Frédéric Paulin

Mark Pollicott

Logarithm laws for equilibrium states in negative curvature

Let M be a pinched negatively curved Riemannian manifold, whose unit tangent bundle is endowed with a Gibbs measure m F associated to a potential F . We compute the Hausdorff dimension of the conditional measures of m F . We study the m F -almost sure asymptotic penetration behaviour of locally geodesic lines of M into small neighbourhoods of closed geodesics, and of other compact (locally) convex subsets of M . We prove Khintchine-type and logarithm law-type results for the spiraling of geodesic lines around these objects. As an arithmetic consequence, we give almost sure Diophantine approximation results of real numbers by quadratic irrationals with respect to general Hölder quasi-invariant measures. 1

Introduction

Let M be a complete connected Riemannian manifold with pinched sectional curvature at most -1, and let (g t ) t∈R be its geodesic flow. In this paper, we consider for instance a closed geodesic D 0 in M , and we want to study the spiraling of geodesics lines around D 0 . Given an ergodic probability measure m invariant under (g t ) t∈R , whose support is the nonwandering set Ω of the geodesic flow, m-almost every orbit is dense in Ω. Two geodesic lines, having at some time their unit tangent vectors very close, remain close for a long time. Hence m-almost every geodesic line will stay for arbitrarily long periods of times in a given small neighbourhood of D 0 . In what follows, we make this behaviour quantitative when m is any equilibrium state.

Let F : T1 M → R be a potential, that is, a Hölder-continuous function. Let M be the set of probability measures m on T 1 M invariant under the geodesic flow, for which the negative part of F is m-integrable, and let h m (g 1 ) be the (metric) entropy of the geodesic flow with respect to m. The pressure of the potential F is

P = P (F ) = sup m∈M h m (g 1 ) + T 1 M F dm .
(1)

Let m F be a Gibbs measure on T 1 M associated to the potential F (see [PPS] and Section 2). When finite and normalised to be a probability measure (and if the negative part of F is m F -integrable), it is the unique equilibrium state, that is, it attains the upper bound defining the pressure P (F ) (see [START_REF] Paulin | Equilibrium states in negative curvature[END_REF]Theo. 6.1], improving [OP] when F = 0). For instance, m F is (up to a constant multiple) the Bowen-Margulis measure m BM if F = 0, and is the Liouville measure if F is the strong unstable Jacobian v → -d dt |t=0 log Jac g t |W su (v) (v) and M is compact (see [START_REF] Paulin | Equilibrium states in negative curvature[END_REF]Theo. 7.2] for a more general case). We will use the construction of m F by Paulin-Pollicott-Schapira [PPS] (building on work of Hamenstädt, Ledrappier, Coudène, Mohsen) via Patterson densities (µ F

x ) x∈ M on the boundary at infinity ∂ ∞ M of a universal cover M of M associated to the potential F .

We first prove (see Section 3) the following result relating measure theoretic invariants of m F and µ F

x , which extends Ledrappier's result [START_REF] Ledrappier | Entropie et principe variationnel pour le flot géodésique en courbure négative pincée[END_REF]§4] when F = 0.

Theorem 1.1 If m F is finite and F is m F -integrable, the Hausdorff dimension of the Patterson measure µ F x (with respect to the Gromov-Bourdon visual distance on ∂ ∞ M ) is equal to the metric entropy of the Gibbs measure m F (for the geodesic flow).

Let D 0 be a closed geodesic in M of length ℓ 0 . If v 0 ∈ T 1 M is tangent to D 0 , let

P 0 = P (F |T 1 D 0 ) = max{ ℓ 0 0 F (g t v 0 ) dt, ℓ 0 0 F (g t (-v 0 )) dt} ℓ 0 . (2)
We will prove that P 0 < P if m F is finite. Let ǫ 0 > 0 and let ψ : [0, +∞[ → [0, +∞[ be a Lipschitz map. As introduced in [START_REF] Hersonsky | On the almost sure spiraling of geodesics in negatively curved manifolds[END_REF], let E(ψ) be the set of (ǫ 0 , ψ)-Liouville vectors around D 0 , that is the set of v ∈ T 1 M such that there exists a sequence (t n ) n∈N in [0, +∞[ converging to +∞ such that for every t ∈ [t n , t n + ψ(t n )], the footpoint of g t v belongs to the ǫ 0 -neighbourhood N ǫ 0 D 0 of 0.

The Khintchine-type result describing the spiraling around the closed geodesic D 0 is the following (simplified version of the) main result of this paper (see Section 4).

Theorem 1.2 Assume that M is compact. If the integral +∞ 0 e ψ(t)(P 0 -P ) dt diverges (resp. converges) then m F -almost every (resp. no) point of T 1 M belongs to E(ψ).

When F = 0 (that is, when m F is the Bowen-Margulis measure), this result is due to Hersonsky-Paulin [START_REF] Hersonsky | On the almost sure spiraling of geodesics in negatively curved manifolds[END_REF]. As m F can be taken to be the Liouville measure, this result answers a question raised in loc. cit. This result, in this particular case when D 0 is a closed geodesic, can be restated as a well-approximation type of result of points in the limit set of the fundamental group of Γ by an orbit of a loxodromic fixed point, see for instance [FSU] for very general results (their measure on the limit set corresponds to F = 0, though an extension might be possible), and the references of [FSU] for historical motivation and partial results. This result is a shrinking target problem type, and our main tool is the mixing property of the geodesic flow of M for Gibbs measure (see [PPS]).

We stated this result as such to emphasize its novelty even in the compact case, but it is true in a much more general setting, both from M and D 0 (see Theorem 4.1). For instance, when M is a geometrically finite locally symmetric orbifold, when F has finite pressure P (F ) and finite Gibbs measure m F , when D 0 is a compact totally geodesic suborbifold (of positive dimension and codimension), the result still holds. When M is the quotient of real hyperbolic 3-space by a geometrically finite Kleinian group Γ, when F has finite pressure P (F ) and finite Gibbs measure m F , and when D 0 is the convex hull of the limit set of a precisely invariant quasi-fuschian closed surface subgroup Γ 0 of Γ, the result still holds. See Section 4 for more examples.

When F = 0, the following logarithm law for the almost sure spiraling of geodesic lines around D 0 is due to Hersonsky-Paulin [START_REF] Hersonsky | On the almost sure spiraling of geodesics in negatively curved manifolds[END_REF]. Let π : T 1 M → M be the unit tangent bundle. Define the penetration map p : T 1 M × R → [0, +∞] of the geodesic lines inside N ǫ 0 D 0 by p(v, t) = 0 if π(φ t v) / ∈ N ǫ 0 D 0 , and otherwise p(v, t) is the maximal length of an interval I in R containing t such that π(φ s v) ∈ N ǫ 0 D 0 for every s ∈ I.

Corollary 1.3 Under the assumptions of Theorem 1.2, for m F -almost every v ∈ T 1 M , we have lim sup t→+∞ p(v, t) log t = 1 P -P 0 .

In Section 5, we will give arithmetic applications of Theorem 1.2. We will in particular generalise to a huge class of measures on R the Khintchine-type result of approximation of real numbers by quadratic irrationals over Q, that was proved in [START_REF] Parkkonen | Spiraling spectra of geodesic lines in negatively curved manifolds[END_REF] for the Lebesgue measure, and prove other 0-1-laws of approximations of real numbers by arithmetically defined points. To conclude this introduction, we give one example of such a result.

Let a, b ∈ N -{0} be positive integers such that the equation x 2a y 2b z 2 = 0 has no nonzero integer solution (for instance a = 2 and b = 3). Let Γ a, b be

x + y √ a z -t √ a b(z + t √ a)
xy √ a : (x, y, z, t) ∈ Z 4 and x 2a y 2b z 2 + ab t 2 = 1 , which is a discrete subgroup of SL 2 (R), whose action by homographies on P 1 (R) = R∪{∞} is denoted by • . If α ∈ R is a solution of the equation γ • X = X for some γ ∈ Γ a, b , then α is quadratic over Q( √ a), and if furthermore α / ∈ Q( √ a), we denote by α σ its Galois conjugate over Q( √ a). Given γ ∈ Γ a, b with trace tr γ = 0, ±2, we denote by γ + and γ - the attractive and repulsive fixed points of γ in R ∪ {∞}. Given a continuous action of a discrete group G on a compact metric space (X, d), recall that a Hölder quasi-invariant measure (see for instance [START_REF] Ledrappier | Structure au bord des variétés à courbure négative[END_REF][START_REF] Hamenstädt | Cocycles, Hausdorff measures and cross ratios[END_REF]) on X (for the action of G) is a probability measure µ such that for every g ∈ G, the measure g * µ is absolutely continuous with respect to µ, and the Radon-Nykodim derivative d g * µ d µ coincides µ-almost everywhere with a Hölder-continuous map on X, that we will still denote by d g * µ d µ . The next result is a Khintchine-type of result, under a huge class of measures, for the Diophantine approximation of real numbers by quadratic irrationals over Q( √ a) in a (dense) orbit under the arithmetic group Γ a, b (extended by the Galois conjugation).

Corollary 1.4 Let µ be a Hölder quasi-invariant measure on R ∪ {∞} for the action by homographies of Γ a, b . Let γ 0 be a primitive element in Γ a, b with tr(γ 0 ) = 0, ±2. For µ-almost every x ∈ R, we have

lim inf α∈Γ a, b •{γ - 0 ,γ + 0 } : |α-α σ |→0 |x -α| |α -α σ |(-log |α -α σ |) -s = 0 (resp. = +∞) if s ≤ 1 δ-δ 0 (resp. s > 1 δ-δ 0 ), where δ = lim sup s→+∞ 1 2 log s log γ∈Γ a, b , 2<| tr(γ)|≤s d(γ -1 ) * µ d µ (γ + ) and δ 0 = 1 2 arcosh( | tr γ 0 | 2 ) max d(γ -1 0 ) * µ d µ (γ + 0 ), d(γ 0 ) * µ d µ (γ - 0 ) .
We refer to Section 5 for more general results, in particular for approximations with congruence properties and for the approximation of complex numbers by quadratic irrationals over an imaginary quadratic extension of Q.

Acknowledgments:

The second author thanks the École Normale Supérieure, for an Invited Professor position in 2009 where this work was started, and the Université Paris-Sud (Orsay) for an Invited Professor position in 2014 where this work was completed.

A summary of the Patterson-Sullivan theory for Gibbs states

Most of the content of this section is extracted from [PPS], to which we refer for the proofs of the claims and for more details.

Let M be a complete simply connected Riemannian manifold with (dimension at least 2 and) pinched negative sectional curvature -b 2 ≤ K ≤ -1, and let x 0 ∈ M be a fixed basepoint. For every ǫ > 0 and every subset A of M , we denote by

N ǫ A the closed ǫ-neighbourhood of A in M .
We denote by π : T 1 M → M the unit tangent bundle of M , where T 1 M is endowed with Sasaki's Riemannian metric. Let ∂ ∞ M be the boundary at infinity of M . We denote by ΛG the limit set of any discrete group of isometries G of M , and by C ΛG the convex hull in M of ΛG, if ΛG has at least two elements.

Let Γ be a nonelementary (not virtually nilpotent) discrete group of isometries of M . Let M and T 1 M be the quotient Riemannian orbifolds Γ\ M and Γ\T 1 M , and let again π : T 1 M → M be the map induced by π : T 1 M → M . We denote by (g t ) t∈R the geodesic flow on T 1 M , as well as its quotient flow on T 1 M .

For

every v ∈ T 1 M , let v -∈ ∂ ∞ M and v + ∈ ∂ ∞ M
, respectively, be the endpoints at -∞ and +∞ of the geodesic line g v : R → M defined by v (that is, such that ġv (0

) = v). Let ∂ 2 ∞ M be the subset of ∂ ∞ M ×∂ ∞ M which consists of pairs of distinct points at infinity of M . Hopf 's parametrisation of T 1 M is the homeomorphism which identifies T 1 M with ∂ 2 ∞ M × R, by the map v → (v -, v + , t), where t is the signed distance of the closest point to x 0 on g v (R) to π(v). Let ΩΓ be the Γ-invariant set of v ∈ T 1 M such that v -, v + ∈ ΛΓ, whose image in T 1 M is the nonwandering set of the geodesic flow of T 1 M .
Let ι : T 1 M → T 1 M be the (Hölder-continuous) antipodal (flip) map of T 1 M defined by ιv = -v. We denote the quotient map of ι again by ι :

T 1 M → T 1 M . Let F : T 1 M → R be a fixed Hölder-continuous Γ-invariant function, called a potential on T 1 M . It induces a Hölder-continuous function F : T 1 M → R, called a potential on T 1 M .
Two potentials F and F * on T 1 M (or their induced maps on T 1 M ) are cohomologous if there exists a Hölder-continuous Γ-invariant map G : T 1 M → R, differentiable along every flow line, such that

F * (v) -F (v) = d dt |t=0 G(φ t v) . (3)
For any two distinct points x, y ∈ M , let v xy ∈ T 1 x M be the initial tangent vector of the oriented geodesic segment [x, y] in M that connects x to y; define

y x F = d(x,y) 0 F (g t v xy ) dt and x
x F = 0 for all x ∈ M . Given a hyperbolic element γ ∈ Γ with translation axis A γ , the period of γ for F is, for any x ∈ A γ , [START_REF] Paulin | Equilibrium states in negative curvature[END_REF]Lem. 3.3]). We assume that δ Γ, F < +∞ (this is for instance satisfied if F is bounded, see [START_REF] Paulin | Equilibrium states in negative curvature[END_REF]Lem. 3.3]). By [START_REF] Paulin | Equilibrium states in negative curvature[END_REF]Theo. 6.1], the critical exponent δ Γ, F is equal to the pressure P (F ) of F on T 1 M , defined in Equation (1). The Poincaré series

