ˘X C geometrically connected smooth projective curve over F q c A complementary set of a subset A cpgq period for a system of conductances c of a closed orbit g for the geodesic flow C c pZq space of real-valued continuous maps with compact support on Z C α b pZq space of bounded α-Hölder-continuous real-valued functions on Z C k, α b pZq space of real-valued functions on Z with bounded α-Hölder-continuous derivatives of order at most k along the flow C α c pZq space of α-Hölder-continuous real-valued functions with compact support on Z C k, α c pZq space of real-valued functions on Z with bounded α-Hölder-continuous derivatives of order at most k along the flow and compact support C c pN q space of real-valued C -smooth functions with compact support on a smooth manifold N codeg D pxq codegree of a vertex x with respect to a subtree D codeg D pxq codegree of a vertex x with respect to a family of subtrees D 297 19/12/2016 cov m, n pφ, ψq n-th correlation coefficient of φ, ψ for the measure m under a transformation cov µ, t pψ, ψ 1 q correlation coefficient of ψ, ψ 1 under a flow at time t for the measure µ cov µv, g correlation coefficient for g P G v and a measure µ v on ΓzG v C ΛΓ convex hull in X of the limit set ΛΓ of Γ x conjugate of a quaternion x ra, b, c, ds crossratio of pairwise distinct points a, b, c, d |a, b, c, d| v absolute crossratio of pairwise distinct points a, b, c, d for a valuation v B 8 X boundary at infinity of X B e X set of points at infinity of the geodesic rays whose initial (oriented) edge is e BV D boundary of set of vertices of a simplicial subtree D BD maximal subgraph with set of vertices BV D B 1 ´D inner unit normal bundle of a closed convex subset D B 1 `D outer unit normal bundle of a closed convex subset D deg v degree of valuation v, equal to dim Fq k v δ " δ Γ, F critical exponent of pΓ, F q ∆ c ˘Laplacian operator associated to a system of conductances c ˘92 ∆ x unit Dirac mass at a point x Disc D reduced discriminant of a quaternion algebra D over Q d D distance-like map on B 8 X ´B8 D associated with closed convex subset D d " d p G X distance on space of generalised geodesic lines d " d T 1 X distance on space of germs of geodesic lines d H Hamenstädt's distance at infinity associated to an horoball H d x visual distance on B 8 X seen from x P X d W ˘pwq Hamenstädt's distance on the strong stable/unstable ball of w P G ˘X EX set of edges of a graph X ϕ Rv Euler function of fonction ring R v f ´negative part of a real-valued map f f D fibration over B 1

  By Lemma 2.4, we have dpg s , p wq ď c d W `pwq pg s , p wq ď c η 1 .

		p
	By the definition of the distance on	G X (see Equation (2.4)), we have
		ż sw
	dp p w, wq ď	´8 |s

w ´t| e ´2|t| dt " Ope sw q .

  ˘, be as in the statement. By the definition of the sets V η, R pB 1 ¯D˘q , there exist geodesic rays w ˘P B 1 ¯D˘, geodesic lines p w ˘P G X extending w ˘, and s ˘P s´η, `ηr, such that ˘" pw ¯q˘a nd d W ˘pw ¯qpg ¯t 2 ¯s¯ , p w ¯q ď R . Let x ˘be the closest point to w ˘p0q on . By the definition of Hamenstädt's distances, we have dpw ˘p0q, x

˘q " dp p˘t 2 ˘s˘q , x ˘q ď ln R , and in particular x ˘" w ˘p0q if and only if p˘t 2 ˘s˘q " w ˘p0q. As t ě 2 ln R `4 and |s ˘| ď 2η ď 2, the points p´t 2 ´s´q , x ´, p0q, x `, p t 2

  If furthermore E 1 " R and f is bounded, then

	sup x,yPE, x‰y	|f pxq ´f pyq| dpx, yq α	"	sup x,yPE, dpx, yqą	|f pxq ´f pyq| dpx, yq α	ď	2 α }f } 8 .

  Let r ą 0, x, y P X and ξ P B 8 X be such that ξ P O x Bpy, rq. Let p be the closest point to y on rx, ξr , so that dpp, yq ď r. By Equations (3.8) and (3.7), we have

	(4) ˇˇC ξ px, yq	`ż y	p r F ˘´δq ˇˇ" ˇˇC ξ px, yq ´Cξ px, pq	´ż p	p r F ˘´δq	`ż y	p r F ˘´δq ˇď
		x						x	x
				| C ξ pp, yq|	`ˇˇż	p	p r F ˘´δq	´ż y	p r F ˘´δq ˇˇ.	(3.10)
						x		x
	First assume that X is an R-tree. Then by Assertion (2) and by Lemma 3.3, we deduce
	from Equation (3.10) that
	ˇˇC ξ px, yq	`ż y x	p r F ˘´δq ˇˇď 2 dpy, pq max π ´1pry,psq	| r F ˘´δ| ď 2r max π ´1pBpy, rqq
	49							19/12/2016

Patterson density for the pair pΓ, r F ˘q if

		γ ˚µx " µ γx	(4.1)
	for all γ P Γ and x P X, and if the following Radon-Nikodym derivatives exist for all x, y P X
	and satisfy for (almost) all ξ P B 8 X
	dµ dµ	x y pξq " e	´Cξ px, yq .	(4.2)

  Patterson densities by Corollary 4.6, we have that m F is finite if and only if m F ˚is finite, and then m F

	F	}m ˚} "	m F }m F }	.	(4.9)
	59				19/12/2016

  etq dµ ópeq p ´q dµ òpeq p `q dt .Since ´, opeq, e t , `are in this order on the geodesic line with ´P B e X and `P B e X, we have C ´ | }µ ópeq } }µ òpeq } " xφ μ ˝o, φ μ ˝oy 2 ď }φ μ ˝o} 2 }φ μ ˝o} 2 ,

	e X `popeq, ´popeq, e t q `C` ż `PBeX ż λpeq 0 e C ´ ´popeq, etq`C ` `popeq, e t q " 0 by Equation (3.8). Hence
	}m F } "	ÿ resPΓzEX	λpeq |Γ e |	µ ópeq pB e Xq µ òpeq pB e Xq
	ÿ resPΓzEX |Γ e which finishes the proof. λpeq ď	l

  For every y P Bpx, nq, again since tO x pzq : z P Spx, nqu is a Borel partition of B 8 X, and by Equation (6.7), we have By Steps 1 and 3, we have G c ¨f1 pyq " G c ¨f2 pyq for every y P Bpx, nq. Let now y P V X ´Bpx, nq. Since x P Bpx, nq, we have as just said G c ¨f1 pxq " G c ¨f2 pxq.

	Step 3: 1 " }ν y } "	ÿ zPSpx,nq	ν y pO x pzqq "	ÿ zPSpx,nq	G c py, zq	ν x pO x pzqq G c px, zq	" G c ¨f2 pyq .	(6.8)
	Step 4: Hence by
	the first claim of Lemma 6.5 (3), we have
			ÿ
	G c ¨f1 pyq "			G c py, zq f 1 pzq
			zPSpx,nq
		"	ÿ zPSpx,nq	e cpy, xq`δcpdpx, zq´dpy, zqq φ µ pxq φ µ pyq	G
									(6.7)
	101								19/12/2016

z n P C x pzqs , so that ν y pO x pzqq ν x pO x pzqq " G c py, zq G c px, zq . c px, zq f 1 pzq " e cpy, xq`δcpdpx, zq´dpy, zqq φ µ pxq φ µ pyq G c ¨f1 pxq " e cpy, xq`δcpdpx, zq´dpy, zqq φ µ pxq φ µ pyq G c ¨f2 pxq " G c ¨f2 pyq . This proves that G c ¨f1 " G c ¨f2 , thereby concluding the proof of Theorem 6.7. l 102 19/12/2016

  when F " 0.Proposition 7.2. Let ρ : r0, `8r Ñ X be the geodesic ray starting from any point of the boundary of H and converging to ξ. The following weak-star limit of measures on B 8 X ´tξu dµ H pηq " lim F ˘´δq dµ ρptq pηq exists, and it does not depend on the choice of ρ. The measure µ H is invariant under the elements of Γ preserving H , and it satisfies, for every x P X and (almost) every η P B 8 X tξu, ) with x replaced by ρptq and y by the present x, by the cocycle equation (3.7) and by Equation (3.8) as z t P rρptq, ηr , we have As t Ñ `8, note that z t converges to P H pηq and that by the HC-property (and since r F is bounded on any compact neighbourhood of P H pηq), we have The result then follows by the continuity of the Gibbs cocycle (see Proposition 3.10 (3)). l Using this proposition and the cocycle property of C ˘in the definition (4.3) of the Gibbs measure, we obtain, for every P G X such that ˘‰ ξ, `pP H p `q, p0qq dµ H p ´q dµ H p `q dt .

	d r m F p q " e	C ´ ´pP H p ´q, p0qq `C`	(7.3)
	Note that it is easy to see that for every ρ P B 1 ˘H , we have
				dr σ H pρq " dµ H pρ ˘q .	(7.4)
	When F " 0, we obtain Hamenstädt's measure
				µ H " lim tÑ`8	e δ Γ t µ ρptq	(7.5)
	on B
				tÑ`8	e	´şP H pηq ρptq	p r
			dµ
					ρp0q
	ξ	ρptq		z t	η
						p H pηq
	Using Equation (4.2e ´şP H pηq ρptq p r F ˘´δq dµ ρptq pηq " e	´şP H pηq ρptq	p r F ˘´δq e	´Cη pρptq, xq dµ x pηq
				" e	´şP H pηq ρptq	p r F ˘´δq e	´Cη pρptq, ztq e	´Cη pzt, xq dµ x pηq
				" e	´şP H pηq ρptq	p r F ˘´δq`ş	z t ρptq p r F ˘´δq e ´Cη pzt, xq dµ x pηq .
		ˇˇż	P H pηq	p r F ˘´δq	´ż zt	p r F ˘´δq ˇˇÑ 0 .
		ρptq			ρptq
	104					19/12/2016

H dµ

x pηq " e ´Cη pP H pηq, xq .

