¸ois Ledrappier
email: fledrapp@nd.edu

Regularity of the entropy for random walks on hyperbolic groups

Keywords:

We consider non-degenerate, finitely supported random walks on a finitely generated Gromov hyperbolic group. We show that the entropy and the escape rate are Lipschitz functions of the probability if the support remains constant.

Introduction

This paper is an extension of [L2] to finitely generated Gromov hyperbolic groups (see [GH] and section 2 below for the definition of hyperbolic groups). Let p be a finitely supported probability measure on an infinite group G and define inductively, with p (0) being the Dirac measure at the identity e, p (n) (x) = [p (n-1) ⋆ p](x) = y∈G p (n-1) (xy -1 )p(y).

Define the entropy h p and the escape rate ℓ S p by

h p := lim n - 1 n x∈G p (n) (x) ln p (n) (x), ℓ S p := lim n 1 n x∈G |x|p (n) (x),
where |.| is the word metric defined by some symmetric generating set S. The entropy h p was introduced by Avez ( [Av]) and is related to bounded solutions of the equation on G f (x) = y∈G f (xy)p(y) (see e.g. [KV]). Erschler and Kaimanovich have shown that, on Gromov hyperbolic groups, the entropy and the escape rate depend continuously on the probability p with finite first moment ( [EK]). Here we are looking for a stronger regularity on a more restricted family of probability measures. We fix a finite set F ⊂ G such that ∪ n F n = G and we consider probability measures in P(F ), where P(F ) is the set of probability measures p such that p(x) > 0 if, and only if, x ∈ F . The set P(F ) is naturally identified with an open subset of the probabilities on F, which is a contractible open polygonal bounded convex domain in R |F |-1 . We show:

Theorem 1.1. Assume G is a Gromov hyperbolic group and F is a finite subset of G such that ∪ n F n = G. Then, with the above notation, the functions p → h p and p → ℓ S p are Lipschitz continuous on P(F ).

If the infinite hyperbolic group G is amenable, G is virtually cyclic and the entropy is vanishing on P(F ). Moreover, it follows from the formula in [KL] that the escape rate is Lipschitz continuous in on P(F ) (see the remark after Formula (4) below). If G is a non-abelian free group, and F a general finite generating set, then p → h p is real analytic ([L2], Theorem 1.1) and p → ℓ S p as well [START_REF] Gilch | Rate of Escape of Random walks on Regular Languages and Free Products by Amalgamation of Finite Groups[END_REF]). This holds more generally for free products (see [START_REF] Gilch | Rate of Escape of Random walks on Regular Languages and Free Products by Amalgamation of Finite Groups[END_REF] and [START_REF] Gilch | Asymptotic Entropy of Random walks on Free Products[END_REF] for the precise conditions). A general non-amenable hyperbolic group has many common geometric features with non-abelian free groups and our proof follows the scheme of [L2]. For Gromov hyperbolic groups, Ancona ( [An]) proved that the Martin boundary of the random walk directed by the probability p is the Gromov geometric boundary. Let K ξ (x) be the Martin kernel associated to a point ξ of the geometric boundary. Our main technical result, Proposition 4.1, uses the description of the Martin kernel by Ancona (see also [W], [INO]) to prove that ln K ξ (x) is a Lipschitz continuous function of p as a Hölder continuous function on the geometric boundary. Then, like in [L2], we can express h p in terms of the exit measure p ∞ of the random walk on the geometric boundary ∂G and the Martin kernel. Unfortunately, it is not clear in that generality that the measure p ∞ , seen as a linear functional on Hölder continuous functions on the geometric boundary, depends smoothly on the probability in M (F ). We use a symbolic representation of ∂G (see [CP]) to express p ∞ as an combination of a finite number of symbolic measures. Each of these symbolic measures depends Lipschitz of p and the entropy h p is the maximum of a finite family of Lipschitz functions. The escape rate is expressed by an analogous formula: it is the maximum of the integrals of the Busemann kernel with respect of the stationary measures on the Busemann boundary. It turns out that the Busemann boundary can be described in terms of the same symbolic representation and the Lipschitz regularity of the escape rate follows. It is likely that both entropy and rate of escape are more regular than what is obtained here, but this is what we can prove for the moment. Observe that for G = Z, S = {±1} and F a finite generating subset, the function p → ℓ S p = | F ip i | is Lipschitz continuous on P(F ), but not C 1 . For another example in the same spirit, we recall that Mairesse and Matheus ( [MM]) have shown that for the braid group B 3 = a, b|aba = bab and F = {a, a -1 , b, b -1 }, p → ℓ F p is Lipschitz, but not C 1 on P(F ). The entropy is constant 0 in the case of Z, the regularity of the entropy for the braid group is unknown.