Per F (γ) = γx x F . The critical exponent of (Γ, F ) is δ Γ, F = lim sup n→+∞ 1 n log γ∈Γ, n-1<d(x,γy)≤n e γy x F . When F = 0, then δ Γ, F is the standard critical exponent δ Γ of Γ. Note that y x F = x y F • ι , (4)
that δ Γ, F = δ Γ, F •ι > -∞ and that δ Γ, F +c = δ Γ, F + c for every constant c > 0 (see
Q Γ, F, x 0 (s) = γ∈Γ e γy x ( F -s) of (Γ, F ) converges if s > δ Γ, F and diverges if s < δ Γ, F . We say that (Γ, F ) is of divergence type if Q Γ, F, x 0 (s) diverges at s = δ Γ, F . The (normalised) Gibbs cocycle of F is the function C F : ∂ ∞ M × M × M → R defined by (ξ, x, y) → C F ξ (x, y) = lim t→+∞ ξt y ( F -δ Γ, F ) - ξt x ( F -δ Γ, F ) ,
where t → ξ t is any geodesic ray with endpoint ξ ∈ ∂ ∞ M . We have

C F +c = C F for every constant c ∈ R. If F = 0, then C F = δ Γ β, where β is the Busemann cocycle.
By [START_REF] Paulin | Equilibrium states in negative curvature[END_REF]Lem. 3.2,3.4], there exists a constant c 1 > 0 (depending only on the Hölder constants of F and on the bounds of the sectional curvature of M ) such that for all x, y, z ∈ M , we have

z x F - z y F ≤ c 1 e d(x, y) + d(x, y) max π -1 (B(x, d(x, y))) | F | , (5)
and, for every

ξ ∈ ∂ ∞ M , C F ξ (x, y) ≤ c 1 e d(x, y) + d(x, y) max π -1 (B(x, d(x, y))) | F | . (6)
A family (µ F x ) x∈ M of finite measures on ∂ ∞ M , whose support is the limit set ΛΓ of Γ, is a Patterson density for the pair (Γ, F ) if

γ * µ F x = µ F γx
for all γ ∈ Γ and x ∈ M , and if the following Radon-Nikodym derivatives exist for all x, y ∈ M and satisfy for almost all

ξ ∈ ∂ ∞ M dµ F x dµ F y (ξ) = e -C F ξ (x, y) .
A Gibbs measure on T 1 M for (Γ, F ) is the measure m F on T 1 M given by the density [START_REF] Paulin | Equilibrium states in negative curvature[END_REF]§3.6], their construction, whence the existence of m F , requires only Γ to be nonelementary and δ Γ, F < +∞). The Gibbs measure m F is independent of x 0 , its support is ΩΓ, and it is invariant under the actions of the group Γ and of the geodesic flow. Thus (see [START_REF] Paulin | Equilibrium states in negative curvature[END_REF]§2.6]), it defines a measure m F on T 1 M which is invariant under the quotient geodesic flow, called a Gibbs measure on T 1 M . For every constant c > 0, note that (µ F

d m F (v) = e C F •ι v -(x 0 , π(v)) + C F v + (x 0 , π(v)) dµ F •ι x 0 (v -) dµ F x 0 (v + ) dt (7) in Hopf's parametrisation. Patterson densities (µ F x ) x∈ M and (µ F •ι x ) x∈ M exist (see
x ) x∈ M is also a Patterson density for the pair (Γ, F + c), thus m F is also a Gibbs measure for (Γ, F + c). If m F is finite, then the Patterson densities are unique up to a common multiplicative constant (see [START_REF] Paulin | Equilibrium states in negative curvature[END_REF]§5.3]); hence the Gibbs measure of m F is uniquely defined, up to a multiplicative constant, and, when normalised to be a probability measure, it is the unique equilibrium state for the potential F , if the negative part of F is m F -integrable, see [START_REF] Paulin | Equilibrium states in negative curvature[END_REF]Theo. 6.1].

By its definition as a quasi-product, the Gibbs measure m F satisfies the following property, used without mention in what follows: for every x ∈ M , the premiage by v → v + of a set of measure 0 (respectively > 0) for µ F

x has measure 0 (respectively > 0) for m F . We refer to [START_REF] Paulin | Equilibrium states in negative curvature[END_REF]§8] for finiteness criteria of m F , in particular satisfied if M is compact. Babillot [START_REF] Babillot | On the mixing property for hyperbolic systems[END_REF]Thm. 1] showed that if m F is finite, then it is mixing for the geodesic flow on T 1 M if the length spectrum of Γ is nonarithmetic (that is, if the set of translation lengths of the elements of Γ is not contained in a discrete subgroup of R). This condition, conjecturally always true, is known, for example, if Γ has a parabolic element, if ΛΓ is not totally disconnected (hence if M is compact), or if M 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].

For every subset A of M and every point x in M ∪ ∂ ∞ M , the shadow of A seen from x is the set O x A of points at infinity of the geodesic rays or lines starting from x and meeting A. By Mohsen's shadow lemma (see [START_REF] Paulin | Equilibrium states in negative curvature[END_REF]Lem. 3.10]), for every x ∈ M , if R > 0 is large enough, there exists c = c(R) > 0 such that for every γ ∈ Γ, we have

1 c e γy x ( F -δ Γ, F ) ≤ µ F x (O x (B(γy, R))) ≤ c e γy x ( F -δ Γ, F ) . (8)
Here is a new consequence of Mohsen's shadow lemma which will be useful in this paper. Recall that a discrete group G of isometries of M is convex-cocompact if its limit set ΛG contains at least two points, and if the action of G on the convex hull C ΛG in M of ΛG has compact quotient.

Lemma 2.1 Let Γ 0 be a convex-cocompact subgroup of Γ such that δ Γ 0 , F 0 < δ Γ, F , where F 0 : Γ 0 \T 1 M → R is the map induced by F . Then µ F

x 0 (ΛΓ 0 ) = 0.

Proof. Since Γ 0 is convex-cocompact, if R is big enough, for every n ∈ N, we have

ΛΓ 0 ⊂ γ∈Γ 0 , d(x 0 , γx 0 )≥n O x 0 (B(γx 0 , R)) .
Hence, by Equation ( 8), there exists c > 0 such that for every n ∈ N,

µ F x 0 (ΛΓ 0 ) ≤ γ∈Γ 0 , d(x 0 , γx 0 )≥n µ F x 0 (O x 0 (B(γx 0 , R))) ≤ c γ∈Γ 0 , d(x 0 , γx 0 )≥n e γx 0 x 0 ( F -δ Γ, F ) .
The Poincaré series Q Γ 0 , F 0 , x 0 (δ Γ, F ) converges, as δ Γ 0 , F 0 < δ Γ, F . Since the remainder of a converging series tends to 0, this proves the result.

A parabolic subgroup of Γ is a maximal infinite subgroup Γ 0 of Γ whose limit set ΛΓ 0 is a singleton. It is bounded if Γ 0 \(ΛΓ -ΛΓ 0 ) is compact. For every bounded parabolic subgroup Γ 0 , if ΛΓ 0 = {ξ 0 }, there exists (see for instance [Bow]) a unique Γ-equivariant family (H αξ 0 ) α∈Γ/Γ 0 of maximal closed horoballs in M with pairwise disjoint interiors and with H ξ 0 centred at ξ 0 . The horoball H ξ 0 is precisely invariant under Γ, that is, its stabiliser in Γ is Γ 0 and the inclusion

• H ξ 0 ⊂ M induces an injection Γ 0 \ • H ξ 0 → Γ\ M . Note that if Γ 0 is a parabolic subgroup of Γ and if m F is
finite, then we also have µ F

x 0 (ΛΓ 0 ) = 0 (see [START_REF] Paulin | Equilibrium states in negative curvature[END_REF]Lem. 5.15]).

Hausdorff dimension of Patterson measures of potentials

Let 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 M . Let F : T 1 M → R be a Hölder-continuous Γ-invariant function. Assume that δ = δ Γ, F is finite. Let m F be the Gibbs measure on T 1 M associated to a pair of Patterson densities

(µ F •ι x ) x∈ M , (µ F x ) x∈ M for (Γ, F • ι)
and (Γ, F ). We use the notation introduced in Section 2.

We fix in this section a point x in M . We denote by d x the Gromov-Bourdon visual distance on ∂ ∞ M seen from x, defined (see [Bou]) by

d x (ξ, η) = lim t→+∞ e 1 2 (d(ξt, ηt)-d(x, ξt)-d(x, ηt)) , (9)
where t → ξ t , η t are any geodesic rays ending at ξ, η respectively. We endow from now on ∂ ∞ M with the distance d x . The aim of this section is to compute the Hausdorff dimension of the Patterson measure µ F

x associated to the potential F (which will be independent of x). Recall that the Hausdorff dimension dim H (ν) of a finite nonzero measure ν on a locally compact metric space X is the greatest lower bound of the Hausdorff dimensions dim H (Y ) of the Borel subsets Y of X with ν(Y ) > 0.

Let us give a motivation for such a computation. As mentioned in the introduction, we are interested in this paper in studying whether the set E(ψ) of vectors of T 1 M that are well-spiraling, as quantified by ψ, around a given closed geodesic D 0 has full or zero measure for the Gibbs measure m F . Varying the potential F may be useful to estimate the Hausdorff dimension of E(ψ): if )-P (F )) dt diverges, as we will prove in Section 4, the set E(ψ) has full measure for m F , and hence dim H (E(ψ)) ≥ dim H (m F ). Note that the Hopf parametrisation of T 1 M is Hölder-continuous (though usually not Lipschitz, except in particular when M is a symmetric space), and m F is in the same measure class as the product measure dµ F •ι

x ⊗ dµ F x ⊗ dt. Hence dim H (m F ) may be estimated using dim H (µ F x ) (that we will prove to be equal to dim H (µ F •ι

x )), and is in fact equal to

2 dim H (µ F x ) + 1 if M is a symmetric space.
The main result of this section, proving Theorem 1.1 in the introduction, is the following one. To simplify the notation, let h(m) = h m m (g 1 ) be the (metric) entropy of the geodesic flow with respect to a finite nonzero (g t ) t∈R -invariant measure m on T 1 M normalised to be a probability measure.

Theorem 3.1 If the Gibbs measure m F is finite and if F is m F -integrable, then the Haus- dorff dimension of the Patterson measure µ F x on (∂ ∞ M , d x ) associated to F satisfies dim H (µ F x ) = dim H (µ F •ι x ) = h(m F ) ≤ δ Γ . ( 10
)
If M is convex-cocompact, then the last inequality is an equality if and only if F -P (F ) is cohomologous to the zero potential.

We think that the convex-cocompact assumption in the last claim could be improved (see the comment at the end of this section).

The first claim is a generalisation of a result of Ledrappier [START_REF] Ledrappier | Entropie et principe variationnel pour le flot géodésique en courbure négative pincée[END_REF], who proved the theorem in the particular case F = 0. Then µ 0

x is the standard Patterson measure of Γ and the associated Gibbs measure m F is the Bowen-Margulis measure m BM . Let Λ c Γ denote the conical (or radial) limit set, that is, the set of ξ ∈ ∂ ∞ M for which lim inf t→+∞ d(ρ(t), Γx) < +∞, where ρ is any geodesic ray with point at infinity ξ. Let h top (g 1 ) be the topological entropy of the geodesic flow on T 1 M . If m BM is finite, then Ledrappier [Led2,Theo. 4.3] proves furthermore that

dim H (µ 0 x ) = h(m BM ) = h top (g 1 ) = dim H (Λ c Γ) = δ Γ .
The second equality is due to Otal-Peigné [OP]. The last equality, which does not require the assumption that m BM is finite, is due to Bishop-Jones in constant curvature, to Hamenstädt and to the first author (see [Pau])in general.

Proof. Up to normalising µ F x , which does not change its Hausdorff dimension nor m F m F , we may assume that µ F

x is a probability measure. The proof will follow from a series of lemmas and propositions. The following is a well known useful alternative characterisation of the dimension of the measure, which was also used by Ledrappier [Led2,Prop. 2.5].

Lemma 3.2 For any finite nonzero measure ν on a compact metric space X, the Hausdorff dimension dim H (ν) is the ν-essential greatest lower bound on x ∈ X of

lim inf ǫ→0 log ν(B(x, ǫ)) log ǫ .
For every ξ ∈ ∂ ∞ M , let ρ ξ : [0, +∞[ → M be the geodesic ray with ρ ξ (0) = x and ρ ξ (+∞) = ξ. The next lemma compares shadows of balls in M with (visual) balls in ∂ ∞ M . Lemma 3.3 (Bourdon [Bou]) For sufficiently large R > 0, there exists

D = D(R) such that, for all ǫ > 0 and ξ ∈ ∂ ∞ M , O x (B(ρ ξ (log(1/ǫ) + D), R)) ⊂ B dx (ξ, ǫ) ⊂ O x (B(ρ ξ (log(1/ǫ) -D), R)) .
Our first step in proving the theorem is the following result.

Proposition 3.4 (1) If (Γ, F ) is of divergence type then dim H (µ F x ) ≤ dim H (Λ c Γ) = δ Γ ; (2) If m F is finite and if F is m F -integrable, then dim H (µ F x ) ≤ P (F )-1 m F T 1 M F dm F .
Proof. By [START_REF] Paulin | Equilibrium states in negative curvature[END_REF]Theo. 5.12], if (Γ, F ) is of divergence type, then the set Λ c Γ has full µ F xmeasure, and thus the inequality in Part (1) follows immediately from the definition of the Hausdorff dimension of measures. The equality in Part (1) has already been mentioned.