Proof. We prove all three assertions simultaneously. Let us fix x P X. For all t ě 0 and η P B 8 X ´tξu, let z t be the closest point to P H pηq on the geodesic ray from ρpyq to η. 8 X ´tξu associated with the horoball H , which is independent of the choice of the geodesic ray ρ starting from a point of the horosphere BH and converging to ξ. Note that for every t ě 0, if H rts is the horoball contained in H whose boundary is at distance t from the boundary of H , we then have µ H rts " e ´δΓ t µ H . (7.6)

  Proposition 7.4. Let D and D 1 be nonempty closed convex subsets of X and h ˘" h D,D 1 . The measures ph ˘q˚r σ D and r σ D1 on P D1 pA D,D 1 q are absolutely continuous one with respect to the other, with dph ˘q˚r σ ¯HB ˘pwq the canonical homeomorphism defined in Section 2.5, we define the skinning measures µ W ˘pwq on the strong stable or strong unstable leaves W ˘pwq by µ W ˘pwq " ppN w q ´1q ˚r σ HB ˘pwq ,

	so that	dµ W ˘pwq p q " e	C ¯ ¯px 0 , p0qq dµ x0 p ¯q	(7.9)
	for every P W ˘pwq. By Proposition 7.3 (ii) and the naturality of N w , for every γ P Γ, we
	have
		γ ˚µW ˘pwq " µ W ˘pγwq .	(7.10)
	By Proposition 7.3 (iv), the support of µ W C ¯ ¯p	ptq, p0qq	" e C w˘p p0q, ptqq .	(7.11)
	106				19/12/2016

D

dr σ D1 pw 1 q " e ´Cw ˘pπpwq, πpw 1 qq , for (almost) all w P P D pA D,D 1 q and w 1 " h ˘pwq. l

Let w P G ˘X . With N w : W ˘pwq Ñ B 1 ˘pwq is t P W ˘pwq : ¯P ΛΓu. For all t P R and P W ˘pwq, we have, using Equations (3.7) and (3.8), and since ˘" w ˘, dpg ´tq ˚µW ˘pwq d µ W ˘pg t wq pg t q " e

  Hopf's parametrisation with time parameter t and the definitions of r m F (see Equation (4.3)) and of U D (see Equation (2.10)),

	•
	107	19/12/2016

  Since µ opeq pB e Xq " e ´hX µ tpeq pB e Xq and by homogeneity, we have pT πq ˚mBM "

		ÿ resPΓzEX	1 |Γ e |	deg opeq deg opeq ´1	}µ opeq }	deg tpeq deg tpeq ´1	}µ tpeq } e ´hX ∆ res
	"	ÿ resPΓzEX	1 |Γ e |	}µ opeq } }µ tpeq } pp `1qpq `1q ? pq	∆

πq ˚mBM " ÿ resPΓzEX 1 |Γ e | µ opeq pB 8 X ´Be Xq µ opeq pB e Xq ∆ res . res " Tvol Γz zX .

  Example 8.3. Let G be a connected graph without vertices of degree ď 2 and let X be its universal cover, with covering group Γ. If C is a cycle in G " ΓzX and if D is the family of geodesic lines in X lifting C, then m D pxq " 1 for all x P V X whose image in G " ΓzX belongs to C if C is a simple cycle (that is, if C passes through no vertex twice).

	116	19/12/2016

1 See Section 12.2 for explanations on the terminology.

  P t0, . . . , |w| ´2u and x, y P ∆ w , we have dpσ n `x, σ n `yq " e n dpx, yq ď e |w|´2 dpx, yq ă dpF pxq, F pyqq ;

x, y have the same |w| first components, we have dpF pxq, F pyqq " dpσ |w|´1 `x, σ |w|´1 `yq " e |w|´1 dpx, yq ě e dpx, yq ; 128 19/12/2016 ' for all w P W , n

  rpσ i xq `s1 and 0 ď s 1 ď rpσ n xq .

	This proves Assertion (1) since π `˝σ " σ `˝π `.
	n´1
	ÿ
	t `s "
	i"0
	Hence
	pσ `qt r `˝π
	135	19/12/2016

r `prx, ssq " pσ `qt r `prπ `pxq, ssq " rσ n `π`p xq, s 1 s and π r `˝σ t r prx, ssq " π r `prσ n x, s 1 sq " rπ `pσ n xq, s 1 s .

  BW prx, ss, rx, s sq ě C 3 e ´T inf λ , then d BW pry, s 1 s, ry, s 1 sq ď 1 `sup λ ď BW prx, ss, rx, s sq ă C 3 e ´T inf λ . Assume that d BW prx, ss, rx, s sq " dpx, xq `|s ´s| ,

	1 `sup λ C 3	e
	Therefore Equation (9.9) follows from Equation (9.10) whenever C 2 ě p1`sup λq α 3 C α	.
	Conversely, suppose that d the other possibilities are treated similarly. Since
	s `T "

1 α d BW pry, s 1 s, ry, s 1 sq α . (9.10) 136 19/12/2016 If d T inf λ d BW prx, ss, rx, s sq .

  19/12/2016 11.1 Part I of the proof of Theorem 11.1: the common partStep 1: Reduction. By additivity, by the local finiteness of the families D ˘, and by the definition of r σ D¯" ř kPI ¯{" r σ Dk , we only have to prove, for all fixed i P I ´and j P I `, that

					ş
	lim tÑ`8	δ }m F } e ´δ t	γPΓ : 0ăλ i, γj ďt ÿ	e	α i,γj	r

  r t qq " e Opηq µ xγ pB dx γ pξ γ , R e Opηq p2ηq 2 µ xγ pB dx γ pξ γ , Re

	Opηq)
	j η, γ pT q "	´λγ 2 q 4η 2 e ´λγ e Ope 2 qq
	" e Opη`e ´λγ 2 q ,

´λγ 2 qq . (11.15) Thus, by the above and by Equations (11.13) and (11.14), (and noting that Opηq `Opηq " ´λγ 2 qqµ xγ pB dx γ pξ γ , Re ´λγ 2 qq µ xγ pB dx γ pξ γ , Re ´λγ 2 qqµ xγ pB dx γ pξ γ , Re

  The remainder of the proof of Theorem 11.10 consists in proving versions of the equidistribution result Equation (11.25) respectively in p G odd ˆp G odd , p G even ˆp G odd , p G odd ˆp G even , and in summing these four contributions.By applying Equation (11.25) by replacing x 0 by a vertex x 1 0 in V odd X, which exchanges V even X and V odd X,

	p	p
	G even and	G odd , as well as r σ D¯, even and r σ D¯, odd , we have
	lim tÑ`8
	.
	176	19/12/2016

  k, β b pΓzG Xq we have as t Ñ `8 cov mc, t pψ, ψ 1 q " Opt ´N n }ψ} k, β }ψ 1 } k, β q ,

	where N P N ´t0u is a constant which will be made precise later on. p
	Let us first prove that for all r ψ ˘P C k, β c	p G Xq, we have, as T Ñ `8,
		δ c }m c } e ´δc T	ÿ	e cpαγ q	r ψ ´pα γ q r ψ `pα γ q
				γPΓ, 0ăλγ ďT
		ż
	"	`D´r B 1 ψ ´dr σ	´D`r `żB 1 ψ `dr σ ´`OpT ´n } r ψ ´}k, β } r ψ `}k, β q .
					(12.18)
	206				19/12/2016

  ), respectively if Y is bipartite or not.

	Chapter 13	l
	209 210	19/12/2016 19/12/2016

  uniformiser of F . The residual field of the valuation v isk v " O v {m v . When k v is finite, the valuation v defines a (normalised, non-Archimedean) absolute value | ¨|v on F by |x| v " |k v | ´vpxq ,with the convention that |k v | ´8 " 0. This absolute value induces an ultrametric distance on F by px, yq Þ Ñ |x ´y| v .

	which is called a
	ÿ gPPer 1 ptq	e cpgq "	e δc t δ c t
	as t Ñ `8.
	(2) If λ " 1 and if the Gibbs measure m c is finite and mixing for the discrete time geodesic
	flow, then
	ÿ gPPer 1 pnq	e cpgq "	e δc e δc ´1 e δc n n
	as n Ñ `8.			l
	10 See the introduction of [Sha] for comments.
	217			19/12/2016 19/12/2016

  The outer/inner skinning measures of the singleton t˚vu are given by d r σ t˚vu pρq " dµ ˚v pρ ˘q " maxt1, |ρ ˘|v u ´2 d Haar Kv pρ ˘q on the set of ρ P B 1 ˘t˚vu such that ρ ˘‰ 8. (2) The total mass of the Patterson density is

	(1)
	232	19/12/2016

  1, we have ξ P O v if and only if P H8 pξq " ˚v.10 Recall that P H8 : BvXv ´t8u Ñ BH8 is the closest point map to the horoball H8, see Section 2.5.For every ξ P K v ´Ov , by Equations (15.2) and (2.8), we have |ξ| v " d H8 p0, ξq ln qv " q On the set of geodesic rays ρ P B 1 ˘t˚vu such that ρ ˘‰ 8, by Equation (7.2), by the last claim of Proposition 7.2, by Equation (15.8), 11 since P H8 pρ ˘q belongs to the geodesic ray r˚v, ρ ˘r (even when ρ ˘P O v ), and by Equation (15.9), we have dr σ t˚vu pρq " dµ ˚v pρ ˘q " e Cρ ˘pP H8 pρ ˘q, ˚vq dµ H8 pρ ˘q " e δ Γ βρ ˘pP H8 pρ ˘q, ˚vq dµ H8 pρ Haar Kv pρ ˘q .

		1 2 dp˚v, P H8 pξqq v	.	(15.10)
		˘q
	" q d Therefore, if ρ P B 1 ´dpP H8 pρ ˘q, ˚vq v ˘t˚vu is such that
	ρ ˘P O
	8
			H 8
	´vpξq
	˚v	p H8 pξq
			BH 8
	0	ξ	K v " B 8 X v ´t8u
	10 233			19/12/2016

v " tξ P K v : |ξ| v ď 1u " tξ P K v : P H8 pξq " ˚vu , then d r σ t˚vu pρq " d Haar Kv pρ ˘q . If ρ ˘P K v ´Ov , Equation (15.10) gives the claim.

  1 as t Ñ 8. Let d D pξ, ξ 1 q " lim The distance-like map d D is invariant by the diagonal action of the isometries of X preserving D. If D consists of a single point x, then d D is the visual distance 15 d x on B 8 X based at x. If D is a horoball with point at infinity ξ 0 , then d D is Hamenstädt's distance 16 on B 8 X ´tξ 0 u.

	tÑ`8	e	1 2 dpξt, ξ 1 t q´t .	(15.14)
	As seen in [HeP4, §2.2, Ex. (4)], if X is a metric tree, then

  Furthermore, since ρptq " ρ 1 ptq for t ď n and dpρptq, ρ 1 ptqq " 2pt ´nq otherwise, we have, using the change of variables u " 2pt ´nq,

	dpρ, ρ 1 q "	ż `8 n	2 pt ´nq e ´2 t dt " e ´2 n	ż `8 0	u e ´u du 2	"	e ´2 n 2	.
	This proves Equation (15.16).
	Let β P s0, 1s and ψ P C β b pB 8 X ´B8 Dq. We have }ψ ˝B`} 8 " }ψ} 8 since B `is a
	homeomorphism, and, by Equation (15.16),
	}ψ ˝B`}1 β
	2
	239							19/12/2016

8 e 1 2 dpρptq, ρ 1 ptqq´t " e ´n .