In this note, the letter C stands for a real number independent of the other variables, but which may vary from line to line. The lower case c 0 , c 1 will be constants which might depend only on p ∈ P(F ). In the same way, the letter O p stands for a neighborhood of p in P(F ) which may vary from line to line.

preliminaries

2.1. Hyperbolic groups. We first recall basic facts about hyperbolic groups ([GH]). Let G be a finitely generated group with a symmetric finite set of generators S. Let d(x, y) = |x -1 y| be the word metric on G associated to S. For a subset F ⊂ G, we denote:

N (F, R) := {x ∈ G : d(x, F ) ≤ R} and ∂F = {x ∈ G : d(x, F ) = 1}.
For x, y, z ∈ G, the Gromov product (x|y) z is defined by the formula:

(x|y) z = 1 2 (d(x, z) + d(y, z) -d(x, y)) .
We write (x|y) for (x|y) e , where e is the unit element. Let δ > 0. The group G is said to be δ-hyperbolic if, for all x, y, z, w ∈ G,

(1) (x|y) w ≥ min{(x|z) w , (y|z) w } -δ.
If G is δ-hyperbolic, then every geodesic triangle ∆ = {α, β, γ} in G is 4δ-slim, i.e.:

α ⊂ N (β ∪ γ, 4δ), β ⊂ N (γ ∪ α, 4δ), γ ⊂ N (α ∪ β, 4δ).
A sequence {x n } n≥1 is said to converge to infinity if lim n,m→∞ (x n |x m ) = ∞. Two sequences {x n } n≥1 and {y n } n≥1 converging to infinity are said to be equivalent if lim n→∞ (x n |y n ) = ∞. The geometric boundary ∂G is defined as the set of equivalence classes of sequences converging to infinity. The Gromov product extends to G ∪ ∂G by setting

(ξ|η) = sup lim inf n,m→∞ (x n |y m ),
where the sup runs over all sequences {x n } n≥1 converging to ξ and {y m } m≥1 converging to η. Recall that G ∪ ∂G is compact equipped with the base {N ({x}, r)} ∪ {V r (ξ)}, where

V r (ξ) := {η ∈ G ∪ ∂G : (η|ξ) > r}.
One can introduce a metric ρ on ∂G such that, for some a > 1 and C > 0,

a -(ξ|η)-C ≤ ρ(ξ, η) ≤ a -(ξ|η)+C .

Another boundary is the Busemann boundary

∂ B G. Define, for x ∈ G the function Ψ x (z) on G by: Φ x (z) = d(x, z) -d(x, e).
The assignment x → Ψ x is continuous, injective and takes values in a relatively compact set of functions for the topology of uniform convergence on compact subsets of G. The Busemann compactification G of G is the closure of G for that topology. The Busemann compactification G is a compact G-space. The Busemann boundary ∂ B G := G \ G is made of Lipschitz continuous functions h on G such that h(e) = 0 and such that the Lipschitz constant is at most 1. Moreover, they are horofunctions in the sense of [CP]: they have the property that for all λ ≤ h(x), the distance of a point x to the set h -1 (λ) is given by h(x) -λ (see subsection 5.1 for more about horofunctions).

2.2. Random walks. Let Ξ be a compact space. Ξ is called a G-space if the group G acts by continuous transformations on Ξ. This action extends naturally to probability measures on Ξ. We say that the measure ν on Ξ is stationary if x∈G (x * ν)p(x) = ν. The entropy of a stationary measure ν is defined by:

(2)

h p (Ξ, ν) = - x∈G Ξ ln dx -1 * ν dν (ξ)dν(ξ) p(x).
The entropy h p and the escape rate ℓ p are given by variational formulas over stationary measures (see [KV], Section 3, for the entropy, [KL], Theorem 18, for the escape rate):

h p = max{h p (Ξ, ν); Ξ G-space and ν stationary on Ξ} (3) ℓ S p = max x∈G G h(x -1 )dν(h) p(x); ν stationary on G . (4)
Moreover, the stationary measures in (4) are supported by ∂ B G. In particular, in the case when G is virtually cyclic, ∂ B G is finite not reduced to a point 1 and ℓ S p is given by the maximum of a finite number of linear functions of p.