In order to prove Part (2), note that (Γ, F ) is of divergence type if m F is finite, by [START_REF] Paulin | Equilibrium states in negative curvature[END_REF]Coro. 5.15]. It hence suffices by Lemma 3.2 to show that for µ F

x -almost every ξ in the full µ F

x -measure subset Λ c Γ, we have

lim inf ǫ→0 log µ F x (B dx (ξ, ǫ)) log ǫ ≤ P (F ) - 1 m F T 1 M F dm F . Let ξ ∈ Λ c Γ. By the definition of Λ c Γ, there exist K ≥ 0, a sequence (γ n ) n∈N in Γ and a sequence (t n ) n∈N converging to +∞ in [0, +∞[ such that d(ρ ξ (t n ), γ n x) ≤ K.
By the triangle inequality, we have d(x, γ n x) ≤ t n + K and the ball B(ρ ξ (t n ), R) contains the ball B(γ n x, R -K), for every R ≥ K. Let us apply the inclusion on the left in Lemma 3.3 with ǫ n = e -tn+D(R) , which tends to 0 as n → +∞ (hence in particular may be assumed to be in ]0, 1]). We have

log µ F x (B dx (ξ, ǫ n )) log ǫ n ≤ log µ F x (O x (B(ρ ξ (t n ), R))) log ǫ n ≤ log µ F x (O x (B(γ n x, R -K))) log ǫ n . (11)
By Mohsen's shadow lemma (see Equation ( 8)) and by [START_REF] Paulin | Equilibrium states in negative curvature[END_REF]Theo. 6.1] which says that

P (F ) = δ Γ, F , if R is large enough, there exists c > 0 such that, for every n ∈ N, µ F x (O x (B(γ n x, R -K))) ≥ 1 c e γnx x ( F -P (F ))
.

By Equation ( 5), we have

γnx x F = tn 0 F ( ρξ (s)) ds + O(1) as n → +∞. Thus Equation (11) gives, as n → +∞, log µ F x (B dx (ξ, ǫ n )) log ǫ n ≤ γnx x ( F -P (F )) -log c -t n + D(R) ≤ P (F ) - 1 t n tn 0 F ( ρξ (s)) ds (1 + o(1)) .
By [START_REF] Paulin | Equilibrium states in negative curvature[END_REF]Theo. 5.4], since (Γ, F ) is of divergence type, the geodesic flow in

T 1 M is ergodic for m F . Since F is m F -integrable on T 1 M
, since m F is finite and by the quasi-product structure of m F in Hopf's parametrisation, for µ F x -almost every ξ, we have by Birkhoff's ergodic theorem

lim n→+∞ 1 t n tn 0 F (g s ρξ (0)) ds = 1 m F T 1 M F dm F .
This proves Proposition 3.4.

We next want to show that the reverse inequality holds.

Proposition 3.5 If m F is a finite measure and if F is m F -integrable, then dim H (µ F x ) ≥ P (F ) -1 m F T 1 M F dm F .
Proof. To prove the result, by Proposition 3.2, we only need to show that for µ F

x -almost every ξ, we have

lim inf ǫ→0 log µ F x (B dx (ξ, ǫ)) log ǫ ≥ P (F ) - 1 m F T 1 M F dm F .
As in the proof of [START_REF] Ledrappier | Entropie et principe variationnel pour le flot géodésique en courbure négative pincée[END_REF]Prop. 4.6], since m F is finite and by the quasi-product structure of m F , by Poincaré's recurrence theorem and Birkhoff's ergodic theorem, for µ F

x -almost every ξ, there exist K > 0, a sequence (γ n ) n∈N in Γ and an increasing sequence

(t n ) n∈N , converging to +∞ in [0, +∞[, such that d(ρ ξ (t n ), γ n x) ≤ K
, and such that the limit lim n→+∞ t n /n exists and is positive.

Let R be big enough and let c = c(R + K) be as in Mohsen's shadow lemma (see Equation ( 8)), so that, for every n ∈ N,

µ F x (O x (B(γ n x, R + K))) ≤ c e γnx x (F -P (F ))
.

By the triangle inequality, the ball B(γ n x, R+K) contains the ball B(ρ ξ (t n ), R). For every n ∈ N, let ǫ n = e -tn-D(R) , which decreases to 0. For every ǫ ∈ ]0, 1] small enough, let n = n(ǫ) ∈ N be such that ǫ n ≥ ǫ > ǫ n+1 . By the inclusion on the right in Lemma 3.3 and by the same arguments as in the end of the proof of the previous proposition, we have

log µ F x (B dx (ξ, ǫ)) log ǫ ≥ log µ F x (B dx (ξ, ǫ n )) log ǫ n+1 ≥ log µ F x (O x (B(ρ ξ (t n ), R))) log ǫ n+1 = -log µ F x (O x (B(ρ ξ (t n ), R))) t n+1 + D(R) ≥ -log µ F x (O x (B(γ n x, R + K))) t n+1 + D(R) ≥ - γnx x (F -P (F )) -log c t n+1 + D(R) ≥ t n P (F ) -tn+o(tn) m F T 1 M F dm F + O(1) t n+1 + D(R)
.

Taking the inferior limit as ǫ → 0, since lim n→+∞ tn t n+1 = 1, the result follows. Now, by the Variational Principle [START_REF] Paulin | Equilibrium states in negative curvature[END_REF]Theo. 6.1], since m F is finite and since F is m F -integrable, we have 10) in Theorem 3.1 follows from Propositions 3.4 and 3.5 applied to both F and F • ι.

P (F ) = h(m F ) + 1 m F T 1 M F dm F . Since ι : T 1 M → T 1 M conjugates (g t ) t∈R to (g -t ) t∈R , and since m F •ι = ι * m F , we have h(m F •ι ) = h(m F ). Hence Equation (
If Γ is convex-cocompact, then m F and m BM = m 0 are finite and F is integrable for m F and m 0 . By the uniqueness in the Variational Principle (see [START_REF] Paulin | Equilibrium states in negative curvature[END_REF]Theo. 6

.1]), if h(m F ) = δ Γ = h(m 0 ), then m F m F = m 0 m 0 .
By the Hamenstädt-Ledrappier correspondence (see [START_REF] Ledrappier | Structure au bord des variétés à courbure négative[END_REF][START_REF] Hamenstädt | Cocycles, Hausdorff measures and cross ratios[END_REF][START_REF] Schapira | On quasi-invariant transverse measures for the horospherical foliation of a negatively curved manifold[END_REF] and the following proposition) saying that if Γ is convex-cocompact, the cohomology class of a potential with zero pressure is determined by its associated Gibbs measure, the last claim of Theorem 3.1 follows.

We end this section by a comment on the correspondence between the potentials and their associated Patterson measures, which will be used at the end of this paper.

Proposition 3.6 (Hamenstädt-Ledrappier) If Γ is convex-cocompact, the map F → µ = µ F
x 0 induces a bijection from the set of Γ-invariant Hölder maps F : ΩΓ → R with zero pressure P (F ) = 0, up to cohomologous maps, to the set of measure classes of Hölder quasi-invariant measures µ on (ΛΓ, d x ) endowed with the action of Γ. Furthermore, for every hyperbolic element γ ∈ Γ with attractive fixed point γ + ∈ ΛΓ, the period of γ for F satisfies

Per F (γ) = log d(γ -1 ) * µ d µ (γ + ) . ( 12
)
Proof. The reader who is not interested in seeing how this result can be deduced from [START_REF] Ledrappier | Structure au bord des variétés à courbure négative[END_REF] (whose arguments extend from the cocompact to the convex-cocompact case, as observed in [Sch]) may skip this proof.

Recall that ∂ ∞ M has a unique Hölder structure such that for every x ∈ M , the map v → v + from T 1

x M to ∂ ∞ M (whose inverse will be denoted by ξ → v x, ξ ) is a Hölder homeomorphism.

The following definitions are are taken from [START_REF] Ledrappier | Structure au bord des variétés à courbure négative[END_REF]. A Hölder cocycle for the action of Γ on ∂ ∞ M is a map c : Γ × ΛΓ → R, which is Hölder-continuous in the second variable, such that c(γγ ′ , ξ) = c(γ, γ ′ ξ) + c(γ ′ , ξ) for all γ, γ ′ ∈ Γ and ξ ∈ ΛΓ. The period for c of a hyperbolic element γ of Γ is c(γ, γ + ), where γ + is the attractive fixed point of γ. Two Hölder cocycles c and c ′ are cohomologous if there exists a Hölder-continuous map

U : ΛΓ → R such that c(γ, ξ) -c ′ (γ, ξ) = U (γξ) -U (ξ) for all γ ∈ Γ and ξ ∈ ΛΓ. Given a Hölder quasi-invariant measure µ, its associated Hölder cocycle is c µ : (γ, ξ) → -log d(γ -1 ) * µ d µ (ξ).
The verification that this is indeed a Hölder cocycle is immediate. [START_REF] Paulin | Equilibrium states in negative curvature[END_REF]Prop. 3.5 (2)] for its Hölder-continuity, F being bounded since Γ\ ΩΓ is compact). Hence, by the definition of a Patterson density, given a potential F : ΩΓ → R, the measure µ F

Fix x 0 ∈ ΩΓ. Given a potential (that is, a Γ-invariant Hölder map) F : ΩΓ → R, the map c F : (γ, ξ) → C F ξ (γ -1 x 0 , x 0 ) is a Hölder cocycle (see
x 0 is a Hölder quasiinvariant measure, whose associated Hölder cocycle is c F . If two potentials F and F * are cohomologous, then their associated Hölder cocycles c F and c F * are cohomologous: it is easy to check that if G : ΩΓ → R is Hölder-continuous, Γ-invariant, differentiable along every flow line, and satisfies Equation (3), then the map

U : ΛΓ → R defined by ξ → G(v x 0 , ξ ) is Hölder-continuous and satisfies c F * (γ, ξ) -c F (γ, ξ) = U (γξ) -U (ξ) for all γ ∈ Γ and ξ ∈ ΛΓ.
Let us relate the periods of a potential F to the periods of the Hölder cocycle c F . Let γ be a hyperbolic element of Γ, with translation axis A γ , translation length ℓ(γ) and attractive fixed point γ + . By the Γ-invariance and the cocycle property of C F , if p is the closest point to x 0 on A γ , we have

C F γ + (γ -1 x 0 , x 0 ) = C F γ + (γ -1 p, p).
Hence, by the definition of C F , with t → ξ t the geodesic ray from p to γ + , we have (note that there are sign differences with [START_REF] Ledrappier | Structure au bord des variétés à courbure négative[END_REF])

c F (γ, γ + ) = C F γ + (γ -1 x 0 , x 0 ) = C F γ + (γ -1 p, p) = lim t→+∞ ξt p ( F -P (F )) - t γ -1 p ( F -P (F )) = - p γ -1 p ( F -P (F )) = P (F ) ℓ(γ) -Per F (γ) . (13)
By [Led1, Théo. 1.c], two Hölder quasi-invariant measures have the same measure class if and only if their associated Hölder cocycles are cohomologous, and this holds if and only if the periods of these Hölder cocycles are the same. By Livšic's theorem (see [START_REF] Paulin | Equilibrium states in negative curvature[END_REF]Rem. 3.1], two potentials F and F * are cohomologous if and only they have the same periods. By Equation ( 13), the periods of two potentials F and F * with zero pressure are the same if and only if the periods of the associated Hölder cocycles c F and c F * are the same. Hence the map which associates to the cohomology class of a potential F the measure class of the Hölder quasi-invariant measure µ F x 0 is well-defined, and is injective. To prove that it is surjective, we start with a Hölder quasi-invariant measure µ, we consider its associated Hölder cocycle c µ , the proof of [START_REF] Ledrappier | Structure au bord des variétés à courbure négative[END_REF]Théo. 3] shows that there exists a potential F such that the Hölder cocycle c F is cohomologous to c µ , and we apply again [START_REF] Ledrappier | Structure au bord des variétés à courbure négative[END_REF]Théo. 1.c] to get that µ F

x 0 and µ have the same measure class. In order to prove (12), if F is a potential with P (F ) = 0, we have, by Equation ( 13),

log d(γ -1 ) * µ F x 0 d µ F x 0 (γ + ) = -c F (γ, γ + ) = Per F (γ) .
It would be interesting to know if one could remove the assumption that Γ is convexcocompact, up to adding the requirements on F that δ Γ, F is finite and (Γ, F ) is of divergence type, and on µ that µ is ergodic. This would improve correspondingly the last claim of Theorem 3.1 and simplify the statement of the requirement on the class of measures under consideration in Theorem 5.1. Γ 0 is a bounded parabolic group (in which case Γ 0 is also malnormal with infinite index in Γ) and C 0 is a precisely invariant closed horoball centred at the singleton ΛΓ 0 . We restrict to the above case for simplicity.

Let F 0 : Γ 0 \T 1 M → R be the map induced by F , and let δ 0 = δ Γ 0 , F 0 be the critical exponent of (Γ 0 , F 0 ). Note that -∞ < δ 0 ≤ δ < +∞ by [START_REF] Hersonsky | On the almost sure spiraling of geodesics in negatively curved manifolds[END_REF]Lem. 3.3 (iii)].

Let ψ : [0, +∞[ → [0, +∞[ be a measurable map, such that there exist c 2 , c 3 > 0 such that for every s, t ≥ c 2 , if s ≤ t + c 2 , then ψ(s) ≤ ψ(t) + c 3 . Recall (see for instance [START_REF] Hersonsky | Counting orbit points in coverings of negatively curved manifolds and Hausdorff dimension of cusp excursions[END_REF]§5]) that this condition is for instance satisfied if ψ is Hölder-continuous; it implies that e ψ is locally bounded, hence it is locally integrable; and for every α > 0, the series n∈N e α ψ(n) converges if and only if the integral +∞ 0 e α ψ(t) dt converges. Note that the constant c 2 and c 3 are unchanged by replacing ψ by ψ + c for any c ∈ R.

Fix ǫ 0 > 0. With the terminology of [START_REF] Hersonsky | On the almost sure spiraling of geodesics in negatively curved manifolds[END_REF], let E(ψ) be the set of (ǫ 0 , ψ)-Liouville vectors for (Γ, Γ 0 ) in T 1 M , that is, the set of v ∈ T 1 M such that there exist sequences

(t n ) n∈N in [0, +∞[ converging to +∞ and (γ n ) n∈N in Γ such that for every t ∈ [t n , t n + ψ(t n )], we have g v (t) ∈ γ n N ǫ 0 C 0 . Note that E(ψ) is invariant under the geodesic flow and under Γ.
If E is a set and f, g : E → ]0, +∞[ are maps, we write f ≍ g if there exists c > 0 such that 1 c f ≤ g ≤ c f . The aim of this section is to prove the following result.