  Lemma 15.9. For every P s0, 1s, for every -locally constant function ψ : ΓzG X v Ñ R, if n " r´1 2 ln s, then the map ψ ˝pG : ΓzG v Ñ R is U n -invariant and }ψ ˝pG } alc ď q v 2 }ψ} lc, ln qv .(15.19)Proof. Let , ψ, n be as in the statement. Let us first prove that if , 1 P G X v satisfy r´n,`ns " 1 r´n,`ns , then dp , 1 q ď . If r´n,`ns " 1 r´n,`ns , then dp ptq, 1 ptqq " 0 for t P r´n, ns and by the triangle inequality dp ptq, 1 ptqq ď 2p|t| ´nq if |t| ě n, hence

	dp , 1 q ď 2	ż `8 n	2 pt ´nq e ´2 t dt " 2 e ´2 n	ż `8 0	u e ´u du 2	" e ´2 n
						.18)
	241					19/12/2016

  .20) For every g P G v , we denote by R g : ΓzG v Ñ ΓzG v the right action of g, and for all bounded continuous functions r ψ, r ψ 1 on ΓzG v , we define Note that byEquations (15.20) and (15.18), for all bounded continuous functions ψ, ψ 1 : ΓzG X v Ñ R and n P Z, we have 19

	ż			ż
	cov µv, g p r ψ, r ψ 1 q "	r ψ ˝Rg r ψ 1 dµ v	´żΓzGv	r ψ dµ v	r ψ 1 dµ v .
	ΓzGv			ΓzGv
	242				19/12/2016

  In the proof of Theorem 16.1, we apply Theorem 15.4 with Γ :" G, D ´:" H 8 and D `:" g r H 8 . Recall that H 8 is the horoball in X v centred at 8 whose boundary contains ˚v (see Section 15.3).

1 where Kv is endowed with the distance px, yq Þ Ñ |x ´y|v 248 19/12/2016

,

  Equation (16.3) gives, with the appropriate error term,

	lim sÑ`8

  Haar Kv pzq |z ´α| v |z ´ασ | v

	(5) The measure
		dµpzq "	d
		¨x ´g ¨y| 2 v " |x ´y| 2 v jgpxq jgpyq
	and	jgpzq "	dpg ´1q ˚Haar Kv d Haar Kv	pzq .
	262				19/12/2016

8 This is a particular case of Proposition 17.3 (2) below.

11

  Proposition 17.8. Let D be a quaternion algebra over Q which splits over Q p and does not split over R. If O max is a maximal Z " 1

	p	‰	-order in D Qp containing O, then
	VolpΓ 1 O z zX p q " rO 1 max : O 1 s	p 12	ź q| Disc D	pq ´1q .

  T pGz GL 2 pQ p qq µ T pGL 2 pZ p qq " µ T pGz SL 2 pQ p qq µ T pSL 2 pZ p qq

	,
	we have
	VolpGz zX p q " µ " ÿ rxsPGzV Xp 1 |G x | " ź p q| Disc D 24 pq ´1q .

  by homographies of the elements in the image of θpDq are quadratic over Qp An immediate application of Theorem 17.1, using Proposition 17.8, gives the following result of equidistribution of quadratic elements in Q p over Qp Theorem 17.9. Let Γ be a finite index subgroup of Γ 1 O , and let γ 0 P Γ be a loxodromic element of Γ. Then as s Ñ `8, pp `1q 2 ś q| Disc D pq ´1q rO 1 max : O 1 s rΓ 1 O : Γs 24 p VolpΓ γ 0 z z Ax γ 0 q s Qp containing O, and there exists κ ą 0 such that as s Ñ `8

	over Qp ?	? aq. ? a, ? bq. More precisely, z ? b is quadratic a, ? bq.

´1 ÿ α P Γ¨γ 0 , hpαqďs ∆ α á Haar Qp , where O max is a maximal Z " 1 p ‰ -order in D

  Theorem 17.10. Let D " p ´1,´1Q q be Hamilton's quaternion algebra over Q. Let p P N be a positive rational prime with p " 1 mod 4 such that p 2 ´1 4 is not of the form 4 a p8b `7q for a, b P N and let O be the Z " 1

	p	‰ -order 16
	O " x P Z " 1 p	‰	`Z" 1 p	‰	i	`Z" 1 p	‰	j	`Z" 1 p	‰	k : x " 1 mod 2	(
	in D Qp . Let Γ be a finite index subgroup of Γ 1 O . Then as s Ñ `8,
	pp `1q 2 rΓ 1 O : Γs 2 p 2 k Γ	s	´1	α P Γ¨α 0 , hpαqďs ÿ	∆ α á Haar Qp ,
				hpαq "	1 |α ´ασ | p
	the complexity of α.
	270												19/12/2016

  UsingEquations (17.11) and (17.12), the result now follows from Theorem 17.1. l 18 This group is denoted by Γp2q in [Lub2, page 95].

	272	19/12/2016

  of quadratic irrational α in K v hpαq complexity of quadratic irrational α in Q p h α pβq relative height of loxodromic fixed point β with respect to α h α pβq relative height of quadratic irrational β with respect to α Haar Kv normalised Haar measure of pK v , `q H rts horoball contained in H whose boundary is at distance t from the boundary of H HB `pwq stable horosphere of w P G `X HB ´pwq unstable horosphere of w P G ´X H `pwq stable horosphere of w P G `X H ´pwq unstable horosphere of wP G ´X ι antipodal map w Þ Ñ tt Þ Ñ wp´tqu of p G X ι G pβq G-reciprocity index of a quadratic irrational β IsompXq isometry group of X I v set of classes of fractional ideals of R v K global function field over F q K v completion of function field K for the valuation v k v residual field of the valuation v on K λpγq translation length of an isometry γ of X ΛΓ limit set of a discrete group of isometries Γ of X Λ c Γ conical limit set of a discrete group of isometries Γ of X L 2 pY, G ˚q Hilbert space of square integrable maps on V Y for the measure vol pY,G˚q lk links of vertices in simplicial trees L Γ length spectrum of action of Γ on X ln natural logarithm (with lnpeq " 1) ˘positive/negative endpoint of geodesic line ˚standard basepoint in space of geodesic lines G X v 299 19/12/2016 m D pxq multiplicity of a vertex x with respect to an equivariant family D r m F Gibbs measure on the space of geodesic lines G X m F Gibbs measure on the quotient space of geodesic lines ΓzG X r m F Gibbs measure on the space of discrete geodesic lines G X m F Gibbs measure on the quotient space of discrete geodesic lines ΓzG X m F renormalised Gibbs measure m F {||m F || on ΓzG X m c renormalised Gibbs measure m c {||m c || on ΓzG X pµ x q xPX (normalised) Patterson density for the pair pΓ, r F ˘q µ W ˘pwq skinning measures on the strong stable or strong unstable leaf W ˘pwq pµ Haus x q xPX Hausdorff measures of the visual distances d x on ΛΓ N pIq (absolute) norm of a nonzero ideal I in a Dedekind ring N w homeomorphism between stable/unstable leaves and inner/outer normal bundles of horoballs N A closed -neighbourhood of a subset A of a metric space npβq relative norm of quadratic irrational β Npxq reduced norm of a quaternion x ν w conditional measure on the (weak) stable/unstable leaf W 0˘p wq of wP G ˘X O x A shadow of a subset A of X seen from x P X Y B 8 X O v valuation ring of of v in K v π footpoint projection w Þ Ñwp0q on (generalised) geodesic lines π v uniformizer of a valuation v of a function field K over F q πpY, G ˚q fundamental groupoid of a graph of groups pY, G ˚q P D closest point map to a convex subset D P D closest point map homeomorphism from B 8 X ´B8 D to outer/inner normal bundle of D r φ μ total mass function of Patterson density pµ x q xPX P φ pressure of a potential φ under a transformation P ψ pressure of a potential ψ under a flow 9, 87 P φ pmq metric pressure for a potential φ of a probability measure m invariant under a transformation P ψ pmq metric pressure for a potential ψ of a flow-invariant probability measure m 9, 87Q " Q Γ, F, x, y Poincaré series of pΓ, F q q v order of residual field k v Galois conjugate of a quadratic irrational β in K v α σ Galois conjugate of a quadratic irrational α in Q p r σ D skinningmeasure on outer/inner normal bundle of convex subset D σ D inner skinning measure on Γz p G X of a family of closed convex subsets D σ D outer skinning measure on Γz p G X of a family of closed convex subsets D skinning measure of a family Ω " pΩ i q iPI of subsets of pB 1 ˘Di q iPI T π first edge map of a geodesic line T 1 X space of germs at t " 0 of geodesic lines in X trpβq relative trace of quadratic irrational β Trpxq reduced trace of a quaternion x Tvol pY, G˚q volume form on the set of edges of a graph of finite groups pY, G ˚q TvolpY, G ˚, λq volume form of the set of edges of a metric graph of finite groups pY, G ˚, λq TVolpY, G ˚q total volume of the set of edges of a graph of finite groups pY, G ˚q TVolpY, G ˚, λq total volume of the set of edges of a metric graph of finite groups pY, G ˚, λq U D domain of the fibration f D V w, η, η 1 dynamical neighbourhoods of a point w P G ˘X V η, η 1 pΩ ¯q dynamical neighbourhood of a subset Ω ¯of G ˘X v germ at t " 0 of geodesic line v 8 valuation at infinity of F q pY q vol pY, G˚q volume form on the set of vertices of a graph of finite groups pY, G ˚q VolpY, G ˚q volume of a graph of finite groups pY, G ˚q V X set of vertices of a graph X w ˘positive/negative endpoint of generalised geodesic line w W `pwq strong stable leaf of w P G `X W 0`p wq stable leaf of w P G `X W ´pwq strong unstable leaf of w P G ´X W 0´p wq unstable leaf of w P G ´X |X| λ geometric realization of a metric tree pX, λq X vBruhat-Tits tree of pPGL 2 , K v q

	β σ
			p
	r σ D Index	outer/inner skinning measure on subsets D	G X of a family of closed convex
	R v r σ Ω absolute crossratio, 273 affine algebra of the affine curve C ´tvu Dedekind's zeta function of a function field K value, 222 acylindrical, 76, 149, 212 adjacency matrix, 98 admissible, 67 alc-norm, 240, 243 algebraic lattice, 40 algebraically locally constant, 240, 243 almost precisely invariant, 117 anti-reversible, 50 antipodal map, 28, 30 bipartite, 36 biregular, 36 boundary, 116 bounded parabolic limit point, 26 Bowen ball, 55 Bowen-Margulis measure, 55 Bowen-Walters distance, 79 Busemann cocycle, 27 closest point map, 32 cocycle Busemann, 27 Gibbs, 47 outer/inner ζ K psq codegree
	300 301 302		19/12/2016 19/12/2016 19/12/2016

itself building on the works of Ledrappier[Led],Hamenstädt, Coudène, Mohsen

see for instance[START_REF] Ruelle | Thermodynamic formalism: The mathematical structure of equilibrium statistical mechanics[END_REF][START_REF] Keller | Equilibrium states in ergodic theory[END_REF][START_REF] Zinsmeister | Formalisme thermodynamique et systèmes dynamiques holomorphes[END_REF]

that is, a finite graph of trivial groups with edge lengths 1

a very strong assumption that we do not want to make in this text

not in the orbifold sense, hence this excludes for instance the case of graphs of groups with some nontrivial vertex stabiliser

More precisely, ptq is the closest point to x0 on .

See[START_REF] Hersonsky | On the rigidity of discrete isometry groups of negatively curved spaces[END_REF] Appendix] and compare with[START_REF] Hamenstädt | A new description of the Bowen-Margulis measure[END_REF].

This assumption, though not necessary at this stage, will be used repeatedly in this text, hence we prefer to add it to the definition.

An automorphism g of a graph has an inversion if there exists an edge e of the graph such that ge " e.

This is in particular the case for the sequence spaces of symbolic dynamical systems, see Sections 5.2 and 9.2.