Let Ω = G N be the space of sequences of elements of G, M the product probability p N . The random walk is described by the probability P on the space of paths Ω, the image of M by the mapping:

(ω n ) n∈Z → (X n ) n≥0
, where X 0 = e and X n = X n-1 ω n for n > 0.

In particular, the distribution of X n is the convolution p (n) . We have:

Theorem 2.1 ( [An], Corollaire 6.3, [K], Theorem 7.5). There is a mapping X ∞ : Ω → ∂G such that for M -a.e. ω,

lim n X n (ω) = X ∞ (ω).
The action of G over itself by left multiplications extends to ∂G and makes ∂G a Gspace. The image measure p ∞ := (X ∞ ) * M is the only stationary probability measure on ∂G and (∂G, ν) achieves the maximum in (3) ([K], Theorem 7.6):

(5)

h p = h p (∂G, p ∞ ) = - x∈F ∂G ln dx -1 * p ∞ dp ∞ (ξ)dp ∞ (ξ) p(x).
The Green function G(x) associated to (G, p) is defined by

G(x) = ∞ n=0 p (n) (x)
1 The restriction of each limit function to a Z coset is of the form ±x + a, where a can take a finite number of values and there is at least one Z coset where both signs appear.

(see for example Proposition 2.2 for the convergence of the series). For y ∈ G, the Martin kernel K y is defined by

K y (x) = G(x -1 y) G(y) .
Ancona ( [An], Théorème 6.2) showed that y n → ξ ∈ ∂G if, and only if, the Martin kernels K yn converge towards a function K ξ called the Martin kernel at ξ. We have

(6) dx * p ∞ dp ∞ (ξ) = K ξ (x).
2.3. Differentiability. We are going to use formula (3) and first show that the mapping p → -ln K ξ (x) is Lipschitz continuous from a neighbourhood O p of p in P(F ) into a space of Hölder continuous functions on ∂G. The following properties are obtained exactly in the same way as in [L2]. For x, y ∈ G, let u(x, y) be the probability of eventually reaching y when starting from x. By left invariance, u(x, y) = u(e, x -1 y). Moreover, by the strong Markov property, G(x) = u(e, x)G(e) so that we have:

(7) K y (x) = u(x, y) u(e,

y) .

By definition, we have 0 < u(x, y) ≤ 1. The number u(x, y) is given by the sum of the probabilities of the paths going from x to y which do not visit y before arriving at y. The next two results are classical:

Proposition 2.2. Let p ∈ P(F ). There are numbers C and ζ, 0 < ζ < 1 and a neighborhood O p of p in P(F ) such that for all q ∈ O p , all x ∈ G and all n ≥ 0,

q (n) (x) ≤ Cζ n .
Proof. Let q ∈ P(F ). Consider the convolution operator P q in ℓ 2 (G, R) defined by:

P q f (x) = y∈F f (xy -1 )q(y).
Derriennic and Guivarc'h ( [DG]) showed that for p ∈ P(F ), P p has spectral radius smaller than one. In particular, there exists n 0 such that the operator norm of P n 0 p in ℓ 2 (G) is smaller than one. Since F and F n 0 are finite, there is a neighborhood O p of p in P(F ) such that for all q ∈ O p , P n 0 q 2 < λ for some λ < 1 and P k q 2 ≤ C for 1 ≤ k ≤ n 0 . It follows that for all q ∈ O p , all n ≥ 0, [DG]] Let p ∈ P(F ). There are numbers C and δ > 0 such that for all q ∈ O p , all x, y ∈ G, G(x, y) ≤ Ce -δ|x -1 y| .

P n q 2 ≤ Cλ [n/n 0 ] . In particular, for all x ∈ G, q (n) (x) = [P n q δ e ](x) ≤ |P n q δ e | 2 ≤ Cλ [n/n 0 ] |δ e | 2 ≤ Cλ [n/n 0 ] . Corollary 2.3. [
Proof. We have q (n) (x -1 y) = 0 for n ≤ 1 r |x -1 y|; take δ = 1 r ln 1 ζ .