Theorem 4.1 Assume that the measure m F is finite, and that there exists κ > 0 such that γ∈Γ : t≤d(x,γy)<t+κ e e ψ(t)(δ 0 -δ) dt diverges (resp. converges) then m F -almost every (resp. no) point of T 1 M belongs to E(ψ).

Remarks.

(1) If the length spectrum of Γ is nonarithmetic, then as said in Section 2, the measure m F is mixing for the geodesic flow on T 1 M , hence by [START_REF] Paulin | Equilibrium states in negative curvature[END_REF]Coro. 9.7], we have γ∈Γ : d(x, γy)≤t e γy x F ∼ c e t δ as t → +∞, for some c > 0, a stronger requirement than the first asymptotic hypothesis. Similarly, if the length spectrum of Γ 0 is nonarithmetic (this implies that Γ 0 is nonelementary), then the Gibbs measure m F 0 of (Γ 0 , F 0 ), being finite since Γ 0 is convex-cocompact, is mixing, and the second asymptotic hypothesis holds. The fact that the second asymptotic hypothesis holds when Γ 0 is elementary (that is, when C 0 is the translation axis of a loxodromic element of Γ) is given by [START_REF] Paulin | Equilibrium states in negative curvature[END_REF]Lem. 3.3 (ix)].

(2) The above theorem implies Theorem 1.2 in the introduction. Indeed, M being compact, the measure m F is finite and the length spectrum of Γ is nonarithmetic. Hence the two asymptotic hypotheses of Theorem 4.1 hold by the previous remark. Note that if C 0 is the translation axis of a loxodromic element of Γ, if D 0 is its image by M → M , then δ 0 = P (F |T 1 D 0 ) by [START_REF] Paulin | Equilibrium states in negative curvature[END_REF]Lem. 3.3 (ix)]. We have δ = P (F ) by [START_REF] Paulin | Equilibrium states in negative curvature[END_REF]Theo. 6.1]. Hence the conclusion of Theorem 4.1 does imply Theorem 1.2.

Proof of Theorem 4.1. Before starting this proof, let us give more informations on Γ 0 . Recall that C 0 is a non-compact, closed convex subset of M such that:

(1) C 0 is Γ 0 -invariant and Γ 0 \C 0 is compact; up to replacing Γ 0 by Stab Γ C 0 , in which Γ 0 has finite index and which remains almost malnormal (see the caracterisation [HP2, Prop. 2.6 (3)]), so that δ 0 and the validity of the second asymptotic hypothesis of Theorem 4.1 are unchanged, we may and we will assume that Γ 0 = Stab Γ C 0 ;

(2) by [HP2, Prop. 2.6 (2), (4)], the limit set ΛΓ 0 is precisely invariant (that is, γΛΓ 0 ∩ ΛΓ 0 = ∅ for every γ ∈ Γ -Γ 0 ), and there exists κ 0 > 0 such that for every γ ∈ Γ -Γ 0 , the diameter of

N ǫ 0 C 0 ∩ γN ǫ 0 C 0 is at most κ 0 . Lemma 4.2 If Γ 0 is a convex-cocompact subgroup of Γ, then δ 0 < δ.
Proof. Since the Gibbs measure m F 0 is finite, by [START_REF] Paulin | Equilibrium states in negative curvature[END_REF]Coro. 6.1], the probability measure

m F 0 m F 0 is an equilibrium state for the potential F 0 on Γ 0 \T 1 M , whose support is contained in the nonwandering set ΩΓ 0 = Γ 0 \ ΩΓ 0 of the geodesic flow on Γ 0 \T 1 M . Since Γ 0 is malnormal in Γ, the canonical map p : Γ 0 \T 1 M → Γ\T 1 M ,
m F 0 = m F m F .
Since Γ 0 is convex-cocompact and has infinite index in Γ, its limit set ΛΓ 0 is a precisely invariant (by the above property (2)) nonempty closed subset with empty interior in ΛΓ. Hence ΓΛΓ 0 is a proper subset of ΛΓ by Baire's theorem. Therefore the support of p * m F 0 , which is the image by

p of Γ 0 \ ΩΓ 0 = Γ 0 \{v ∈ T 1 M : v -, v + ∈ ΛΓ 0 }, is a proper subset of the support Γ\ ΩΓ = Γ\{v ∈ T 1 M : v -, v + ∈ ΛΓ} of m F , a contradiction.
We start the proof of Theorem 4.1 by two reductions of the statement.

(i) Up to adding a big enough constant to F , which does not change m F , nor δ 0δ, nor the asymptotics of the series in the above statement, we assume that δ 0 > 0. In particular, δ is finite and positive.

(ii) Let x 0 ∈ C 0 be a basepoint. Let R 0 > 0 and let U 0 = π -1 (B(x 0 , R 0 )) be the set of the unit tangent vectors in T 1 M based at a point at distance less that R 0 of x 0 . If R 0 is big enough, then m F ( U 0 ) > 0. Since m F is finite, it is ergodic under the action of the geodesic flow on T 1 M (see [START_REF] Paulin | Equilibrium states in negative curvature[END_REF]Coro. 5.15]). Hence the result is equivalent to proving that, when R 0 is big enough, if +∞ 0 e ψ(t)(δ 0 -δ) dt diverges (resp. converges) then m F -almost every (resp. no) point of U 0 belongs to E(ψ) ∩ U 0 .

We now define the various subsets of U 0 that we will study during the proof of Theorem 4.1.

Let E 0 be the set of [γ] ∈ Γ/Γ 0 such that d(x 0 , γC 0 ) ≤ R 0 + ǫ 0 . Since Γ is discrete, and since Γ 0 acts cocompactly on C 0 , only finitely many distinct images of C 0 under Γ meet a given compact subset of M . In particular, the set E 0 is finite.

Since Γ 0 \C 0 is compact, let ∆ 0 > 0 be such that the restriction to the ball B(x 0 , ∆ 0 ) of the canonical projection C 0 → Γ 0 \C 0 is onto. Choose and fix once and for all a representative γ of [γ] ∈ Γ/Γ 0 -E 0 such that if p γ is the closest point to x 0 on γC 0 , then d(p γ , γx 0 ) ≤ ∆ 0 . We will use this representative whenever a coset is considered. For every

[γ] ∈ Γ/Γ 0 -E 0 , define D γ = d(x 0 , γC 0 ) = d(x 0 , p γ ) > 0 .
Remark 4.3 Note that by an argument similar to [START_REF] Hersonsky | On the almost sure spiraling of geodesics in negatively curved manifolds[END_REF]Lem. 4.1], for every λ ∈ R, there are only finitely many F ≍ e δ 0 t .

[γ] ∈ Γ/Γ 0 -E 0 such that D γ ≤ λ.
Then there exists κ ′ ≥ 1 such that [γ]∈Γ/Γ 0 : t≤Dγ <t+κ ′ e γx 0 x 0

F ≍ e δ t .

Proof. We start by proving that there exist c 4 , c 5 > 0 such that for every

([γ], α) ∈ Γ/Γ 0 × Γ 0 , we have D γ ≤ d(x 0 , γx 0 ) ≤ D γ + ∆ 0 , (14)
d(x 0 , γx 0 ) + d(x 0 , αx 0 ) -c 4 ≤ d(x 0 , γαx 0 ) ≤ d(x 0 , γx 0 ) + d(x 0 , αx 0 ) , ( 15
) γαx 0 x 0 F - γx 0 x 0 F - αx 0 x 0 F ≤ c 5 . (16)
Equation ( 14), as well as the inequality on the right hand side of Equation ( 15), follow by the triangle inequality:

D γ = d(x 0 , γC 0 ) ≤ d(x 0 , γx 0 ) ≤ d(x 0 , p γ ) + d(p γ , γx 0 ) ≤ D γ + ∆ 0 .
By the convexity of γC 0 , the angle at p γ of the geodesic segments [p γ , x 0 ] and [p γ , γαx 0 ] (if they are non-trivial) is at least π 2 . By hyperbolicity, the point

p γ is hence at distance at most log(1+ √ 2) from a point in [x 0 , γαx 0 ]. Thus γx 0 is at distance at most ∆ 0 +log(1+ √ 2) from a point u in [x 0 , γαx 0 ]
. By the triangle inequality, the inequality on the left hand side of Equation ( 15) follows with c 4 = 2(∆ 0 + log(1 + √ 2)). Let us apply Equation (5) twice, with x = u, y = γx 0 and with either z = x 0 or z = γαx 0 . Since d(γx 0 , u) ≤ ∆ 0 + log(1 + √ 2), Equation ( 16) follows with

c 5 = 2 c 1 e ∆ 0 +log(1+ √ 2) + (∆ 0 + log(1 + √ 2)) max π -1 (B(x 0 , ∆ 0 +log(1+ √ 2))) | F | .
We are now going to use the following lemma. By the first asymptotic assumption in Lemma 4.4, there exists c > 0 such that, for every t ≥ κ, γ∈Γ, t-κ≤d(x 0 , γx 0 )<t

e γx 0 x 0 F ≥ 1 c e δ t .
We will use Lemma 4.5 by taking, for every k ∈ N,

a k = [γ]∈Γ/Γ 0 , k≤Dγ<k+1 e γx 0 x 0 F and b k = α∈Γ 0 , k-κ-c 4 ≤d(x 0 , αx 0 )<k e αx 0 x 0 F .
By Equation ( 14) and by the first asymptotic assumption in Lemma 4.4, there exists C ′ > 0 such that, for every k ∈ N,

a k ≤ γ∈Γ, k≤d(x 0 , γx 0 )<k+1+∆ 0 e γx 0 x 0 F ≤ c ′ e δ k . (17)
By the second asymptotic assumption in Lemma 4.4, there exists c ′′ > 0 such that, for every 15), we hence have

k ∈ N, b k ≤ c ′′ e δ 0 k . Let n ≥ κ + c 4 and ([γ], α) ∈ Γ/Γ 0 × Γ 0 satisfy n -κ -c 4 ≤ d(x 0 , γ ′ x 0 ) < n -c 4 where γ ′ = γα. Let k = ⌊D γ ⌋ be the integral part of D γ . By Equation (
0 ≤ k ≤ D γ ≤ d(x 0 , γx 0 ) ≤ d(x 0 , γαx 0 ) -d(x 0 , αx 0 ) + c 4 ≤ n ,
and

n -κ -c 4 -k ≤ d(x 0 , γαx 0 ) -d(x 0 , γx 0 ) ≤ d(x 0 , αx 0 ) ≤ d(x 0 , γαx 0 ) -d(x 0 , γx 0 ) + c 4 ≤ n -k .
Therefore, respectively by the definition of a k and b n-k , and by Equation ( 16), we have

n k=0 a k b n-k = n k=0 [γ]∈Γ/Γ 0 , k≤Dγ<k+1 α∈Γ 0 , n-k-κ-c 4 ≤d(x 0 , αx 0 )<n-k e γx 0 x 0 F e αx 0 x 0 F ≥ e -c 5 γ ′ ∈Γ, n-κ-c 4 ≤d(x 0 , γ ′ x 0 )<n-c 4 e γ ′ x 0 x 0 F ≥ e -c 5 1 c e δ (n-c) .
Applying Lemma 4.5 with A = max{c ′ , c ′′ , c e c 5 +δc 4 } gives the lower bound required to prove Lemma 4.4. The upper bound follows from Equation (17).

For every r > 0 and β ∈ Γ, let

A β (r) = {v ∈ U 0 : g v ([0, +∞[) ∩ B(βx 0 , r) = ∅} .
Let us fix a positive constant c 6 ≥ κ (depending only on ǫ 0 , ∆ 0 , R 0 , κ and ψ) to be made precise later on. For every k ∈ N, define I k to be the set of

[γ] ∈ Γ/Γ 0 such that k ≤ D γ < k + 1, and let J k = J k (ψ) be the set of pairs ([γ], α) ∈ Γ/Γ 0 × Γ 0 such that k ≤ D γ < k + κ ′
(where κ ′ is given by Lemma 4.4) and ψ(k) ≤ d(x 0 , αx 0 ) < ψ(k) + c 6 . For every k ∈ N, let

A k (r, ψ) = ([γ], α)∈J k A γα (r) .
These sets are related to the set E(ψ) that we want to study by the following result. Recall that if (B k ) k∈N is a sequence of subsets of a given set, one defines lim sup

k B k = n∈N k≥n B k . Proposition 4.6 If r ≥ ǫ 0 + ∆ 0 , there exist c ′ 5 , c ′′ 5 > 0 such that, up to sets of m F -measure zero, lim sup k A k (r, ψ + c ′′ 5 ) ⊂ E(ψ) ∩ U 0 ,
and if ψ(t) ≥ c ′ 5 for t big enough,

E(ψ) ∩ ( U 0 -Z) ⊂ lim sup k A k (r, ψ -c ′ 5 ) .
Proof. Let us first prove the second inclusion. Let

c 0 = ǫ 0 + 2 arsinh(coth ǫ 0 ). Let c ′ 0 = c 3 2R 0 +c 0 c 2
, with c 2 , c 3 the constants appearing in the assumption on ψ.