The standard proof using Arzela-Ascoli's theorem applies with our slightly different definition of the Hölder norms.

Note that a linear combination of -locally constant functions is again a -locally constant function.

A potential in this work is not the analog of a potential in an electric network, we follow the dynamical systems terminology as in for example[PauPS].

That is, X is a geodesically complete proper CATp´1q space, x0 P X is a basepoint, and Γ is a nonelementary discrete group of isometries of X.

2 See also the previous works[START_REF] Ledrappier | A renewal theorem for the distance in negative curvature[END_REF][START_REF] Hamenstädt | Cocycles, Hausdorff measures and cross ratios[END_REF][START_REF] Coudène | Gibbs measures on negatively curved manifolds[END_REF][START_REF] Mohsen | Le bas du spectre d'une variété hyperbolique est un point selle[END_REF].

See for instance[START_REF] Kitchens | Symbolic Dynamics: One-sided, Two-sided and Countable State Markov Shifts[END_REF][START_REF] Sarig | Thermodynamic formalism for countable Markov shifts[END_REF].

Note that the terminology could be misleading, a topological Markov shift comes a prori without a measure, and many probability measures invariant under the shift do not satisfy the Markov chain property that the probability to pass from one state to another depends only on the previous state, not of all past states.

Note that some references have a stronger notion of Gibbs measure (see for instance[START_REF] Sarig | Existence of Gibbs measures for countable Markov shifts[END_REF]), with the constant C independent of E.

with the convention that max H " 0 when we consider the empty set H as a subset of N.

See previous footnote.

See for instance[Sel, GuL], which require other minor hypothesis that are not relevant here.

Personal communication.

with conditional measure on the fiber tφtpyq : 0 ď t ă τ pyqu over y P Y the image of the Lebesgue measure on r0, τ pyqr by t Þ Ñ φtpyq 80 19/12/2016

See the definition in Sections 4.2 and 4.3.

See after the proof of Theorem 5.12 for a comment on cohomology classes.

This definition is given for transformations and not flows, and for possibly unbounded potentials, contrarily to the one the Introduction.

For every a P A , for the constant ca required by the definition of the weak Gibbs property in Equation (5.13), take the constant CE given by the definition (see Equation (5.1)) of the Gibbs property with E " tau.

See Section 2.4.

See Section 2.7.

See Proposition 3.11 with the edge length map λ constant equal to 1

that is, Cardtn P N : Z x n " yu " 8 for every y P I (or equivalently, there exists y P I such that Cardtn P N : Z x n " yu " 8)

See for instance[START_REF] Paulin | Equilibrium states in negative curvature[END_REF] §2.6] for details on the definition of the induced measure when Γ may have torsion.11019/12/2016

Although it is standard to denote the one-sided shift by σ in the same way as the two-sided shift, we use σ`for readability.

See Section 3.1 for the definition of the Hölder norm } ¨}α.

Actually, only a much weaker assumption is required, such as a Hölder-continuity property of this Jacobian, see[START_REF] Young | Recurrence times and rates of mixing[END_REF].

Actually, there is in[START_REF] Young | Recurrence times and rates of mixing[END_REF] (see also[CyS]) a control on the constant in terms of some norms of the test functions, but these norms are not the ones we are interested in.

and by probabilists in order to study the statistics of cusp excursions (see for instance[EF])

See[START_REF] Dolgopyat | Prevalence of rapid mixing in hyperbolic flows[END_REF], and more precisely[START_REF] Melbourne | Rapid decay of correlations for nonuniformly hyperbolic flows[END_REF] Def. 2.2] whose definition is slightly different but implies the one given in this paper by the Principle of Uniform Boundedness argument of[ChCS, Appendix B] already used in Section 9.2.

They have a positive lower bound by definition, see Section 2.7.

The result for general Y 1 follows by averaging.

See Section 4.1 for definitions.

In fact, ergodicity is sufficient to have this.

See Section 3.5.

By abuse, we will still denote by D ˘the geometric realisation |D ˘|λ .

We leave to the reader the extension to more general locally finite families of subtrees, as for instance finite unions of those above. 190 19/12/2016

19/12/2016

See definition in Section 9.3.

See Section 3.5.

See Section 3.2.

19/12/2016

See for instance[START_REF] Serre | Arbres, amalgames, SL 2 . 3ème éd. corr[END_REF] II.2 Notation]. We refer for instance to[START_REF] Narkiewicz | Elementary and analytic theory of algebraic numbers[END_REF] §1.1] for background on Dedekind rings.

Other normalisations are useful when considering Fourier transforms, see for instance Tate's thesis[Tat].22519/12/2016

VolpPGL 2 pR v qz zX v q " pq ´1q VolpGL 2 pR v qz zX v q " 2 ζ K p´1q .(15.7)If K " F q pY q is the rational function field over F q and if we consider the valuation at infinity v " v 8 of K, then the Nagao lattice 7 Γ v " PGL 2 pF q rY sq acts transitively on P 1 pKq. Its quotient graph of groups Γ v z zX v is the following modular ray (with associated edge-indexed graph)Γ ´1 Γ 0 Γ 1 Γ 2 Γ 1

See for instance[BasL] for a profusion of geometrically infinite lattices in simplicial trees.

See Section 2.7 for a definition.

This lattice was studied by Nagao in[Nag], see also[Moz, BasL]. It is called the modular group in[START_REF] Weil | On the analogue of the modular group in characteristic p[END_REF].23119/12/2016

Since the potential is zero, the Gibbs cocycle is the critical exponent times the Busemann cocycle.

See Proposition 4.14 (2).

Note that the existence of a nonuniform lattice in Gv " PGL2pKvq forces the characteristic to be positive, see for instance[START_REF] Lubotzky | Lattices in rank one Lie groups over local fields[END_REF].

See Section 15.1.

See for example[START_REF] Rosen | Number theory in function fields[END_REF] §1].25519/12/2016

See Section 15.1.

See Section 2.2. 257 19/12/2016

The groups GL2pKvq and PGL2pKvq act on P 1 pKvq " Kv Y t8u by homographies. See Section 15.1.

See Equation (14.3).

that is, such that ˚v R γD

6Itconnects ˚v to its closest point P γD `p˚v q on γD `.

This is equivalent to q being odd.

This order plays an important role in the construction of Ramanujan graphs by Lubotzky, Phillips and Sarnak[START_REF] Lubotzky | Ramanujan conjectures and explicit construction of expanders[END_REF][START_REF] Lubotzky | Ramanujan graphs[END_REF] (see also[START_REF] Lubotzky | Discrete groups, expanding graphs and invariant measures[END_REF] §7.4]), and in the explicit construction of free subgroups of SOp3q in order to construct Hausdorff-Banach-Tarsky paradoxical decompositions of the 2-sphere, see for instance[START_REF] Lubotzky | Discrete groups, expanding graphs and invariant measures[END_REF] page 11].

The group O ˆ{Z pO ˆq is denoted by Λp2q in [Lub2, page 11]. 271 19/12/2016

The factor |α ´ασ |v in the denominator, that did not appear in[START_REF] Parkkonen | On the arithmetic of crossratios and generalised Mertens' formulas. Numéro Spécial "Aux croisements de la géométrie hyperbolique et de l'arithmétique[END_REF] in the analogous definition for the case when Kv is R or C, is there in order to simplify the statements below.

using, for all u, v, c ą 0 and n P N, the notation u "c ˘nv if 1 c n v ď u ď c n v

Acknowledgements: This work was partially supported by the NSF grants no 093207800 and DMS-1440140, while the third author was in residence at the MSRI, Berkeley CA, during the Spring 2015 and Fall 2016 semesters. The second author thanks Université Paris-Sud, Forschungsinstitut für Mathematik of ETH Zürich, and Vilho, Yrjö ja Kalle Väisälän rahasto for their support during the preparation of this work. This research was supported by the CNRS PICS n 0 6950 "Equidistribution et comptage en courbure négative et applications arithmétiques". We thank, for interesting discussions on this text, Y. Benoist, J. Buzzi (for his help in Sections 5.2 and 9.2, and for kindly agreeing to insert 292 19/12/2016

dpx 0 , C γ q " dpx 0 , γx 0 q ´λpγq 2 .

(13.1)

By the equivariance properties of the skinning measures, the total mass of the skinning measure 4 σ D where D " pγ 1 C γ q γ 1 PΓ{Γ Cγ depends only on the conjugacy class K of γ in Γ, and will be denoted by }σ Ḱ }. This quantity, called the skinning measure of K, is positive unless B 8 C γ " ΛΓ, which is equivalent to γ P Z virt pΓq (and implies in particular that γ is elliptic). Furthermore, }σ Ḱ } is finite if γ is loxodromic, and it is finite if γ is elliptic and Γ Cγ zpC γ XC ΛΓq is compact. This last condition is in particular satisfied if C γ X C ΛΓ itself is compact, and this is the case for instance if, for some k ě 0, the action of Γ on X is k-acylindrical (see for instance [Sel, GuL]), that is, if any element of Γ fixing a segment of length k in C ΛΓ is the identity.

For every γ P Γ ´teu, we define

which is a natural multiplicity of γ, and equals 1 if the stabiliser of x 0 in Γ is trivial (for instance if Γ is torsion-free). Note that for every α P Γ, the real number m αγα ´1 depends only on the double coset of α in Γ x 0 zΓ{Γ Cγ . The centraliser Z Γ pγq of γ in Γ is contained in the stabiliser of C γ in Γ. The index i K " rΓ Cγ : Z Γ pγqs depends only on the conjugacy class K of γ; it will be called the index of K. The index i K is finite if γ is loxodromic (the stabiliser of its translation axis C γ is then virtually cyclic), and also finite if C γ is compact (as for instance if the action of Γ on X is k-acylindrical for some k ě 0). We define

where pe 1 , . . . , e k q is the shortest edge path from x 0 to C γ .

We finally define the orbital counting function in conjugacy classes, counting with multiplicities and weights coming from the system of conductances, as N K, x 0 ptq " ÿ αPK, dpx 0 , αx 0 qďt m α e cα . for t P r0, `8r (simply t P N in the simplicial case). When the stabiliser of x 0 in Γ is trivial and when the system of conductances c vanishes, we recover the definition of the Introduction.

Theorem 13.1. Let K be the conjugacy class of a nontrivial element γ 0 of Γ, with finite index i K , and with positive and finite skinning measure }σ Ḱ }. Assume that δ c is finite and positive.

4 See Chapter 7 for definitions and notation.

212

19/12/2016

(1) Assume that m c is finite and mixing for the continuous time geodesic flow on ΓzX. Then, as t Ñ `8,

If Γ Cγ zpC γ X C ΛΓq is compact when γ P K is elliptic and if there exists β P s0, 1s such that the continous time geodesic flow on pΓzG X, m c q has superpolynomial decay of β-Hölder correlations, then the error term is O `t´n e δc 2 t ˘for every n P N.

(2) Assume that λ " 1 and that m c is finite and mixing for the discrete time geodesic flow on ΓzG X. Then, as n Ñ `8, N K, x 0 pnq " e δc i K }µ x0 } }σ Ḱ } pe δc ´1q }m c } e δct n´λpγ 0 q 2 u .