Fix p ∈ P(F ) and let ∆ be a subset of G. We can define G ∆ (x, y), u ∆ (x, y) by considering only the paths of the random walk which remain inside ∆. Clearly,

G ∆ ≤ G, u ∆ ≤ u. For x ∈ G, V a subset of G and v ∈ V , let α V
x (v) be the probability that the first visit in V of the random walk starting from x occurs at v (α V x (v) = u G\V ∪{v} (x, v)). We have 0 ≤ v∈V α V x (v) ≤ 1 and Proposition 2.4. Fix x and V . For all s > 1, the mapping

p → α V x (v) is a C ∞ function from P(F ) into ℓ s (V ). Moreover, ∂α V
x ∂p i s is bounded independently of x and V . Proof. By Proposition 2.2, there is a neighbourhood O p of p in P(F ) and numbers C, ζ, 0 < ζ < 1, such that for q ∈ O p and for all y ∈ G,

q (n) (y) ≤ Cζ n .
The number α V x (v) can be written as the sum of the probabilities α n,V x (v) of entering V at v in exactly n steps. We have:

α n,V x (v) ≤ q (n) (x -1 v) ≤ Cζ n . Moreover, the function p → α n,V
x (v) is a homogeneous polynomial of degree n on P(F ):

α n,V x (v) = E q i 1 q i 2 • • • q in , where E is the set of paths {x, xi 1 , xi 1 i 2 , • • • , xi 1 i 2 • • • i n = v} of length n made of steps in F which start from x and enter V in v. It follows that for all α = {n 1 , n 2 , • • • , n |B| , n i ∈ N ∪ {0}}, all v ∈ V , | ∂ α ∂p α α n,V x (v)| ≤ n |α| (inf i∈F p i ) |α| α n,V x (v) ≤ Cn |α| (inf i∈F p i ) |α| ζ n , where |α| = i∈F n i . Therefore, v∈V | ∂ α ∂p α α n,V x (v)| s ≤ Cn s|α| (inf i∈F p i ) s|α| ζ (s-1)n v∈V α n,V x (v) ≤ Cn s|α| (inf i∈F p i ) s|α| ζ (s-1)n . Thus, q → ∂ α ∂p α α V x (v) is given locally by a uniformly converging series in ℓ s (V ) of derivatives. It follows that q → α V x (v) is a C ∞ function from P(F ) into ℓ s (V ). From the above computation, it follows that ∂ α ∂p α α V x (v) s ≤ n Cn s|α| (inf i∈F p i ) s|α| ζ (s-1)n , independently of x, V.
Proposition 2.5. There exists T large enough that for t > T , for any y and V , the

mapping p → α {y} v (y) is a C ∞ function from P(F ) into ℓ t (V ). Moreover, v → ∂α {y} v (y) ∂p i is bounded in ℓ t (V ) independently of y and V .
Proof. It suffices to show that there is T such that the function v → α {y} v (y) ∈ ℓ T (V ) and to apply the same arguments as in the proof of Proposition 2.4. Consider the probability p with support F -1 defined by p(x) = p(x -1 ) and define all quantities (p) (n) , Ǧ(x), ǔ(x, y).

Observe that, since it is the sum of the same probabilities over the same set of paths, G(v, y) = Ǧ(y, v). Therefore, we have, using Corollary 2.3 for the p random walk,

α {y} v (y) ≤ G(v, y) = Ǧ(y, v) ≤ e -δ|y -1 v| .
The group G has exponential growth: there is a v such that there are less than Ce vR elements of G at distance less than R from y. It follows that for

T > v/ δ, the function v → α {y} v (y) ∈ ℓ T (V ).
2.4. Projective contractions on cones. In this subsection, we recall Birkhoff Theorem about linear maps preserving convex cones. Let C be a convex cone in a Banach space and define on C the projective distance between half lines as

ϑ(f, g) := ln[τ (f, g)τ (g, f )],
where τ (f, g) := inf{s, s > 0, sf -g ∈ C}. Let D be the space of directions in C. Then, ϑ defines a distance on D. Let A be an operator from C into C and let T : D → D be the projective action of A. Then, by [Bi]:

(8) ϑ(T f, T g) ≤ βϑ(f, g), where β = tanh 1 4 Diam T (D) .
In some cases, ϑ-diameters are easy to estimate: for example, in C t = {f ∈ ℓ t ; f ≥ 0}, the set U (g, c) := {f : c -1 g ≤ f ≤ cg}, where g ∈ C and c ≥ 1, has ϑ-diameter 4 ln c. Moreover, the following observation is useful:

Lemma 2.6. [[Li], Lemma 1.3] Let f, g ∈ C t , f t = g t . Then, f -g t ≤ e ϑ(f,g) -1 f t .