Let c ′ 5 = ǫ 0 + 2∆ 0 + R 0 + c 0 + c ′ 0 . Assume that ψ(t) ≥ c ′ 5 for t big enough. Let v ∈ E(ψ)∩ U 0 . For every n ∈ N, there exist sequences (t n ) n∈N in [0, +∞[ converging to +∞ and ([γ n ]) n∈N in Γ/Γ 0 such that for every t ∈ [t n , t n + ψ(t n )], we have g v (t) ∈ γ n N ǫ 0 C 0 . Let n ∈ N.
The geodesic line g v enters in γ n N ǫ 0 C 0 at a time t - n at most t n . Up to extracting a subsequence, we may assume, by Remark 4.3, that [γ n ] / ∈ E 0 , so that

D γn > R 0 + ǫ 0 and t - n > 0. Let k n = ⌊D γn ⌋. Let us prove that k n → +∞ as n → +∞, up to sets of m F -measure zero of elements v ∈ E(ψ)∩ U 0 . Otherwise, up to extracting a subsequence, (γ n ) n∈N is con- stant by Remark 4.3. Hence v + belongs to the set γ 0 ∂ ∞ C 0 of accumulation points of γ 0 C 0 in ∂ ∞ M . By Lemma 2.1, the µ F x 0 -measure of ∂ ∞ C 0 = ΛΓ 0 is zero.
Hence, since the action of Γ preserves the sets of µ F

x 0 -measure zero by the properties of the Patterson densities, we have µ F x 0 β∈Γ β∂ ∞ C 0 = 0. By the decomposition of m F in Hopf's parametrisation (see Equation ( 7)), the m F -measure of the set of v ∈ E(ψ) such that v + ∈ β∈Γ β∂ ∞ C 0 is zero. This proves the above claim.

g v (t - n ) γ n x 0 D γn γ n α n x 0 π(v) x 0 R 0 γ n N ǫ0 C 0 p γn γ n C 0 q n v g v (t n + ψ(t n )) g v (t ′ n ) g v (t n ) r
Let q n be the closest point to π(v) on γ n C 0 . It satisfies d(p γn , q n ) ≤ R 0 , since closest point maps do not increase the distances. Note that the point q n is at distance ǫ 0 from the entry point in γ n N ǫ 0 C 0 of the geodesic segment from π(v) to q n . By the penetration properties of geodesic rays in ǫ 0 -neighbourhoods of convex subsets of CAT(-1) metric spaces (see [PaP1, Lem. 2.3]), we have d(q n , g v (t - n )) ≤ c 0 = ǫ 0 + 2 arsinh(coth ǫ 0 ). Hence, by the triangle inequality,

d(γ n x 0 , g v (t - n )) ≤ d(γ n x 0 , p γn ) + d(p γn , q n ) + d(q n , g v (t - n )) ≤ ∆ 0 + R 0 + c 0 . ( 18
)
Again by the triangle inequality, we have

k n ≤ D γn = d(x 0 , p γn ) ≤ d(x 0 , π(v))+t - n +d(g v (t - n ), q n )+d(q n , p γn ) ≤ t n +2R 0 +c 0 . (19)
Up to extracting a subsequence, we may assume that ψ(k n ) ≥ c ′ 5 and that t n , k n ≥ c 2 . By the assumption on ψ and since c ′ 0 = c 3 2R 0 +c 0 c 2 , we have by Equation ( 19),

t - n + ψ(k n ) ≤ t n + ψ(t n ) + c ′ 0 . Let t ′ n = t - n + ψ(k n ) -c ′ 0 , which belongs to [t - n , t n + ψ(t n )], since ψ(k n ) ≥ c ′ 5 ≥ c ′ 0 . By convexity, the point g v (t ′ n ) belongs to N ǫ 0 C 0 . Let α n be an element of Γ 0 such that d(g v (t ′ n ), γ n α n x 0 ) ≤ ǫ 0 + ∆ 0 , (20)
which exists by the definition of ∆ 0 . By the triangle inequality, and by Equation ( 18), we have

d(γ n x 0 , γ n α n x 0 ) -d(g v (t - n ), g v (t ′ n )) ≤ d(g v (t ′ n ), γ n α n x 0 ) + d(γ n x 0 , g v (t - n )) ≤ ǫ 0 + 2∆ 0 + R 0 + c 0 . Hence d(x 0 , α n x 0 ) -ψ(k n ) = |d(γ n x 0 , γ n α n x 0 ) -|t ′ n -t - n | -c ′ 0 | ≤ ǫ 0 + 2∆ 0 + R 0 + c 0 + c ′ 0 = c ′ 5 . (21)
Define c 6 = max{2c ′ 5 , κ} (which only depends on ǫ 0 , ∆ 0 , R 0 , κ and ψ). Assume that r ≥ ǫ 0 + ∆ 0 . For every n ∈ N, we hence have v ∈ A γnαn (r) by Equation ( 20). Besides,

([γ n ], α n ) ∈ J kn (ψ -c ′ 5
) since k n = ⌊D γn ⌋ and κ ′ ≥ 1, and by Equation ( 21). Therefore v ∈ A kn (r, ψc ′ 5 ). This proves the second inclusion in Proposition 4.6.

Let us now prove the first inclusion. By hyperbolicity and an argument of (strict) convexity (see for instance [PaP1, Lem. 2.2]), there exists c ′′ 0 = c ′′ 0 (ǫ 0 , r) such that if a geodesic segment has endpoints at distance at most max{R 0 , r} + log(1 + √ 2) from two points in C 0 at distance at least c ′′ 0 one from the other, then this geodesic segment enters

N ǫ 0 C 0 . Let c ′′ 5 = max c ′′ 0 + ∆ 0 , c 3 + ∆ 0 + R 0 + 2c 0 + r + 2R 0 +c 0 +1 c 2 .
Let v ∈ U 0 and let (k n ) n∈N be a sequence in N converging to +∞. Assume that v ∈ A kn (r, ψ + c ′′ 5 ) for every n in N. Let ([γ n ], α n ) ∈ J kn (ψ + c ′′ 5 ) be such that v ∈ A γnαn (r): there exists τ n ≥ 0 such that g v (τ n ) ∈ B(γ n α n x 0 , r). Since d(π(v), x 0 ) ≤ R 0 , by the properties of closest point projections in CAT(-1)-space, there exists τ

′ n ∈ [0, τ ] such that d(g v (τ ′ n ), p γn ) ≤ max{R 0 , r} + log(1 + √ 2)
. By the definition of c ′′ 0 and since

d(p γn , γ n α n x 0 ) ≥ d(γ n x 0 , γ n α n x 0 ) -∆ 0 ≥ ψ(k n ) + c ′′ 5 -∆ 0 ≥ c ′′ 0 , the geodesic line g v enters γ n N ǫ 0 C 0 . q ′ n γ n x 0 D γn γ n α n x 0 x 0 R 0 γ n N ǫ0 C 0 p γn γ n C 0 v g v (t - n ) g v (t + n ) g v (τ n ) r
Let t - n be the entry time of g v inside γ n N ǫ 0 C 0 , which satisfies, by Equation ( 18), [START_REF] Parkkonen | Prescribing the behaviour of geodesics in negative curvature[END_REF]Lem. 2.3] and since closest point maps do not increase the distances, if q ′ n is the closest point to g v (τ n ) on C 0 , we have

d(g v (t - n ), γ n x 0 ) ≤ ∆ 0 + R 0 + c 0 . Let t + n be either τ n if g v (τ n ) ∈ γ n N ǫ 0 C 0 or the exit time of g v out of N ǫ 0 C 0 otherwise. Again by
d(g v (t + n ), γ n α n x 0 ) ≤ d(g v (t + n ), q ′ n ) + d(q ′ n , γ n α n x 0 ) ≤ c 0 + r .
As in Equation ( 19), we have

k n ≥ D γn -1 ≥ t - n -2R 0 -c 0 -1. Hence ψ(k n ) ≥ ψ(t - n ) -c 3 2R 0 + c 0 + 1 c 2 .
By the triangle inequality and since

([γ n ], α n ) ∈ J kn (ψ + c ′′ 5 ), we have t + n -t - n ≥ d(γ n x 0 , γ n α n x 0 ) -d(g v (t - n ), γ n x 0 ) -d(g v (t + n ), γ n α n x 0 ) ≥ ψ(k n ) + c ′′ 5 -∆ 0 -R 0 -2c 0 -r ≥ ψ(t - n ) ,
where c ′′ 5 is a constant depending only on ∆ 0 , R 0 , ǫ 0 , r and ψ. Hence v belongs to E(ψ), which proves the result.

In a series of lemmae and propositions, we now state the required properties of the sets A γα (r) for ([γ], α) ∈ Γ/Γ 0 × Γ 0 and A k (r, ψ) for k ∈ N.

We start by the following estimate on the mass of the A γα (r)'s. Before stating it, let us motivate it. Let d ′ be the distance on T 1 M induced by Sasaki's Riemannian metric on T M (when Γ is cocompact, any Riemannian distance on T 1 M is allowed). Recall that, for ǫ > 0 and T ≥ 0, the dynamical (ǫ, T )-ball centred at a point

v ∈ T 1 M is B ǫ, T (v) = {w ∈ T 1 M : ∀ t ∈ [0, T ], d ′ (g t w, g t v) ≤ ǫ} .
We proved in [START_REF] Paulin | Equilibrium states in negative curvature[END_REF]Prop. 3.16] (which was in fact written after the first version of this paper), using a minor modification of these dynamical balls, that Gibbs measure satisfy the Gibbs property (when Γ is torsion free and cocompact, see for instance [START_REF] Bowen | The ergodic theory of Axiom A flows[END_REF]Theo. 3.3] for the lower bound, and [START_REF] Katok | Introduction to the modern theory of dynamical systems[END_REF]Lem. 20.3.4] in the discrete time case): for every ǫ > 0, for all v ∈ T 1 M and T ≥ 0 such that v, g T (v) map to a given compact subset of Γ\T 1 M , we have

m F B ǫ, T (v) ≍ e T 0 F (g t v) dt -T P (F ) . Now, A γα (r)
is almost such a dynamical ball. Indeed, let v γα be the unit tangent vector at x 0 of the geodesic segment from x 0 to γαx 0 , and let T γα = d(x 0 , γαx 0 ) (see the figure below). Our set A γα (r) contains B ǫ -,Tγα (v γα ) and is contained in B ǫ + ,Tγα (v γα ) for some positive constants ǫ ± depending only on R 0 , r. The following result (or rather Equation ( 25)) is hence closely related to this Gibbs property.

Proposition 4.7 If r and R 0 are big enough, there exists c 7 = c 7 (r) > 0 such that for all but finitely many ([γ], α) ∈ Γ/Γ 0 × Γ 0 , we have

1 c 7 e γx 0 x 0 ( F -δ) e αx 0 x 0 ( F -δ) ≤ m F A γα (r) ≤ c 7 e γx 0 x 0 ( F -δ) e αx 0 x 0 ( F -δ) . x 0 v R 0 v γα r γαx 0 γx 0 ≥ π/2 p γ Proof. For every R 0 > 0 and β ∈ Γ, define B R 0 , β = z ′ ∈B(βx 0 , r) O z ′ B(x 0 , R 0
3 ), which is contained in (and is a pertubation of) the shadow O βx 0 B(x 0 , R 0

3 ) (see the picture below). Since Γ is nonelementary, the support of the Patterson measures is not reduced to one point,

hence m = inf ξ∈∂∞ M µ F •ι x 0 -µ F •ι
x 0 ({ξ}) is positive. By hyperbolicity (as first remarked by Sullivan), for every ξ ∈ ∂ ∞ M , the family c O ξ B(x 0 , R) R>0 is a fundamental system of neighbourhoods of ξ. By compacteness and discreteness, there exists hence R 0 > 0 such that for all but finitely many β ∈ Γ, we have

µ F •ι x 0 B R 0 , β ≥ m 2 . ( 22
)
By the definition of A β (r), the set of points v + for v in A β (r) is exactly

A β (r) + = z∈B(x 0 , R 0 ) O z B(βx 0 , r) ,
which is a bit larger than the shadow O x 0 B(βx 0 , r) (see the picture below). By a minor modification of Mohsen's shadow lemma (see Equation ( 8)), we have, if r is big enough,

µ F x 0 (A β (r) + ) ≍ e βx 0 x 0 ( F -δ) . ( 23
) R0 3 x 0 ≈ A β (r) + ≈ r B R0, β O βx0 B(x 0 , R 0 /3) βx 0 O x0 B(βx 0 , r)
Let us first prove, using Hopf's parametrisation defined by the point x 0 and the definition of A β (r), that for all but finitely many β ∈ Γ, we have

B R 0 , β × A β (r/2) + × [-R 0 /3, R 0 /3] ⊂ A β (r) ⊂ ∂ ∞ M × A β (r) + × [-R 0 , R 0 ] . ( 24
)
The inclusion on the right hand is immediate. To prove the other one, let v ∈ T 1 M be such that if p is the closest point to x 0 on the geodesic ray

g v ([0, +∞[), then v -∈ B R 0 , β , v + ∈ A β (r/2) + and d(π(v), p) ≤ R 0 /3 (see the picture below). Let us prove that v ∈ A β (r). z r R 0 z ′ v - R0 3 x 0 z ′′ p v + r 2 βx 0 v
By the definition of A β (r/2) + , let z ∈ B(x 0 , R 0 ) be such that the geodesic ray [z, v + [ meets B(βx 0 , r/2) at a point z ′ . By the definition of B R 0 , β , the geodesic ray [z ′ , v -[ meets B(x 0 , R 0 /3) at a point z ′′ . Since d(z, z ′′ ) ≤ 2R 0 , and since by discreteness, for all but finitely many β in Γ, the distance d(z, z ′ ) is big, the angle at z ′ between the geodesic segments [z ′ , z] and [z ′ , z ′′ ] is small, hence the angle at z ′ between the geodesic rays [z ′ , v -[ and [z ′ , v + [ is close to π. Hence the geodesic line g v between v -and v + is close to the union of these two rays. In particular, since g v passes close to z ′ ∈ B(βx 0 , r/2), it enters the ball B(γx 0 , r), and since it passes close to z ′′ ∈ B(x 0 , R 0 3 ), the point p belongs to B(x 0 , R 0 2 ) and hence π(v) belongs to B(x 0 , R 0 ), which proves the result. Now, for every v ∈ A β (r), since d(π(v), x 0 ) ≤ R 0 , the point x 0 is at distance at most R 0 from a point on the geodesic line between the endpoints v -, v + . Hence by Equation ( 6), there exists c ′ 6 ≥ 0 (depending only on R 0 , on max U 0 | F | < +∞, on the Hölder constants of F and on the bounds of the sectional curvature of M ) such that, for every v ∈ A β (r), we have

-c ′ 6 ≤ C F •ι v -(x 0 , π(v)), C F v + (x 0 , π(v)) ≤ c ′ 6 .
Therefore, by definition of the Gibbs measure m F , we have, using Equations ( 22) and ( 24),

e -2 c ′ 6 m 2 µ F x 0 (A β (r/2) + ) (2R 0 /3) ≤ m F A β (r) ≤ e 2 c ′ 6 µ F •ι x 0 µ F x 0 (A β (r) + ) (2R 0 ) .
Hence, by Equation ( 23), for some constant c ′′ 6 ≥ 1, we have

1 c ′′ 6 e βx 0 x 0 ( F -δ) ≤ m F A β (r) ≤ c ′′ 6 e βx 0 x 0 ( F -δ) . (25)
For all [γ] ∈ Γ/Γ 0 and α ∈ Γ 0 , by Equations ( 15) and ( 16), if β = γα, we have

βx 0 x 0 ( F -δ) - γx 0 x 0 ( F -δ) - αx 0 x 0 ( F -δ) ≤ c 5 + δ c 4 .
The result follows.