If Γ Cγ zpC γ X C ΛΓq is compact when γ P K is elliptic and if there exists β P s0, 1s such that the discrete time geodesic flow on pΓzG X, m c q is exponentially mixing for the β-Hölder regularity, then the error term is O `epδc´κqn{2 ˘for some κ ą 0.

One can also formulate a version of the above result for groups acting on bipartite simplicial trees based on Theorem 12.12 and Remark (ii) following the proof of Theorem 12.17.

The error term in Assertion (1) holds for instance if r c " 0, X is uniform, and either ΓzX is compact and the length spectrum L Γ is 2-Diophantine or Γ is a geometrically finite lattice of X whose length spectrum L Γ is 4-Diophantine, by the Remark following Theorem 12.20. When ΓzX is compact and Γ has no torsion (in particular, Γ has then a very restricted group structure, as it is then a free group), we thus recover a result of [KeS].

The error term in Assertion (2) holds for instance if r c " 0, X is uniform with vertices of degrees at least 3, Γ is a geometrically finite lattice of X with length spectrum equal to Z, by Remark (i) following the proof of Theorem 12.17.

Theorem 1.12 in the introduction follows from this theorem, using Proposition 4.14 (3) and Proposition 4.15.

Proof. We only give a full proof of Assertion (1) of this theorem, Assertion (2) follows similarly using Theorems 12.9 and 12.17 instead of Theorems 12.8 and 12.20.

The proof is similar to the proof of [START_REF] Parkkonen | On the hyperbolic orbital counting problem in conjugacy classes[END_REF]Theo. 8]. Let D ´" tx 0 u and D `" C γ 0 . Let D ´" pγD ´qγ 1 PΓ{Γ D ´and D `" pγD `qγPΓ{Γ D `. By Equation (7.14), we have

By Equation (13.1), by the definition 5 of the counting function N D ´, D `and by the last claim

Assertion (1) without the error term follows, and the error term statement follows similarly from Theorem 12.20. l

Theorem 13.1 (1) without an explicit form of the multiplicative constant in the asymptotic is due to [KeS] under the strong restriction that Γ is a free group acting freely on X and ΓzX is a finite graph. The following result is due to [START_REF] Douma | A lattice point problem on the regular tree[END_REF]Thm. 1] in the very special case when X is a regular tree and the group Γ has no torsion and finite quotient ΓzX.

Corollary 13.2. Let X be a regular simplicial tree with vertices of degree q `1 ě 3, let x 0 P V X, let Γ be a lattice of X such that ΓzX is nonbipartite, and let K be the conjugacy class of a loxodromic element γ 0 P Γ. Then, as n Ñ `8, ÿ αPK, dpx 0 , αx 0 qďn m α " λpγ 0 q rZ Γ pγ 0 q : γ Z 0 s VolpΓz zXq

If we assume furthermore that Γ has no torsion, then the result holds also when ΓzX is bipartite. In this case, we have as n Ñ `8, Cardtα P K : dpx 0 , αx 0 q ď nu " λpγ 0 q |ΓzV X| q t n´λpγ 0 q 2 u .

Proof. Under these assumptions, taking c " 0 in Theorem 13.1 so that the Gibbs measure is the Bowen-Margulis measure, the discrete time geodesic flow on ΓzG X is finite and mixing by Proposition 4.14 (3) and Proposition 4.15. We also have δ c " ln q. Using the normalisation of the Patterson density pµ x q xPV X to probability measures, Proposition 8.1 (3) and Equation (8.11), the result follows, since when γ is loxodromic,

The claim for the bipartite pp `1q-regular case follows from Remark 12.14. l

The value of C 1 given below Theorem 1.12 in the Introduction follows from this corollary.

We leave to the reader an extension with nonzero potential F of the results for manifolds in [START_REF] Parkkonen | On the hyperbolic orbital counting problem in conjugacy classes[END_REF], along the lines of the above proofs.

Fields with discrete valuations

Let p K be a non-Archimedean local field. Basic examples of such fields are the field of formal Laurent series over a finite field, and the field of p-adic numbers (see Examples 14.1 and 14.2). In Part III of this book, we apply the geometric equidistribution and counting results for simplicial trees given in Part II, in order to prove arithmetic equidistribution and counting results in such fields p K. The link between the geometry and the algebra is provided by the Bruhat-Tits tree of pPGL 2 , p Kq, see Chapter 15.1. We will only use the system of conductances equal to 0 in this Part III.

In the present Chapter, before embarking on our arithmetic applications, we recall basic facts on local fields for the convenience of the geometer reader. For more details, we refer for instance to [START_REF] Serre | Corps locaux[END_REF][START_REF] Goss | Basic structures of function field arithmetic[END_REF]. We refer to [BrPP] for an announcement of the results of Part III, with a presentation different from the one in the Introduction.

We will only give results for the algebraic group G " PGL 2 over p K and special discrete subgroups Γ of PGL 2 p p Kq, even though the same methods give equidistribution and counting results when G is any semisimple connected linear algebraic group over p K of p K-rank 1 and Γ any lattice in G " Gp p Kq.

Local fields and valuations

Let F be a field and let F ˆ" pF ´t0u, ˆq be its multiplicative group

We now turn to measure computations for arithmetic lattices Γ in X v in the function field case. We still assume that the Patterson density of Γ is normalised so that µ H8 pO v q " 1, and we denote by m BM the Bowen-Margulis measure of Γ associated with this choice of Patterson density.

Proposition 15.3. Let K be a function field over F q and let v be a valuation of K. Let Γ be a finite index subgroup of Γ v " PGL 2 pR v q, with Patterson density normalised such that µ H8 " Haar Kv .

(1) We have

and if K " F q pY q and v " v 8 is the valuation at infinity of C " P 1 , then }m BM } " 2 rPGL 2 pF q rY sq : Γs q pq ´1q 2 .

(2) Let Γ 8 be the stabiliser in Γ of 8 P B 8 X v , and let D " pγH 8 q γPΓ{Γ8 . We have

The first claim of Assertion (1) hence follows from Equation (15.7).

If K " F q pY q and v " v 8 , then the second claim of Assertion (1) follows either from the first claim where the value of ζ K p´1q is given by Equation (14.5), or from the fact that q v " q and that the covolume VolpPGL 2 pF q rY sqz zX v8 q of the Nagao lattice PGL 2 pF q rY sq is VolpPGL 2 pF q rY sqz zX v8 q " 2 pq ´1qpq 2 ´1q , (15.11) as an easy geometric series computation shows using the description of the modular ray in Section 15.2 (see also [START_REF] Bass | Tree lattices[END_REF]Sect. 10.2]).

(2) Let us prove that

Haar Kv ppΓ v q 8 zK v q " q g´1 q ´1 . (15.12)

The result then follows by Proposition 15.2 (4) since }σ D } " Haar Kv pΓ 8 zK v q " rpΓ v q 8 : Γ 8 s Haar Kv ppΓ v q 8 zK v q .

The stabiliser of 8 " r1 : 0s in Γ v acts on K v exactly by the set of transformations z Þ Ñ az `b with a P pR v q ˆand b P R v . Since pR v q ˆ" pF q q ˆ(see Equation (14.3)) acts freely by left translations on pK v ´Rv q{R v , and by Lemma 14.4, we have Haar Kv ppΓ v q 8 zK v q " 1 q ´1 Haar Kv pK v {R v q " q g´1 q ´1 . This proves Equation (15.12). l 15.4 Exponential decay of correlation and error terms for arithmetic quotients of Bruhat-Tits trees

As in the beginning of Section 15.1, let K v be a non-Archimedean local field, with valuation v, valuation ring O v , choice of uniformiser π v , and residual field k v of order q v . Let Γ be a lattice of the Bruhat-Tits tree X v of pPGL 2 , K v q. In this Section, we discuss the error terms estimates that we will use in Part III.

In part in order to simplify the references, we start by summarizing in the next statement the only results from the geometric Part II of this book, on geometric equidistribution and counting problems, that we will use in this algebraic Part III. We state it with the normalisation which will be useful there (see Section 15.3).

Theorem 15.4. Let Γ be a lattice of X v whose length spectrum L Γ is equal to 2Z. Assume that the Patterson density of Γ is normalised so that }µ x } "

´, γD `qďn

for the weak-star convergence of measures on the locally compact space p G X v . Furthermore, if Γ is geometrically finite, then for every β P s0, 1s, there exists an error term for this equidistribution claim when evaluated on r ψ P C β c p p G Xq of the form Op } r ψ } β e ´κ n q for some κ ą 0.

As recalled at the end of Section 2.7, arithmetic lattices in PGL 2 pK v q are geometrically finite, see [START_REF] Lubotzky | Lattices in rank one Lie groups over local fields[END_REF]. We will hence be able to use the error term in Theorem 15.4 in particular when ' K v is the completion of a function field K over F q with respect to a (normalised discrete) valuation v of K and Γ is a finite index subgroup of PGL 2 pR v q with R v the affine function ring associated with v, 13 as in Chapters 16 and 19, and in Sections 17.2 and 18.2;

' when K v " Q p and Γ is an arithmetic lattice in PGL 2 pK v q derived from a quaternion algebra, see Sections 17.3 and 18.2.

Proof. In order to prove the first claim, we apply Corollary 11.11 with X " X v and p " q " q v . Since L Γ " 2Z, the lattice Γ leaves invariant the partition of V X v into vertices at even distance from a base point x 0 and vertices at odd distance from x 0 . Since the Patterson density is now normalised so that }µ x 0 } " qv`1 qv (instead of }µ x 0 } " qv`1 ? qv in Corollary 11.11), the skinning measures σ D˘a re now 1 ? qv times the ones in the statement of Corollary 11.11. Hence the second assertion of Corollary 11.11 gives

´, γD `qďn

By Equation (2.16), we have

The first claim follows.

The last claim concerning error terms follows from Remark (ii) following the statement of Theorem 12.17. l

In the last four Chapters 16, 17, 18 and 19 of this book, we will need to push to infinity the measures appearing in the statement of Theorem 15.4. We regroup in the following two lemmas the necessary control tools for such a pushing.

The first one is a metric estimate on the extension of geodesic segments to geodesic rays.

Lemma 15.5. Let X be a geodesically complete proper CATp´1q space, let T ě 1, and let α P p G X be a generalised geodesic line which is isometric exactly on r0, T s. For every generalised

13 See Section 14.2 for definitions Corollary 16.8 (Mertens's formula for polynomials). We have

ÿ QPFqrXs, deg Qďn φ q pQq " pq ´1q q q `1 .

Proof. We apply the first claim of Corollary 16.7, in the special case when K " F q pT q and v " v 8 so that g " 0, q v " q and R v " R, and with m " R v , so that m v, x 0 , y 0 " q ´1, in order to obtain the asymptotic value of ψ Rv psq with the change of variable s " q n . The result follows from Equations (16.13) and (14.5). l

The above result is an analog of Mertens's formula when K is replaced by Q and R v by Z, see [START_REF] Hardy | An introduction to the theory of numbers[END_REF]Thm. 330]. See also [START_REF] Grotz | Mittelwert der Eulerschen ϕ-Funktion und des Quadrates der Dirichletschen Teilerfunktion in algebraischen Zahlkörpern[END_REF]Satz 2], [START_REF] Cosentino | Equidistribution of parabolic fixed points in the limit set of Kleinian groups[END_REF]§4.3], as well as [START_REF] Parkkonen | On the arithmetic of crossratios and generalised Mertens' formulas. Numéro Spécial "Aux croisements de la géométrie hyperbolique et de l'arithmétique[END_REF] and [START_REF] Parkkonen | Counting and equidistribution in Heisenberg groups[END_REF]§5] for further developments.