Note, as it will be important later on, that the contributions of γ and of α are decoupled in this Proposition 4.7.

The sets A γα (r) satisfy the following almost disjointness property in shells.

Lemma 4.8 For every r > 0, there exists c 8 = c 8 (r) > 0 such that for every k ∈ N, for every subset

P of J k = J k (ψ), 1 c 8 ([γ], α)∈P m F A γα (r) ≤ m F ([γ], α)∈P A γα (r) ≤ ([γ], α)∈P m F A γα (r) ,
and for every subset

Q of I k 1 c 8 [γ]∈Q m F A γ (r) ≤ m F [γ]∈Q A γ (r) ≤ [γ]∈Q m F A γ (r) .
Proof. The inequality on the right hand side of the first claim is immediate. In order to obtain the one on the left hand side, let us prove that there exists c 8 ∈ N -{0} such that for all k ∈ N and v ∈ T 1 M , the number of

([γ], α) ∈ J k such that v ∈ A γα (r) is at most c 8 , which implies the result. Let ([γ], α), ([γ ′ ], α ′ ) ∈ J k such that v ∈ A γα (r) ∩ A γ ′ α ′ (r)
. By Equation ( 14) and ( 15), and by the definition of J k , we have

k + ψ(k) -c 4 ≤ D γ + ψ(k) -c 4 ≤ d(x 0 , γx 0 ) + d(x 0 , αx 0 ) -c 4 ≤ d(x 0 , γαx 0 ) and d(x 0 , γαx 0 ) ≤ d(x 0 , γx 0 ) + d(x 0 , αx 0 ) ≤ D γ + ∆ 0 + ψ(k) + c 6 ≤ k + κ ′ + ∆ 0 + ψ(k) + c 6 . Similarly k + ψ(k) -c 4 ≤ d(x 0 , γ ′ α ′ x 0 ) ≤ k + ψ(k) + κ ′ + ∆ 0 + c 6 . p ≤ r ≤ r x 0 π(v) ≤ R 0 γ ′ α ′ x 0 p ′ γαx 0 v
Let p and p ′ be the closest points on g v ([0, +∞[) to γαx 0 and γ ′ α ′ x 0 respectively. They satisfy d(p, γαx 0 ),

d(p ′ , γ ′ α ′ x 0 ) ≤ r since v ∈ A γα (r) ∩ A γ ′ α ′ (r).
We may assume, up to permuting γα and γ ′ α ′ , that p ′ belongs to the geodesic segment [π(v), p]. Since closest point maps do not increase the distances, by the triangle inequality, and since v ∈ U 0 , we have

d(p, p ′ ) = d(p, π(v)) -d(π(v), p ′ ) ≤ d(γαx 0 , π(v)) -d(π(v), p ′ ) ≤ d(γαx 0 , x 0 ) + d(x 0 , π(v)) -d(γ ′ α ′ x 0 , x 0 ) + d(π(v), x 0 ) + d(p ′ , γ ′ α ′ x 0 ) ≤ (k + ψ(k) + κ ′ + ∆ 0 + c 6 ) + R 0 -(k + ψ(k) -c 4 ) + R 0 + r .
Hence, again by the triangle inequality,

d(γαx 0 , γ ′ α ′ x 0 ) ≤ κ ′ + ∆ 0 + c 6 + 2R 0 + c 4 + 3r .
Now the first claim follows from the discreteness of Γ, which implies that there are only finitely many elements β in Γ such that βx 0 belongs to a ball of centre x 0 with given radius.

The second claim is proven similarly.

The two results above allow to estimate the mass of the A k (r, ψ)'s, as follows.

Proposition 4.9 Assume that there exists κ > 0 such that γ∈Γ : t≤d(x,γy)<t+κ e γy x F ≍ e δ t and α∈Γ 0 : t≤d(x, αy)<t+κ e αy x F ≍ e δ 0 t .

If r is big enough, there exists c 9 > 0 such that, for every k ∈ N, we have

1 c 9 e ψ(k)(δ 0 -δ) ≤ m F A k (r, ψ) ≤ c 9 e ψ(k)(δ 0 -δ) .
It follows from this proposition and the assumption on the function ψ that the series

k∈N m F A k (r, ψ) converges if and only if the integral +∞ 0 e ψ(t)(δ 0 -δ) dt converges.
Proof. By Equation ( 14) and by the first asymptotic assumption in the statement of Proposition 4.9, there exists c > 0 such that for every k ∈ N,

[γ]∈Γ/Γ 0 , k≤Dγ <k+κ ′ e γx 0 x 0 F ≤ γ∈Γ, k≤d(x 0 , γx 0 )<k+κ ′ +∆ 0 e γx 0 x 0 F ≤ c e δ k . (26)
By the second asymptotic assumption in Proposition 4.9, since c 6 ≥ κ, there exists c ′ > 0 such that, for every t ∈ [0, +∞[ ,

1 c ′ e δ 0 t ≤ α∈Γ 0 , t≤d(x 0 , γx 0 )<t+c 6 e αx 0 x 0 F ≤ c ′ e δ 0 t . (27)
Let us first prove the inequality in the right hand side in Proposition 4.9. Let r be big enough and k ∈ N. Respectively by Lemma 4.8 with P = J k and the definition of A k (r, ψ), by Proposition 4.7, by the definition of J k , by Equations ( 26) and ( 27), we have

m F A k (r, ψ) ≤ ([γ], α)∈J k m F A γα (r) ≤ c 7 ([γ], α)∈J k e γx 0 x 0 ( F -δ) e αx 0 x 0 ( F -δ) = c 7 [γ]∈Γ/Γ 0 , k≤Dγ<k+κ ′ e γx 0 x 0 ( F -δ) α∈Γ 0 , ψ(k)≤d(x 0 , αx 0 )<ψ(k)+c 6 e αx 0 x 0 ( F -δ) ≤ c 7 (c e δ k ) e -δ k (c ′ e δ 0 ψ(k) ) e -δ ψ(k) = c 7 c c ′ e ψ(k)(δ 0 -δ) .
This proves the inequality on the right hand side in Proposition 4.9.

Let us now prove similarly the inequality on the left hand side in Proposition 4.9. By Lemma 4.4, there exists c ′′ > 0 such that, for every t ∈ [0, +∞[ ,

[γ]∈Γ/Γ 0 : t≤Dγ <t+κ ′ e γx 0 x 0 F ≥ 1 c ′′ e δ t . (28)
Respectively by Lemma 4.8 with P = J k and the definition of A k (r, ψ), by Proposition 4.7, by the definition of J k , by Equations ( 28), ( 14) and ( 27), we have

m F A k (r, ψ) ≥ 1 c 8 ([γ], α)∈J k m F A γα (r) ≥ 1 c 7 c 8 ([γ], α)∈J k e γx 0 x 0 ( F -δ) e αx 0 x 0 ( F -δ) = 1 c 7 c 8 [γ]∈Γ/Γ 0 , k≤Dγ<k+κ ′ e γx 0 x 0 ( F -δ) α∈Γ 0 , ψ(k)≤d(x 0 , αx 0 )<ψ(k)+c 6 e αx 0 x 0 ( F -δ) ≥ 1 c 7 c 8 ( 1 c ′′ e δ k ) e -δ (k+κ ′ +∆ 0 ) ( 1 c ′ e δ 0 ψ(k) ) e -δ(ψ(k)+c 6 ) = 1 c 7 c 8 c ′ c ′′ e δ(κ ′ +∆ 0 +c 6 ) e ψ(k)(δ 0 -δ) .
This proves Proposition 4.9.

The following result is a quasi-independence property of the sets A k (r, ψ) for k ∈ N.

Proposition 4.10 Under the hypotheses of Proposition 4.9, there exists a constant c 10 > 0 such that for every k = k ′ in N, if ψ ≥ c 10 , we have

m F A k (r, ψ) ∩ A k ′ (r, ψ) ≤ c 10 m F A k (r, ψ) m F A k ′ (r, ψ) .
Proof. The proof has two parts, a geometric one and a measure-theoretic one. We state the geometric one as a lemma.

Lemma 4.11 There exist c ′ 10 > 0 and r ′ > r such that for every k < k ′ in N, for every

([γ], α) ∈ J k and ([γ ′ ], α ′ ) ∈ J k ′ , if ψ ≥ c ′ 10 and if A γα (r) meets A γ ′ α ′ (r), then A γ ′ (r) is contained in A γα (r ′ ). Proof. Let k < k ′ in N and ([γ], α) ∈ J k and ([γ ′ ], α ′ ) ∈ J k ′ . If A γα (r) ∩ A γ ′ α ′ (
r) is non empty, there exists v ∈ U 0 such that g v (R) meets B(γαx 0 , r) and B(γ ′ α ′ x 0 , r). Let q, q ′ be the closest points to π(v) on the convex sets γC 0 , γ ′ C 0 . Let p, p ′ be the closest points to q, q ′ on the geodesic ray g v ([0, +∞[). Let x, x ′ be the closest points to γαx 0 and γ

′ α ′ x 0 on g v ([0, +∞[) x γαx 0 r y ′ r x ′ γ ′ α ′ x 0 R 0 π(v) y v p q ′ γ ′ x 0 p ′ γC 0 γ ′ C 0 q p γ γx 0 x 0
We have d(γαx 0 , x) ≤ r and d(γ ′ α ′ x 0 , x ′ ) ≤ r. By the properties of geodesic triangles in CAT(-1)-spaces and by the convexity of C 0 , we have d(p, q), d(p ′ , q ′ ) ≤ r + log(1 + √ 2). By the choice of the representatives of elements in Γ/Γ 0 , the closest point p γ to x 0 on γC 0 is at distance at most ∆ 0 from γx 0 . Hence, since closest point maps do not increase the distances and by the triangle inequality,

d(p, γx 0 ) ≤ d(p, q) + d(q, p γ ) + d(p γ , γx 0 ) ≤ r + log(1 + √ 2) + R 0 + ∆ 0 . (29)
Hence, by Equation ( 14) and the definition of J k , with c = κ ′ +2R 0 +2∆ 0 +r +log(1+ √ 2), we have and d(π(v), p) ≥ kc by the inverse triangle inequality. Similarly,

d(π(v), p) ≤ d(π(v), x 0 ) + d(x 0 , γx 0 ) + d(γx 0 , p) ≤ R 0 + (D γ + ∆ 0 ) + d(γx 0 , p) ≤ k + c ,
k ′ -c ≤ d(π(v), p ′ ) ≤ k ′ + c . By similar arguments, if c ′ = R 0 + ∆ 0 + 2r + log(1 + √ 2) + c 6 , we have ψ(k) -c ′ ≤ d(p, x) ≤ ψ(k) + c ′ and ψ(k ′ ) -c ′ ≤ d(p ′ , x ′ ) ≤ ψ(k ′ ) + c ′ . (30)
Assume first that π(v), x, p ′ are in this order on g v ([0, +∞[). Any geodesic ray, with origin at distance at most R 0 from x 0 and passing at distance at most r from γ ′ x 0 , passes at distance at most 2r +log(1+ √ 2)+R 0 +∆ 0 from p ′ by the analog for γ ′ of Equation ( 29), hence by convexity passes at distance at most c ′′ = max{2R 0 , 2r + log(1 + √ 2) + R 0 + ∆ 0 } from x, thus passes at distance at most c ′′ + r from γαx 0 . Therefore, if

r ′ ≥ c ′′ + r > r, then A γ ′ (r) is contained in A γα (r ′ ).
Assume now that π(v), p ′ , x are in this order on g v ([0, +∞[) (see the picture above). There exists a constant c ′ 10 > 0 (depending only the hyperbolicity constant log(1 + √ 2 and on r) such that if ψ ≥ c ′ 10 , then π(v), p, x and π(v), p ′ , x ′ are in this order on g v ([0, +∞[). Since k ′ ≥ k, either π(v), p, p ′ are in this order on g v ([0, +∞[), or p ′ ∈ [π(v), p] is at distance at most 2c from p, since then

d(p, p ′ ) = d(p, x) -d(p ′ , x) ≤ (k + c) -(k ′ -c) ≤ 2c .
In both cases, by convexity, p ′ is at distance at most 2c + r + log(1 + √ 2) from a point y in γC 0 . Similarly, by Equation ( 30) and since ψ satisfies ψ(s) ≤ ψ(t)

+ c 3 if s ≤ t, either π(v), x, x ′ are in this order on g v ([0, +∞[), or x ′ ∈ [π(v), x] is at distance at most 2c + 2c ′ + c 3 from x. In both cases, x ′ is at distance at most 2c + 2c ′ + c 3 + r + log(1 + √ 2) from a point y ′ in γ ′ C 0 .
If for a contradiction d(p ′ , x) > R for arbitrarily large constants R, then the geodesic segments [y, γαx 0 ] and [q ′ , y ′ ], have endpoints at bounded distance from the long geodesic segment [p ′ , x]. Hence they have their endpoints at bounded distance while being long, if R is large. By hyperbolicity, this implies that N ǫ 0 (γC 0 ) ∩ N ǫ 0 (γ ′ C 0 ) contains a long segment if R is large. Taking R large enough, this contradicts the fact that the diameter of this intersection, since γ = γ in Γ/Γ 0 , is at most the constant κ 0 , as explained in the beginning of the proof of Theorem 4.1.