A much more precise result than Corollary 16.8 can be obtained by purely number theoretical means as follows. The average value of φ q is computed in [START_REF] Rosen | Number theory in function fields[END_REF]Prop. 2.7

This gives ř deg f "n φ q pf q " q 2n pq´1q 2 q , so that

Ov , then the translation axis of β passes at distance at most 2k from ˚v, since it passes through P H8 pπ ´k v q which is the closest point on

Furthermore, we have, by Equations (15.2) and (2.8)

Therefore by the definition of the complexity in Equation ( 17.1), we have for these elements

Since the family D `" pγD `qγPΓ{Γ D `is locally finite, there are only finitely many elements γ P Γ{Γ D `such that γD `" Ax γγ 0 γ ´1 is at distance at most 2k from ˚v. Hence for all but finitely many γ P Γ{Γ D `such that γ ¨γ0 " pγγ 0 γ ´1q ´P π ´k v `Ov , we have γ ¨γ0 " pγγ 0 γ ´1q `P π ´k v `Ov and, using Equation (17.4) with β " γγ 0 γ ´1, hpγ ¨γ0 q " q dpD ´,γD `q´2k v .

Therefore, using the change of variable s " q n´2k v , Equation (17.3) becomes

Note that the stabiliser Γ γ 0 of γ 0 in Γ has index ι γ 0 in Γ D `by the definition of ι γ 0 and that Γ{Γ γ 0 identifies with Γ ¨γ0 by the map γΓ γ 0 Þ Ñ γ ¨γ0 . Since `pπ v q ´k `Ov ˘kPN is a countable family of pairwise disjoint compact-open subsets covering K v , and since the support of any continuous function with compact support is contained in finitely many elements of this family, we have

Hence if instead the Patterson densities are normalised to have total mass qv`1 qv as in Proposition 15.2 (2), then

Note that, since ι γ 0 " rΓ Axγ 0 : Γ γ 0 s,

Equation (17.6) thus gives the equidistribution result in Theorem 17.1. l

In the following two Sections, we use Theorem 17.1 to deduce counting and equidistribution results of elements of non-Archimedean local fields that are quadratic irrational over appropriate subfields, when an appropriate algebraic complexity tends to infinity.

Counting and equidistribution of quadratic irrationals in positive characteristic

Let K be a function field over F q , let v be a (normalised discrete) valuation of K, and let R v be the affine function ring associated with v. We assume in this Section that the characteristic of K is different from 2. 7

An element β P K v is quadratic irrational over K if β R K and β is a root of a quadratic polynomial aβ 2 `bβ `c for some a, b, c P K with a ‰ 0. The Galois conjugate β σ of β is the other root of the same polynomial. Let trpβq " β `βσ and npβq " ββ σ be the relative trace and relative norm of β. It is easy to check that β σ ‰ β since the characteristic of K is different from 2. The next proposition gives a characterisation of quadratic irrationals over K.

Proposition 17.2. Let β P K v . The following assertions are equivalent:

(1) β is quadratic irrational over K,

(2) β is a fixed point of a loxodromic element of PGL 2 pR v q.

Proof. The fact that (2) implies ( 1) is immediate since PGL 2 pR v q acts by homographies. The converse is classical, see for instance [START_REF] Parkkonen | Spiraling spectra of geodesic lines in negatively curved manifolds[END_REF]Lem. 6.2] in the Archimedean case and [BerN] above its Section 5 when K " F q pY q and v " v 8 . l

If β P K v is quadratic irrational over K, its Galois conjugate β σ is the other fixed point of a loxodromic element of PGL 2 pR v q fixing β, hence the notations β σ in this Section and in Section 17.1 coincide.

The actions by homographies of the groups GL 2 pR v q and PGL 2 pR v q on K v Y t8u preserve the set of quadratic irrationals over K. Contrary to the case of rational points, both groups act with infinitely many orbits. 19/12/2016 When β R tα, α σ u, we have

We will use the relative height as a complexity when β varies in a given orbit of Γ (and α is fixed).

The following properties of relative heights are easy to check using the definitions and the invariance properties of the crossratio.

Lemma 18.1. Let α, β P K v be loxodromic fixed points of Γ. Then (1) h α ρ pβ τ q " h α pβq for all ρ, τ P tid, σu.

(2) If β P tα, α σ u, then h α pβq " 1.

(3) h γ¨α pγ ¨βq " h α pβq for every γ P Γ.

(4) h α pγ ¨βq " h α pβq for every γ P Stab Γ ptα, α σ uq. l

The following result relates the relative height of two loxodromic fixed points with the distance between the two translation axes.

Proposition 18.2. Let α, β P K v be loxodromic fixed points of Γ such that β R tα, α σ u. Then

In particular, we have h α pβq ą 1 if and only if the geodesic lines sα, α σ r and sβ, β σ r in X v are disjoint, and h α pβq " 1 otherwise (using Lemma 18.1 (2) when β P tα, α σ u).

Proof. Up to replacing α, β, α σ , β σ by their images under a big enough power γ of a loxodromic element in Γ with attracting fixed point in O v , we may assume that these four points belong to O v . Note that γ exists since ΛΓ " B 8 X v , and it preserves the relative height by Lemma 18.1 (3) as well as the distances between translation axes.

Let A " sα, α σ r and B " sβ, β σ r. Let u be the closest point to ˚v on A, so that vpα ´ασ q " dpu, ˚vq .

We will consider five configurations.

Case 1. First assume that A and B are disjoint. Let ra, bs be the common perpendicular from A to B, with a P A, so that dpA, Bq " dpa, bq .

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First assume that u ‰ a. Up to exchanging α, α σ (which does not change dpA, Bq or h α pβq by Lemma 18.1 (1)), we may assume that a P ru, αr . Then (see the picture on the left above), vpβ ´βσ q " dpb, ˚vq, vpα ´βq " vpα ´βσ q " dpa, ˚vq and vpα σ ´βq " vpα σ ´βσ q " dpu, ˚vq .

Therefore

which proves the result by Equation (18.1).

Assume on the contrary that u " a. Let u 1 P V X v be such that ra, ˚vs X ra, bs " ra, u 1 s. Note that u 1 P r˚v, bs since β, β σ P O v . Then (see the picture on the right above), vpβ ´βσ q " dpb, ˚vq, vpα ´βq " vpα ´βσ q " vpα σ ´βq " vpα σ ´βσ q " dpu 1 , ˚vq .

which proves the result by Equation (18.1).

Case 2. Now assume that A and B are not disjoint, so that dpA, Bq " 0 .

Since β R tα, α σ u, the intersection A X B is a compact segment ra, bs (possibly with a " b) in X v . Up to exchanging α and α σ , as well as β and β σ (which does not change dpA, Bq nor h α pβq by Lemma 18.1 (1)), we may assume that α, a, b, α σ and β, a, b, β σ are in this order on A and B respectively, and that a P ru, αr .

Assume first that b P su, αr . Then (see the picture on the left above), vpα ´βq " dpa, ˚vq, vpα ´βσ q " vpβ ´βσ q " dpb, ˚vq and vpβ ´ασ q " vpβ σ ´ασ q " dpu, ˚vq .

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Therefore |α, β σ , α σ , β| v " q vpα´βq`vpα σ ´βσ q´vpα´α σ q´vpβ´β σ q v " q dpa, ˚vq´dpb, ˚vq v " q dpa,bq v ě 1 , and |α, β, α σ , β σ | v " q vpα´β σ q`vpα σ ´βq´vpα´α σ q´vpβ´β σ q v " q 0 v " 1 " q ´dpA,Bq v , which proves the result by Equation (18.1).

Assume that b P su, α σ r . Then (see the picture in the middle above), vpα ´βq " dpa, ˚vq, vpα σ ´βσ q " dpb, ˚vq and vpα ´βσ q " vpβ ´ασ q " vpβ ´βσ q " dpu, ˚vq .

Therefore |α, β σ , α σ , β| v " q vpα´βq`vpα σ ´βσ q´vpα´α σ q´vpβ´β σ q v " q dpa, ˚vq`dpb, ˚vq´2dpu,˚vq v " q dpa,bq v ě 1 ,

which proves the result by Equation (18.1).

Assume at last that b " u. Let u 1 P V X v be such that rb, ˚vs X rb, β σ r " rb, u 1 s. Then (see the picture on the right above), vpα ´βq " dpa, ˚vq, vpα σ ´βq " dpu, ˚vq and vpα ´βσ q " vpβ ´βσ q " vpα σ ´βσ q " dpu 1 , ˚vq .

Therefore |α, β σ , α σ , β| v " q vpα´βq`vpα σ ´βσ q´vpα´α σ q´vpβ´β σ q v " q dpa, ˚vq´dpu,˚vq v " q dpa,bq v ě 1 ,

which proves the result by Equation (18.1). l

The next result says that the relative height is an appropriate complexity on a given orbit under Γ of a loxodromic fixed point, and that the counting function we will study is well defined. We denote by Γ ξ the stabiliser in Γ of a point ξ P B 8 X v " P 1 pK v q.

Lemma 18.3. Let α, β P K v be loxodromic fixed points of Γ. Then for every s ą 1, the set

Proof. The set E s is well defined by Lemma 18.1 (4). Recall that a loxodromic fixed point is one of the two points at infinity of a unique translation axis. By local finiteness, there are, up to the action of the stabiliser of a fixed translation axis A, only finitely many images under Γ of another translation axis B at distance at most ln s ln qv from A. Since the stabiliser of A contains the stabiliser of either of its points at infinity with index at most 2, the result then follows from Proposition 18.2. l

We now state our main counting and equidistribution result of orbits of loxodromic fixed points, when the complexity is the relative height with respect to a fixed loxodromic fixed point.

Theorem 18.4. Let Γ be a lattice in PGL 2 pK v q `. Let α 0 , β 0 P K v be loxodromic fixed points of Γ. Then for the weak-star convergence of measures on K v ´tα 0 , α σ 0 u, as s Ñ `8,

If Γ is geometrically finite, for every β 1 P s0, 1s, there exists κ ą 0 such that for every ψ P C β 1 c pK v ´tα 0 , α σ 0 uq, where K v ´tα 0 , α σ 0 u is endowed with the distance-like map d sα 0 , α σ 0 r , 2 there is an error term in the equidistribution claim of Theorem 18.4 when evaluated on ψ, of the form Ops ´κ}ψ} β 1 q. This result applies for instance if ψ : K v ´tα 0 , α σ 0 u Ñ R is locally constant with compact support, see Remark 3.2.

Proof. The proof of the equidistribution claim is similar to the one of Theorem 17.1. We now apply Theorem 15.4 with D ´:" sα 0 , α σ 0 r and D `:" sβ 0 , β σ 0 r . Since Γ is contained in PGL 2 pK v q `, the length spectrum L Γ of Γ is equal to 2Z. The families D ˘" pγD ˘qγPΓ{Γ D ȃre locally finite, and }σ D`} is finite and nonzero. Arguing as in the proof of Theorem 17.1, 3 we have

´, γD `qďn

for the weak-star convergence of measures on B 8 X v ´B8 D ´. When Γ is geometrically finite, for every β 1 P s0, 1s, there exists κ ą 0 such that for every β-Hölder-continuous function ψ P C β 1 c pB 8 X v ´B8 D ´q, where B 8 X v ´B8 D ´is endowed with the distance-like map d D

´,

there is an error term in the equidistribution claim of Theorem 18.4 when evaluated on ψ, of the form Ops ´κ}ψ} β 1 q. By Proposition 18.2, we have on the full measure subset K v ´tα 0 , α σ 0 u of B 8 X v . Hence, using the change of variable s " q n v , we have, with the appropriate error term when Γ is geometrically finite,

We again denote by ι α 0 the index ι α 0 " rΓ tα 0 , α σ 0 u : Γ α 0 s , and similarly for β 0 . Since the stabiliser Γ β 0 of β 0 in Γ has index ι β 0 in Γ D `and Γ{Γ β 0 identifies with Γ ¨β0 by the map γ Þ Ñ γ ¨β0 , we have, with the appropriate error term when Γ is geometrically finite,

As in the end of the proof of Theorem 17.1, we have

This proves the equidistribution claim, and its error term when Γ is geometrically finite. In order to obtain the counting claim, we note that since r σ D´i s invariant under the stabiliser in Γ of D ´, hence under Γ α 0 , the measures on both sides of the equidistribution claim in Theorem 18.4 are invariant under Γ α 0 , see Proposition 17.3 (5) for the invariance of the right hand side. By Proposition 15.2 ( 5) and ( 6), and by the definition of ι α 0 , we have

Counting and equidistribution of crossratios of quadratic irrationals

In this Section, we give two arithmetic applications of Theorem 18.4.

Let us first consider an application in positive characteristic. Let K be a function field over F q , let v be a (normalised discrete) valuation of K, and let R v be the affine function ring associated with v. We assume that the characteristic of K is different from 2. Given two quadratic irrationals α, β P K v over K, with Galois conjugates α σ , β σ respectively, such that β R tα, α σ u, we define the relative height of β with respect to α by

The following result says that the orbit of any quadratic irrational in K v over K, by homographies under a given finite index subgroup of the modular group PGL 2 pR v q, equidistributes, when its complexity is given by the relative height with respect to another fixed quadratic irrational α 0 . The limit measure is absolutely continuous with respect to the Haar measure on K v and it is invariant under the stabiliser of α 0 in PGL 2 pR v q by Proposition 17.3 (5).

Theorem 18.5. Let G be a finite index subgroup of GL 2 pR v q. Let α 0 , β 0 P K v be quadratic irrationals over K. Then, as s Ñ `8,

and there exists κ ą 0 such that, as s Ñ `8, Card Γ α 0 ztβ P G ¨β0 : h α 0 pβq ď su " 4 q v pq ´1q |vptr g 0 q| |vptr h 0 q| |ZpGq| pq v `1q ζ K p´1q m 0 n 0 rGL 2 pR v q : Gs s `Ops 1´κ q .

Here g 0 , h 0 P G fixes α 0 , β 0 with vptr g 0 q, vptr h 0 q ‰ 0, and m 0 , n 0 is the index of g Z 0 , h Z 0 in the stabiliser of α 0 , β 0 in G respectively.

Proof. This follows, as in the proof of Theorem 17.5, from Theorem 18.4 using Equations (16.6) and (17.7), as well as Equation ( 18.3) for the counting claim. l

Example 18.6. (1) Theorem 1.16 in the Introduction follows from this result, by taking K " F q pY q and v " v 8 , and by using Equation (14.5) in order to simplify the constant.

(2) If G I is the Hecke congruence subgroup associated with a nonzero ideal I of R v (see Equation (16.11)), using Lemma 16.5, we have, as s Ñ `8,

The second arithmetic application of Theorem 18.4 is in Q p . We use the notation of Section 17.3.

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Chapter 19

Counting and equidistribution of integral representations by quadratic norm forms

In the final Chapter of this text, we give another equidistribution and counting result of rational elements in non-Archimedean local fields of positive characteristic, again using our equidistribution and counting results of common perpendiculars in trees summarized in Section 15.4. In this Chapter, we use a complexity defined using the norm forms associated with fixed quadratic irrationals. In particular, the complexity in this Chapter is different from that used in the Mertens type of results in Section 16.1. We refer for instance to [START_REF] Parkkonen | On the arithmetic of crossratios and generalised Mertens' formulas. Numéro Spécial "Aux croisements de la géométrie hyperbolique et de l'arithmétique[END_REF]§5.3] for motivations and results in the Archimedean case, and also to [GoP] for higher dimensional norm forms.

Let K be a function field over F q , let v be a (normalised discrete) valuation of K, and let R v be the affine function ring associated with v. Let α P K v be a quadratic irrational over K. The norm form n α associated with α is the quadratic form K ˆK Ñ K defined by px, yq Þ Ñ npx ´yαq " px ´yαqpx ´yα σ q " x 2 ´xy trpαq `y2 npαq . See Proposition 17.3 for elementary transformation properties under elements of GL 2 pR v q of this norm form.

A pair px, yq P R v ˆRv is an integral representation of an element z P K by the quadratic norm form n α if n α px, yq " z. The following result describes the projective equidistribution as s Ñ `8 of the integral representations by n α of elements with absolute value at most s. For every px 0 , y 0 q P R v ˆRv , let H px 0 ,y 0 q be the stabiliser of px 0 , y 0 q for the linear action of any subgroup H of GL 2 pR v q on R v ˆRv . We use the notation N xx 0 , y 0 y for the norm of the ideal xx 0 , y 0 y generated by x 0 , y 0 (see Section 14.2) and the notation m v, x 0 , y 0 introduced above Theorem 16.1.

Theorem 19.1. Let G be a finite index subgroup of GL 2 pR v q, let α P K v be a quadratic irrational over K, and let px 0 , y 0 q P R v ˆRv ´tp0, 0qu. Let c 1 " pq v ´1q pq v `1q 2 ζ K p´1q m v, x 0 , y 0 pN xx 0 , y 0 yq 2 rGL 2 pR v q : Gs q 3 v pq ´1q q g´1 rGL 2 pR v q px 0 , y 0 q : G px 0 , y 0 q s .

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Then for the weak-star convergence of measures on K v ´tα, α σ u, we have

For every β P s0, 1s, there exists κ ą 0 such that for every ψ P C β c pK v ´tα, α σ uq, where K v ´tα, α σ u is endowed with the distance-like map d sα, α σ r , 1 as for instance if ψ : K v ´tα, α σ u Ñ R is locally constant with compact support (see Remark 3.2), there is an error term in the equidistribution claim of Theorem 19.1 when evaluated on ψ, of the form Ops ´κ}ψ} β q.

Examples 19.2. (1) Let px 0 , y 0 q " p1, 0q, K " F q pY q and v " v 8 . Theorem 1.17 in the Introduction follows from Theorem 19.1, using Equations (14.5) and ( 16.1) to simplify the constant c 1 .

(2) Let px 0 , y 0 q " p1, 0q and let G " G I be the Hecke congruence subgroup of GL 2 pR v q defined in Equation (16.11). The index in rGL 2 pR v q : G I s is given by Lemma 16.5 and G I satisfies pG I q p1,0q " GL 2 pR v q p1,0q . For every nonzero ideal I of R v , for the weak-star convergence of measures on K v ´tα, α σ u, we have

(3) This example is interesting when the ideal class number is larger than 1. Given any fractional ideal m of R v , taking px 0 , y 0 q P R v ˆRv such that the fractional ideals xx 0 , y 0 y and m have the same ideal class and G " GL 2 pR v q, using the change of variables s Þ Ñ sN pmq 2 in the statement of Theorem 19.1, for the weak-star convergence of measures on K v ´tα, α σ u, with the same error term as for Theorem 19.1, we have

. Before proving Theorem 19.1, let us give a counting result which follows from it. Any subgroup of G acts on the left on any orbit of G. Furthermore, the stabiliser G α of α in G preserves the map px, yq Þ Ñ | npx ´yαq| v , by Proposition 17.3 (3). We may then define a counting function Ψ 1 psq " Ψ 1 G, α, x 0 ,y 0 psq of elements in R v ˆRv in a linear orbit under a finite index subgroup G of GL 2 pR v q on which the absolute value of the norm form associated with α is at most s, as Ψ 1 psq " Card G α z px, yq P Gpx 0 , y 0 q, | npx ´yαq| v ď su .

1 See Equation (15.14).

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Corollary 19.3. Let G be a finite index subgroup of GL 2 pR v q, let α P K v be a quadratic irrational over K, and let px 0 , y 0 q P R v ˆRv ´tp0, 0qu. Let g 0 P G α with vptr g 0 q ‰ 0 and let m 0 be the index of g Z 0 in G α . Let c 2 " 2 q 2 v pq ´1q q g´1 |ZpGq| |vptr g 0 q| rGL 2 pR v q px 0 , y 0 q : G px 0 , y 0 q s pq v 2 ´1q ζ K p´1q |α ´ασ | v m 0 m v, x 0 , y 0 pN xx 0 , y 0 yq 2 rGL 2 pR v q : Gs .

Then there exists κ ą 0 such that, as s Ñ `8, Ψ 1 psq " c 2 s `Ops 1´κ q .

Proof. Using Equation ( 18.3) (with Γ the image of G in PGL 2 pR v q) and Equation (17.7), we have ż

The corollary then follows by applying the equidistribution claim in Theorem 19.1 to the characteristic function of a compact-open fundamental domain of K v ´tα, α σ u modulo the action by homographies of G α . l

Example 19.4. Let px 0 , y 0 q " p1, 0q, K " F q pY q, v " v 8 and G " GL 2 pF q rY sq. Using Equations (14.5) and (16.1), Proposition 17.3 (1), and the fact that |ZpGq| " q ´1 to simplify the constant c 2 of Corollary 19.3, and recalling the expression of the absolute value at 8 in terms of the degree from Section 14.2, we get the following counting result: For every integral quadratic irrational α P F q ppY qq over F q pY q, there exists κ ą 0 such that, as t Ñ `8, Card GL 2 pF q rY sq α I " px, yq PF q rY s ˆFq rY s : xx, yy " F q rY s, degpx 2 ´xy trpαq `y2 npαqq ď t * " 2 m 0 pq ´1q 4 degptr g 0 q q 1´1 2 degptrpαq 2 ´4 npαqq q t `Opq t´κ q , where g 0 P GL 2 pF q rY sq fixes α with degptr g 0 q ‰ 0 and m 0 is the index of g Z 0 in the stabiliser GL 2 pF q rY sq α of α in GL 2 pF q rY sq.