Therefore d(p ′ , x) ≤ R for some R ≥ 0. Any geodesic ray, with origin at distance at most R 0 from x 0 and passing at distance at most r from γ ′ x 0 , passes at distance from γαx 0 at most

r + d(γ ′ x 0 , γαx 0 ) ≤ r + d(γ ′ x 0 , p ′ ) + d(p ′ , x) + d(x, γαx 0 ) ≤ R + 3r + log(1 + √ 2) + R 0 + ∆ 0 ,
by the analog for γ ′ of Equation ( 29). Therefore, if For every

r ′ ≥ R+3r +log(1+ √ 2)+R 0 +∆ 0 > r, then A γ ′ (r) is contained in A γα (r ′ ).
([γ], α) ∈ J k , let I [γ], α, k ′ ⊂ I k ′ be the set of [γ ′ ] ∈ Γ/Γ 0 such that there exists α ′ ∈ Γ 0 with ([γ ′ ], α ′ ) ∈ J k ′ such that the intersection A γα (r) ∩ A γ ′ α ′ (
[γ ′ ]∈I [γ],α,k ′ e γ ′ x 0 x 0 ( F -δ) ≤ [γ ′ ]∈I [γ],α,k ′ c 7 (r ′ ) m F A γ ′ (r ′ ) ≤ c 7 (r ′ ) c 8 (r ′ ) m F [γ ′ ]∈I [γ],α,k ′ A γ ′ (r ′ ) ≤ c 7 (r ′ ) c 8 (r ′ ) m F A γα (r ′ ) ≤ c 7 (r) c 7 (r ′ ) 2 c 8 (r ′ ) m F A γα (r) . ( 31
)
By the assumptions of Proposition 4.10, there exists c > 0 such that for every t ∈ R,

α ′ ∈Γ 0 : d(x 0 ,α ′ x 0 )<t e α ′ x 0 x 0 F ≤ c e t δ 0 . ( 32
)
To simplify the notation, let

A k = A k (r, ψ).
Respectively by the definition of A k , by Proposition 4.7, by Equation ( 32), by Equation ( 31) with c ′ = c c 7 (r) 2 c 7 (r ′ ) 2 c 8 (r ′ ) e c 6 δ 0 , and by Proposition 4.9 and Lemma 4.8 with P = J k , we have

m F (A k ∩ A k ′ ) ≤ ([γ], α)∈J k ([γ ′ ], α ′ )∈J k ′ : Aγα(r)∩A γ ′ α ′ (r) =∅ m F A γ ′ α ′ (r) ≤ ([γ], α)∈J k [γ ′ ]∈I [γ], α, k ′ c 7 (r) e γ ′ x 0 x 0 ( F -δ) α ′ ∈ Γ 0 ψ(k ′ ) ≤ d(x 0 , α ′ x 0 ) < ψ(k ′ ) + c 6 e α ′ x 0 x 0 ( F -δ) ≤ c c 7 (r) e (ψ(k ′ )+c 6 )δ 0 -δψ(k ′ ) ([γ], α)∈J k [γ ′ ]∈I [γ], α, k ′ e γ ′ x 0 x 0 ( F -δ) ≤ c ′ e ψ(k ′ )(δ 0 -δ) ([γ], α)∈J k m F A γα (r) ≤ c ′ c 9 c 8 (r) m F (A k ′ ) m F (A k ) .
This proves Proposition 4.10.

Let us now conclude the proof of Theorem 4.1. The following version of the Borel-Cantelli Lemma is well-known (see for instance [Spr]). Proposition 4.12 Let (Z, ν) be a measured space with finite nonzero measure. Let (A n ) n∈N be a sequence of measurable subsets of Z such that there exists c > 0 with ν

(A n ∩ A m ) ≤ c ν(A n ) ν(A m ) for all distinct n, m in N. Then ν(lim sup n A n ) > 0 if and only if the series n∈N ν(A n ) diverges.
We apply this result with (Z, ν) = ( U 0 , m | U 0 ), which satisfies the hypothesis if R 0 is big enough as in the reductions at the beginnning of the proof of Theorem 4.1. Let r = ǫ 0 + ∆ 0 and let c ′ 5 , c ′′ 5 be given by Proposition 4.6. Assume first that the integral +∞ 0 e ψ(t)(δ 0 -δ) dt diverges, which is still true if a constant is added to ψ. The quasi-independence assumption of Proposition 4.12 is satisfied if A n = A n (r, ψ + c 10 + c ′′ 5 ) ⊂ U 0 , by Proposition 4.10. As claimed after the statement of Proposition 4.9, the series k∈N m F (A k ) diverges. Hence by the above Borel-Cantelli argument, lim sup k A k has positive measure. Since A n (r, ψ + c 10 + c ′′ 5 ) ⊂ A n (r, ψ + c ′′ 5 ) and by the first claim of Proposition 4.6, the set E(ψ) has positive m F -measure. Since it is invariant under the geodesic flow and under Γ, and by ergodicity of the Gibbs measure m F , it has full measure.

Conversely, assume that the integral +∞ 0 e ψ(t)(δ 0 -δ) dt converges, which is still true if a constant is substracted from ψ. Then ψ(t) ≥ c ′ 5 for t big enough. Let

A n = A n (r, ψ - c ′ 5 ) ⊂ U 0 .
Again by the assertion following the statement of Proposition 4.9, the series k∈N m F (A k ) converges. By the standard Borel-Cantelli Lemma, lim sup k A k (r, ψc ′ 5 ) has zero m F -measure. By the second claim of Proposition 4.6, the set E(ψ) ∩ U 0 has zero F -measure. Up to taking R 0 big enough, this implies that E(ψ) has zero F -measure.

Remark. Let us comment on the range of the numerical constant δδ 0 , crucial for the dichotomy in Theorem 4.1, as the potential F varies. We only consider the case when C 0 is the translation axis of a loxodromic element of Γ, so that by Remark (2) following the statement of Theorem 4.1, we have, with C 0 the image of (2) The set of real numbers P (F ) -P (F |T 1 C 0 ), as F varies in the set of Γ-invariant bounded Hölder functions on T 1 M , is equal to ]0, +∞[ .

C 0 in M = Γ\ Γ, δ -δ 0 = P (F ) -P (F |T 1 C 0 ) .
Proof. For the first observation, the 1-Lipschitz dependence of P (F |T 1 C 0 ) on F is immediate by Equation ( 2), and so it suffices to show the same for P (F ). This is a direct consequence of our definition of the pressure in Equation (1). More precisely, given F 1 , F 2 two bounded Γ-invariant Hölder-continuous functions on T 1 M , for every ǫ > 0, we can choose

m 1 , m 2 ∈ M satisfying h(m 1 ) + F 1 dm 1 ≥ P (F 1 ) -ǫ and h(m 2 ) + F 2 dm 2 ≥ P (F 2 ) -ǫ .
Using the definition of pressure again, we have that

P (F 1 ) ≥ h(m 2 ) + F 1 dm 2 and P (F 2 ) ≥ h(m 1 ) + F 2 dm 1 .
Comparing these four inequalities gives that

(F 1 -F 2 ) dm 1 ≥ P (F 1 ) -P (F 2 ) -ǫ and (F 2 -F 1 ) dm 2 ≥ P (F 2 ) -P (F 1 ) -ǫ , from which we deduce |P (F 1 ) -P (F 2 )| ≤ F 1 -F 2 ∞ + ǫ. Letting ǫ → 0, this proves that F → P (F ) is 1-Lipschitz.
For the second observation, first note that P (F ) -P (F |T 1 C 0 ) = δδ 0 is positive by Lemma 4.2. It now suffices to find two potentials F, F ′ for which P (F ) -P (F |T 1 C 0 ) can be arbitrarily large and P (F ′ ) -P (F ′ |T 1 C 0 ) can be arbitrarily close to 0. Given any L > 0 and a second distinct closed geodesic C 1 (which exists since Γ is nonelementary), we can choose a bounded potential F on T 1 M which is constant with values L and 0 on T 1 C 1 and T 1 C 0 , respectively. If m 1 denotes a probability measure supported on T 1 C 1 and invariant under the geodesic flow, then by the definition of the pressure, we have that P

(F |T 1 C 0 ) = 0 and P (F ) ≥ h m C 1 (g 1 ) + F dm C 1 = L, as required.
Finally, given any η > 0, we want to construct a bounded potential

F ′ on T 1 M satis- fying P (F ′ ) -P (F ′ |T 1 C 0 ) < η. For every ǫ > 0, we let A 0 = {v ∈ T 1 M : d(v, T 1 C 0 ) < ǫ}. We choose ǫ ∈ ]0, 1 e [ small enough, so that -(1 -ǫ) log(1 -ǫ) -2 ǫ log ǫ + 4 ǫ log 2 < η.
We choose K > h top (g 1 )/ǫ, and we define a bounded potential

F ′ on T 1 M by F ′ (v) = -K min{d(v, T 1 C 0 ), 1} ≤ 0. Given any m ∈ M, we can consider two cases: Either (a) m(T 1 M -A 0 ) > ǫ or (b) m(A 0 ) ≥ 1 -ǫ. In case (a), we have that h(m) + F ′ dm ≤ h top (g 1 ) + max v∈T 1 M -A 0 F ′ (v) ≤ h top (g 1 ) -Kǫ < 0 .
In case (b), we can choose a measurable partition α = {A n } n∈N of T 1 M , such that:

• α is generating, that is, the Borel σ-algebra is the smallest σ-algebra containing g

t 1 A i 1 ∩ • • • ∩ g t k A i k , for all k, i 1 , • • • , i k ∈ N and t 1 , • • • , t k ∈ R; • for n ≥ 1, we have m(A n ) ≤ ǫ/2 n-1 (note that m +∞ n=1 A n = 1 -m(A 0 ) ≤ ǫ).
If M were compact, then a sufficient condition for the partition to be generating would be that each element A n , for n ≥ 1, has diameter smaller than the injectivity radius of M . (At the level of the geodesic flow, this is related to choosing the diameter smaller than the expansivity constant). More generally, we can assume that each A n is the union of suitably separated components, each of which has diameter smaller than the injectivity radius of points in that component. In particular, with H m (α) the entropy of the partition α with respect to m, we can then bound

h(m) + F ′ dµ ≤ h(m) ≤ H m (α) ≤ -m(A 0 ) log m(A 0 ) - +∞ n=1 m(A n ) log m(A n ) ≤ -(1 -ǫ) log(1 -ǫ) - +∞ n=1 ǫ 2 n-1 log ǫ 2 n-1 = -(1 -ǫ) log(1 -ǫ) -2 ǫ log ǫ + 4 ǫ log 2 < η .
In either case, we have that h(m) + F ′ dm < η and from the definition, P (F ′ ) -P (F ′ |T 1 C 0 ) = P (F ′ ) < η, as required.

Let us now give the main corollary from Theorem 4.1, our logarithm law for Gibbs measures.

Define the penetration map p :

T 1 M × R → [0, +∞] of the geodesic lines inside ΓN ǫ 0 C 0 by p(v, t) = 0 if π(φ t v) /
∈ ΓN ǫ 0 C 0 , and otherwise p(v, t) is the maximal length of an interval I in R containing t such that there exists γ ∈ Γ with π(φ s v) ∈ γN ǫ 0 C 0 for every s ∈ I. The next result implies Corollary 1.3 using Remark (2) following Theorem 4.1.

Corollary 4.14 Under the assumptions of Theorem 4.1, for m F -almost every v ∈ T 1 M , we have

lim sup t→+∞ p(v, t) log t = 1 δ -δ 0 .
Proof. The proof is a standard deduction from Theorem 4.1 using the Lipschitz functions ψ n : t → κ log(1 + t) for κ = 1 δ-δ 0 ± 1 n , see for instance the proof of [START_REF] Hersonsky | On the almost sure spiraling of geodesics in negatively curved manifolds[END_REF]Theo. 5.6].

We end this section by giving a corollary of Theorem 4.1 in the special case when M has constant sectional curvature, in a form which is suitable for arithmetic applications in the next section. We will use the upper halfspace model of the real hyperbolic nspace H n R , whose boundary at infinity is ∂ ∞ H n R = R n-1 ∪ {∞}, and we endow R n-1 with the usual Euclidean norm • and its associated distance. We denote by x 0 the point (0, 1) ∈ R n-1 × ]0, +∞[. If α is a fixed point of a hyperbolic element γ of a given discrete group of isometries of H n R , we denote by α σ its other fixed point, which does not depend on γ.

Corollary 4.15 Let Γ be a nonelementary discrete group of isometries of H n R . Let F : T 1 H n R → R be a Γ-invariant Hölder-continuous map, with δ = δ Γ, F and m F finite. Let γ 0 be a hyperbolic element of Γ, let Γ 0 be the stabiliser in Γ of its translation axis, let F 0 : Γ 0 \ M → R be the map induced by F , and let δ

0 = δ Γ 0 , F 0 . Let R γ 0 be the set of fixed points in R n-1 ∪ {∞} of the conjugates in Γ of γ 0 . Let φ : ]0, 1] → ]0, 1] be a measurable map, such that there exist c ′ 2 , c ′ 3 ∈ ]0, 1[ such that for every s, t ∈ ]0, c ′ 2 ], if s ≥ c ′ 2 t, then φ(s) ≥ c ′ 3 φ(t).
If 1 0 φ δ-δ 0 (s)/s ds diverges (respectively converges), then µ F x 0 -almost every (respectively no) point in R n-1 belongs to infinitely many Euclidean balls of centre α and radius αα σ φ( αα σ ), as α ranges over R γ 0 .

Proof. Recall that the hyperbolic distance between the horizontal horosphere at Euclidean height 1 in H n R and a disjoint geodesic line with endpoints x and y islog x-y 2 , by a standard hyperbolic distance computation. By the triangle inequality and the discreteness of Γ, for every compact subset K of R n-1 , there exists c > 0 such that for every α ∈ R γ 0 ∩K except finitely many of them, we have αα σ ≤ 1 and, with C α the geodesic line with endpoints α, α σ ,

d(x 0 , C α ) -log α -α σ 2 ≤ c . (33)
Let ψ : t →log φ(e -t ) which is a map from [0, +∞[ to [0, +∞[ satisfying the assumption of the beginning of Section 4 (with c 2 =log c ′ 2 > 0 and c 3 =log c ′ 3 > 0). As in [START_REF] Hersonsky | On the almost sure spiraling of geodesics in negatively curved manifolds[END_REF]Lem. 5.2] (and since the Hamenstädt distance on ∂ ∞ H n R -{∞} = R n-1 is a multiple of the Euclidean distance), there exists a constant c ′ ≥ 1 such that for every

v ∈ T 1 M such that v + ∈ K -(R γ 0 ∩ K), we have • if v if (ǫ 0 , ψ)-Liouville for (Γ, Γ 0
), then v + belongs to infinitely many balls of centre α and radius c ′ e -d(x 0 , Cα)-ψ(d(x 0 , Cα)) , as α ranges over R γ 0 .

• if v + belongs to infinitely many balls of centre α and radius 1 c ′ e -d(x 0 , Cα)-ψ(d(x 0 , Cα)) , as α ranges over R γ 0 , then v if (ǫ 0 , ψ)-Liouville for (Γ, Γ 0 ).

By Equation ( 33), there exists c ′′ ≥ 1 such that, for every α ∈ R γ 0 ∩ K,

1 c ′′ α -α σ φ( α -α σ ) ≤ e -d(x 0 , Cα)-ψ(d(x 0 , Cα)) ≤ c ′′ α -α σ φ( α -α σ ) .

Since

+∞ 0 e ψ(t)(δ 0 -δ) dt = 1 0 φ δ-δ 0 (s)/s ds, the result follows from Theorem 4.1, whose hypotheses on sum asymptotics are satisfied by the first remark following its statement (since the curvature of M is constant).

Remark. As in [START_REF] Parkkonen | Spiraling spectra of geodesic lines in negatively curved manifolds[END_REF], replacing H n R by the Siegel domain model of the complex hyperbolic space H n C , replacing R n-1 endowed with the Euclidean distance xy by the Heisenberg group endowed with the Cygan distance d Cyg (x, y), the same result holds.

Arithmetic applications

Let K be either the field Q or an imaginary quadratic extension of Q, and correspondingly, let K be either R or C. Let O K be the ring of integers of K. By quadratic irrational, we mean an element in K which is quadratic irrational over K. For every quadratic irrational α ∈ K, let α σ be its Galois conjugate over K.

The group PSL 2 ( K) acts on P 1 (K) = K ∪ {∞} by homographies, and its subgroup PSL 2 (O K ) preserves the set K and the set of quadratic irrationals. Though it acts transitively on the former set, it does not act transitively on the latter one. Note that, for every quadratic irrational α and every γ ∈ PSL 2 (O K ), we have (γ • α) σ = γ • (α σ ).

Let us fix a finite index subgroup Γ of PSL 2 (O K ), for instance a congruence subgroup. We are interested in the approximation of elements of K by elements in the orbit under Γ of a fixed quadratic irrational and of its Galois conjugate.

For every quadratic irrational α ∈ K, let E α, Γ be the (countable, dense in K) set Γ • {α, α σ }, endowed with its Fréchet filter, and let h(α) = 2 |αα σ | .

We refer to [PaP2, §6.1] and [PaP3, §4.1] for motivations on this complexity h(α) of a quadratic irrational α, as well as for other algebraic expressions and comparisons to other algebraic heights. For instance, if K = Q, Γ = PSL 2 (Z) and α is the Golden Ratio 1+

√ 5 2 , then E α, Γ is the set of real numbers whose continued fraction expansion ends with an infinite string of 1's.

Recall that a map f : [0, +∞[ → ]0, +∞[ is slowly varying if it is measurable and if there exist constants B > 0 and A ≥ 1 such that for every x, y in R + , if |x -y| ≤ B, then f (y) ≤ A f (x). Recall that this implies that f is locally bounded, hence it is locally integrable; also, if log f is Lipschitz, then f is slowly varying.

Theorem 5.1 Let α 0 ∈ K be a fixed quadratic irrrational and let γ 0 ∈ Γ be a primitive element of Γ fixing α 0 with |γ ′ 0 (α 0 )| > 1. Let µ F x 0 be a Patterson measure on K ∪ {∞} associated to a potential F for Γ such that δ = δ Γ, F and m F are finite. Let δ 0 be the critical exponent of γ Z 0 for F . Let ϕ : [0, +∞[ → ]0, +∞[ be a map such that t → ϕ(e t ) is slowly varying. If the integral +∞ 1 ϕ(t) δ-δ 0 /t dt diverges (resp. converges), then for µ F x 0 -almost every x ∈ K, lim inf r∈E α 0 ,Γ h(r) ϕ(h(r))

|x -r| = 0 (resp. = +∞) .

When F = 0, this result is due to [START_REF] Parkkonen | Spiraling spectra of geodesic lines in negatively curved manifolds[END_REF]Theo. 6.4 (4)].

Proof. Let us first give some details on the notation of this theorem. Recall (see for instance [START_REF] Parkkonen | Spiraling spectra of geodesic lines in negatively curved manifolds[END_REF]Lem. 6.2]), that the quadratic irrationals in K are exactly the fixed points of the loxodromic elements of PSL 2 (O K ), hence of Γ, since Γ has finite index in PSL 2 (O K ).

Hence an element γ 0 as in the statement exists and is unique, it is the unique primitive loxodromic element of Γ with attractive fixed point α 0 .

Let M be the real hyperbolic plane H 2 R if K = R and the real hyperbolic space H 3 R if K = C. We fix a point x 0 in M . Note that ∂ ∞ M = K ∪ {∞}, and Γ is a discrete group of isometries (actually an arithmetic lattice) of M , so that a Γ-invariant potential F on T 1 M with δ = δ Γ, F does define a Patterson measure µ F x 0 seen from x 0 (unique up to scalar multiple if m F is finite) on K ∪ {∞}, see Section 2. Let Γ 0 be the stabiliser of {α 0 , α 0 σ } in Γ (that is of the translation axis of γ 0 ), and let F 0 : Γ 0 \T 1 M → R be the map induced by F . Since γ Z 0 has finite index in Γ 0 , the critical exponent δ 0 is equal to δ Γ 0 , F 0 . Note that E α 0 ,Γ is exactly the set of fixed points of the conjugates of γ 0 in Γ.

We may assume that ϕ ≤ 1. Define φ : s → ϕ( 2 s ), which is a measurable map from ]0, 1] to ]0, 1[. The result then follows from Corollary 4.15.

To conclude, let us give a proof of the last statement of the Introduction.

Proof of Corollary 1.4.

It is well known that Γ a,b is a uniform lattice in SL 2 (R) (see for instance [START_REF] Katok | Fuchsian groups[END_REF]§5.2] or [START_REF] Benoist | Systèmes dynamiques élémentaires[END_REF]§8.5]): it is a Fuschian group derived from the quaternion algebra a, b Q over Q, which is a division algebra by the nonexistence of nonzero integer solutions to x 2a y 2b z 2 = 0, hence to x 2a y 2b z 2 + ab t 2 = 0 by [START_REF] Benoist | Systèmes dynamiques élémentaires[END_REF]Lem. 8.17 For all s ≥ 0 and ǫ > 0, let φ : ]0, 1] → ]0, 1] be the map t → min{1, ǫ (-log t) -s }, so that 1 0 φ δ-δ 0 (t)/t dt diverges if and only if s ≤ 1 δ-δ 0 . By Corollary 4.15, we hence have that if s ≤ 1 δ-δ 0 (resp. s > 1 δ-δ 0 ), then, for µ-almost every x ∈ R,

lim inf α∈Γ a, b •{γ - 0 ,γ + 0 } : |α-α σ |→0 |x -α| |α -α σ |(-log |α -α σ |) -s ≤ 1 ǫ (resp. ≥ 1 ǫ ) .
By taking ǫ = k (resp. ǫ = 1 k ) for k ∈ N tending to +∞, this proves the result.

  )(P (F |T 1 D 0

≍

  e t δ and α∈Γ 0 : t≤d(x,αy)<t+κ e αy x F ≍ e t δ 0 . If +∞ 0

Lemma 4. 4 0 x 0 F

 40 Assume that there exists κ > 0 such that γ∈Γ : t≤d(x 0 , γx 0 )<t+κ e γx ≍ e δ t and α∈Γ 0 : t≤d(x 0 , αx 0 )<t+κ e αx 0 x 0

  Lemma 4.5[START_REF] Hersonsky | On the almost sure spiraling of geodesics in negatively curved manifolds[END_REF] Lem. 3.3] For all A, δ 0 , δ > 0, there exists N ∈ N and B > 0 such that for all sequences (a k ) k∈N and (b k ) k∈N such that a n ≤ A e δ n , b n ≤ A e δ 0 n and n k=0 a k b n-k ≥ 1 A e δ n for every n ∈ N big enough, we have N k=0 a n+k ≥ B e δ n for every n ∈ N.

  Now, let us use Lemma 4.11 to prove Proposition 4.10. Let k, k ′ be elements of N with k < k ′ .

  r) is non empty. Then respectively by Proposition 4.7, by the second part of Lemma 4.8 with Q = I [γ], α, k ′ , by Lemma 4.11 and the definition of I [γ], α, k ′ , and by Proposition 4.7 (twice), we have

  Proposition 4.13 (1) The map F → P (F ) -P (F |T 1 C 0 ) is 1-Lipschitz for the uniform norm on bounded potentials.

  ]. Let Γ = Γ a,b be the image of Γ a,b in PSL 2 (R), which is a cocompact group of isometries of M = H 2 R , whose action on ∂ ∞ H 2 R = P 1 (R) is the action by homographies. Ifγ 0 = x + y √ a zt √ a b(z + t √ a) xy √ a with (x, y, z, t) ∈ Z 4 , then tr γ 0 = 2x. Hence | tr γ 0 | > 2 bythe assumptions, that is, the image of γ 0 in Γ, that we again denote by γ 0 , is hyperbolic.It is well known that its translation length ℓ(γ 0 ) satisfies (see for instance[START_REF] Beardon | The geometry of discrete groups[END_REF] page 173]cosh ℓ(γ 0 ) 2 = | tr γ 0 | 2 .Let us fix x 0 ∈ H 2 R . By Proposition 3.6, let F : T 1 M → R be a Γ-invariant Höldercontinuous map such that µ and µ Fx 0 have the same measure class. Since the conclusion of Corollary 1.4 depends only on the measure class of µ, and since d(γ -1 ) * µ ) for every hyperbolic element γ ∈ Γ by[START_REF] Ledrappier | Structure au bord des variétés à courbure négative[END_REF] Théo. 1.c], as seen in the proof of Proposition 3.6, we may assume that µ = µ F x 0 . Since Γ is cocompact, both δ = δ Γ, F and m F are finite. Let F 0 : Γ 0 \T 1 M → R be the map induced by F , and let δ 0 = δ Γ 0 , F 0 . By Remark (2) following the statement of Theorem 4.1 and by Equation (12), we haveδ 0 = max Per F (γ), Per F (γ -1 ) ℓ(γ 0 ) = 1 2 arcosh( | tr γ 0 | 2Since Γ is cocompact, by Bowen's period counting theorem (see for instance[START_REF] Paulin | Equilibrium states in negative curvature[END_REF] Coro. 1.7]), again by Equation (12), and by the change of variable s = 2 cosh t 2 , we haveδ = δ Γ,

  when restricted on the nonwandering sets, is a finite-to-one map, by the above property (2). Hence p * m F 0 m F 0 is another equilibrium state for F on Γ\T 1 M . But by[START_REF] Paulin | Equilibrium states in negative curvature[END_REF] Coro. 6.1], this equilibrium state is unique, hence p *

	m F 0

Keywords: geodesic flow, negative curvature, spiraling, Khintchine theorem, logarithm law, equilibrium states, Gibbs measure, pressure, Hausdorff dimension, Diophantine approximation, quadratic irrational. AMS codes: 37D35, 53D25, 37D40, 53C22, 37A25, 37A45,

11J83, 28D20

Almost sure spiraling for Gibbs statesWe will study in this section the generic asymptotic penetration properties of the geodesic lines, in a negatively curved simply connected manifold, under a discrete group of isometries, of a tubular neighbourhood of a convex subset with cocompact stabiliser, not only as in[START_REF] Hersonsky | On the almost sure spiraling of geodesics in negatively curved manifolds[END_REF] for the Bowen-Margulis measure, but for any Gibbs measure.Let ( M , Γ, F , (µ F •ι x ) x∈ M , (µ F x ) x∈ M, m F ) be as in the beginning of Section 3, with δ = δ Γ, F finite. We again use the notation introduced in Section 2.Recall that a subgroup H of a group G is almost malnormal if, for every g in G -H, the subgroup gHg -1 ∩ H is finite. Let Γ 0 be an almost malnormal and convex-cocompact subgroup of Γ, of infinite index in Γ, let C 0 = C ΛΓ 0 be the convex hull of the limit set of Γ 0 . For instance, C 0 could be the translation axis of a loxodromic element of Γ, and Γ 0 the stabiliser of C 0 in Γ (see[START_REF] Hersonsky | On the almost sure spiraling of geodesics in negatively curved manifolds[END_REF] §4] for an explanation and for more examples). Up to adding assumptions on the behaviour of the potential and on growth properties in cusp neighbourhoods (including a gap property for the pressures), our result should extend when Γ 0 is assumed to be only geometrically finite instead of convex-cocompact, or when

Mathematics Institute, University of Warwick, Coventry, CV4 7AL, UK e-mail: mpollic@maths.warwick.ac.uk