Proof of Theorem 19.1. The proof is similar to that of Theorem 16.1. Let r " x 0 y 0 P K Y t8u. If y 0 " 0, let g r " id P GL 2 pKq, and if y 0 ‰ 0, let

We apply Theorem 15.4 with Γ :" G the image of G in PGL 2 pR v q, D ´:" sα, α σ r the (image of any) geodesic line in X v with points at infinity α and α σ , and D `:" γ r H 8 , where γ r is the image of g r in PGL 2 pR v q.

We have L Γv " 2Z and the family D `" pγD `qγPΓ{Γ D `is locally finite, as seen in the beginning of the proof of Theorem 16.1. The family D ´" pγD ´qγPΓ{Γ D ´is locally finite as seen in the beginning of the proof of Theorem 17.1.

By Proposition 15.2 (5), we have (on the full measure subset

For every γ P Γ{Γ r such that D ´and γD `are disjoint, let ρ γ be the geodesic ray starting from α é, γ p0q and ending at the point at infinity γ ¨r of γD `.

Hence, as in order to obtain Equation (16.2), we have, with an error term for every β P s0, 1s of the form Ops ´κ}ψ} β q for some κ ą 0 when evaluated on ψ

We use the following Lemma to switch from counting over elements γ P Γ{Γ r for which 0 ă dpD ´, γD `q ď t to counting over integral representations with bounded value of the norm form. See [PaP11a, page 1054] for the analogous result for the real hyperbolic 3-space and indefinite binary Hermitian forms.

Lemma 19.5. Let g P GL 2 pR v q and let γ be the image of g in PGL 2 pKq. Let z 0 " y 0 if y 0 ‰ 0 and z 0 " x 0 otherwise. Let px, yq " gpx 0 , y 0 q. If dpD ´, γD `q ą 0, then

Proof. We start by showing that

Indeed, if y 0 ‰ 0, we have g g r p1, 0q " gpr, 1q " 1 y 0 gpx 0 , y 0 q and otherwise g g r p1, 0q " gp1, 0q " 1 x 0 gpx 0 , 0q " 1 x 0 gpx 0 , y 0 q .

In particular,

Note that g g r P GL 2 pKq and | detpg g r q| v " | det g| v | det g r | v " 1 since g P GL 2 pR v q. By Proposition 17.3 (2), we hence have

We use the signed distance dpL, Hq " min xPL β ξ px, x H q between a geodesic line L and a horoball H centred at ξ ‰ L ˘, where x H is any point of the boundary of H. Now, by Equations (15.2) and (2.8), we have dpD ´, γD `q " dpsα, α σ r , γγ r H 8 q " d `spg g r q ´1 ¨α, pg g r q ´1 ¨ασ r , H 8 " v `pg g r q ´1 ¨α ´pg g r q ´1 ¨ασ " ´ln ˇˇpg g r q ´1 ¨α ´pg g r q ´1 ¨ασ ˇˇv ln q v " ln hppg g r q ´1 ¨αq ln q v . ( l By discreteness, there are only finitely many double classes rgs P G α zG{G px 0 ,y 0 q such that D ´" sα, α σ r and gD `" g g r H 8 are not disjoint. Let ZpGq be the centre of G, which is finite. Since ZpGq acts trivially on P 1 pK v q, the map G{G px 0 ,y 0 q Ñ Γ{Γ r induced by the canonical map GL 2 pR v q Ñ PGL 2 pR v q is |ZpGq|-to-1. Using the change of variable A weak Gibbs measure is the unique equilibrium, by J. Buzzi

Abstract

For a transitive topological Markov shift endowed with a Hölder-continuous potential, we prove that a weak Gibbs measure is the unique equilibrium measure.

A.1 Introduction

Let σ : Σ Ñ Σ be a topological Markov shift (possibly one-or two-sided), see for instance Section 5.1. More precisely, we consider the one-sided and two-sided vertex-shifts defined by a countable oriented graph G with set of vertices V G and set of arrows A G Ă V G ˆVG . We assume that Σ is transitive, that is, that G is connected (as an oriented graph). We denote by PpΣq the set of σ-invariant probability measures on Σ and by P erg pΣq the subset of ergodic ones. Recall that, for all n P N, the n-cylinders are the following subsets of Σ, where x varies in Σ:

C n pxq " rx 0 . . . x n´1 s " ty P Σ : @ k P t0, . . . , n ´1u, y k " x k u , so that the 1-cylinders are rvs " ty P Σ : y 0 " vu for all v P V G . The points of Σ admitting n P N as period under the shift σ form the set Fix n pΣq " tx P Σ : σ n x " xu .

We fix a potential on Σ, that is, a continuous function φ : Σ Ñ R. We do not assume that φ is bounded. We define φ ´" maxt´φ, 0u and, for all n P N ´t0u, var n pφq " sup

x,yPΣ, @ kPt0,. Note that cpmq is then unique, called the Gibbs constant of m. Let us stress that we do not assume the so-called Big Image Property [START_REF] Sarig | Existence of Gibbs measures for countable Markov shifts[END_REF] and hence using the above weakened Gibbs property (that is, allowing C to depend on v) is necessary.

Note that if Σ is locally compact, that is, if every vertex of G has finite degree (the number of arrows arriving or leaving from the given vertex), then the above condition is equivalent to the fact that for any nonempty compact subset K in Σ, there exists C ě 1 with @ n ě 1, @ x P Fix n pΣq X K, C ´1 ď mpC n pxqq exp pS n φpxq ´cpmqnq ď C .

The pressure P pφ, νq of an element ν P PpΣq such that ş φ ´dν ă `8 is P pφ, νq " h ν pσq `ż φ dν.

An equilibrium measure µ eq for pΣ, φq is an element µ eq P PpΣq such that ş φ ´dµ eq ă `8 and P pφ, µ eq q " suptP pφ, νq : ν P PpΣq and ż φ ´dν ă `8u .

The Gurevič pressure is If there exists an equilibrium measure µ, then dµ " h dν where h : Σ Ñ R is a continuous, positive function and ν is a positive measure on Σ such that ' L φ h " e P G pφq h, and L φ ν " e P G pφq ν where L φ is the transfer operator defined by L φ u pxq " ř yPσ ´1x e φpyq upyq.

' ν is finite on each cylinder. l

We note that [BuS] assumed sup φ ă 8, but this was only used to justify the variational principle and so this condition can be removed by using Theorem A.2.

We now state the main result of this appendix.

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Theorem A.4. Let pΣ, σq be a one-sided transitive topological Markov shift and let φ : Σ Ñ R be a potential with summable variations. Let m be an invariant probability measure of Σ such that ş φ ´dm ă `8. Then m is a weak Gibbs measure if and only if it is an equilibrium measure. In this case, the Gibbs constant cpmq is equal to the Gurevič pressure and the equilibrium measure is unique.

By a classical argument that follows, this result extends to two-sided topological Markov shifts (up to a slight strengthening of the regularity assumption).

Corollary A.5. Let pΣ, σq be a two-sided transitive topological Markov shift and let φ : Σ Ñ R be a potential with ř ně1 n var n pφq ă 8. Let m be an invariant probability measure of Σ such that ş φ ´dm ă `8. Then m is a weak Gibbs measure if and only if it is an equilibrium measure. In this case, the Gibbs constant cpmq is equal to the Gurevič pressure and the equilibrium measure is unique.

Remark. The case of the full shift N Z has been treated in [START_REF] Pesin | Thermodynamics of towers of hyperbolic type[END_REF]Sec. 3]. More generally, assuming the Big Image Property, the above result follows from [START_REF] Sarig | Existence of Gibbs measures for countable Markov shifts[END_REF] and [BuS] along the lines of [PeSZ].

Proof of Corollary A.5. Let pΣ, σq, φ, and m be as in the statement of this Corollary. Let π : Σ Ñ Σ `with px n q nPZ Þ Ñ px n q nPN be the obvious factor map onto the one-sided topological Markov shift pΣ `, σ `q defined by the same graph G as for pΣ, σq, called the natural extension.

First, we replace φ by a potential φ depending only on future coordinates. The proof of [START_REF] Bowen | Equilibrium states and the ergodic theory of Anosov diffeomorphisms[END_REF]Lemma 1.6] applies to our non-compact setting without changes. To be more precise, for each vertex a P V G , choose z a P Σ with z a 0 " a, and define r : Σ Ñ Σ by rpxq " y with y n " x n for n ě 0 and y n " z x 0 n for n ď 0 and let upxq " Step 5. If the mean entropy H µ pβq " ´řbPβ µpbq ln µpbq is finite, then h µ pσq " ´ż ln dµ dσ ˚µ dµ , where σ ˚µ is the measure on Σ defined by B Þ Ñ ř bPβ µpσpB X bqq, with respect to which µ is absolutely continuous: µ Î σ ˚µ. This is a classical formula which follows from the computation of the entropy in terms of the information function The absolute continuity follows from a direct computation and ensures that the integral above is well-defined.

Step 6. For all a P V G , N ě 1 and µ P P erg pΣq with ş φ ´dµ ă `8, we have We use arguments from the proof of [START_REF] Buzzi | Uniqueness of equilibrium measures for countable Markov shifts and multidimensional piecewise expanding maps[END_REF]Theorem 1.1]: the key is to see that the partition β of ras has finite mean entropy for the induced measure μ using a Bernoulli approximation.

Let us consider the Bernoulli measure µ B for pras, σq defined by µ B `n´1 č i"0 σ ´iB i ˘" n´1 ź i"0 µpB i q for all B i P β. We construct from it an invariant and ergodic measure µ B on pΣ, σq: For every Borel subset A, let This formula extends to h µ pσ N q for all integers N ě 1. Using Abramov's formula this time for µ and µ (since µ is ergodic), we have h µ pσq " µprasq N h µ pσ N q " ´µprasq ż ras 1 N ln dµ dppσ N q ˚µq dµ , as claimed.

Step 7. The entropy of m is equal to cpmq ´ş φ dm.

In order to prove this, we apply Step 6 with µ " m (which is possible, since m has been proven to be ergodic in Step 3). As in the proof of Step 3, the Radon-Nikodym derivative is almost everywhere dµ dppσ N q ˚µq pxq " lim nÑ8 µpβ n pxqq µpσ N pβ n pxqqq " C ˘2 exp `´cpmqτ N pxq `Sτ N pxq φpxq ˘.

Therefore, using Step 6 and the fact that µ |ras " µprasq µ, we have To analyze the second term in Equation (A.4), let φpxq " ř τ pxq´1

k"0 φpσ k xq and observe that, by a variation of the proof of Kac's theorem, φ P L 1 pµq with µp φq " µprasq ´1µpφq. Indeed, passing to the natural extension, one can assume the system to be invertible and use the partition modulo µ given by ď ně1, 0ďkăn σ k ptx P ras : τ pxq " nuq .

Since S τ N pxq φpxq coincides with the Birkhoff sum SN φpxq for the induced system, Birkhoff's ergodic theorem yields, with convergence in L 1 pµq, Step 8. Conclusion: any weak Gibbs measure is an equilibrium measure and cpmq " P G pφq.

Steps 4 and 7 prove that h m pσq `ş φ dm is well-defined and equal to cpmq, which by Step 4 is equal to P G pφq, which is equal to suptP pφ, νq : ν P PpΣq and ş φ ´dν ă `8u by Theorem A.2, so that m is an equilibrium measure. This completes the proof of Theorem A.4. l 295 19/12/2016

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