Obstacles

In this section, we show that the function Φ on ∂G defined by Φ(ξ) := -ln K ξ (x) is Hölder continuous for any fixed x ∈ G. This is not a new result ( [INO]). Nevertheless, we present the construction and the proof in order to introduce the notation used in the next section to show that Φ is Lipschitz in p as a Hölder continuous function on ∂G. Like in [INO], the proof is based on Ancona's Harnack inequality at infinity (see [An], [INO] Proposition 2.1 for the form used here): there exist a number R and a constant c = c(p)

such that if [x, y] is a geodesic segment and z ∈ [x, y], then for any ∆ ⊂ G, N ([x, y], R) ⊂ ∆, we have: (9) c -1 u ∆ (x, z)u ∆ (z, y) ≤ u ∆ (x, y) ≤ cu ∆ (x, z)u ∆ (z, y),
where u ∆ (v, w) is the probability of ever arriving at w starting from v before reaching G \ ∆. Moreover, from the proof of ( 9) in [INO] or [W], it follows that there exists a neighborhood O of p in P(F ) and a constant C such that c(p) ≤ C for p ∈ O.

3.1. Obstacles. Without loss of generality, we may assume that F contains the set of generators and δ is an integer. Set r = max{|x|; x ∈ F, δ}. Fix M large. In particular, M ≥ R + 12r, where R is given by ( 9). For a geodesic γ, we call an obstacle a family

U - 0 ⊂ U 0 ⊂ U - 1 ⊂ U 1 of subsets of G such that U - 0 = {x ∈ G : d(x, γ(-2M )) < d(x, γ(0))}, U 0 = {x ∈ G : d(x, γ(-2M )) < d(x, γ(4r))}, U - 1 = {x ∈ G : d(x, γ(0)) < d(x, γ(2M ))}, U 1 = {x ∈ G : d(x, γ(0)) < d(x, γ(2M + 4r))}.
The subsets U ± i are connected and satisfy

U - 0 ⊂ U 0 ⊂ U - 1 ⊂ U 1 .
More precisely, we have the two following elementary facts:

Lemma 3.1. If x ∈ U - 0 and [x, γ(-2M )] is a geodesic segment, then [x, γ(-2M )] ⊂ U - 0 . Proof. Assume not. Then there is a z ∈ [x, γ(-2M )] such that d(z, γ(-2M )) ≥ d(z, γ(0)).
Adding d(z, x) to both sides of this inequality, we obtain:

d(x, γ(-2M )) = d(x, z) + d(z, γ(-2M )) ≥ d(x, z) + d(z, γ(0)) ≥ d(x, γ(0)), a contradiction to x ∈ U - 0 .
The statements and the proofs are the same for all

U ± i . Lemma 3.2. If x ∈ U - 0 , then B(x, r) ⊂ U 0 ; if x ∈ U 0 , then B(x, M -3r) ⊂ U - 1 . Proof. Let x ∈ U - 0 and x ′ ∈ B(x, r).
Writing (1) with x = x ′ , y = γ(0), z = γ(4r) and w = γ(-2M ), we get:

d(x ′ , γ(-2M )) -d(x ′ , γ(0)) + 2M ≥ min{d(x ′ , γ(-2M )) -d(x ′ , γ(4r)) + 2M + 4r, 4M } -2δ. Since d(x ′ , γ(-2M )) -d(x ′ , γ(4r)) < d(x, γ(-2M )) + r -d(x, γ(0)) + 5r < 6r ≤ 2M -4r, we get d(x ′ , γ(4r)) ≥ d(x ′ , γ(0)) + 4r -2δ > d(x, γ(0)) + r > d(x, γ(-2M )) + r > d(x ′ , γ(-2M )).
Analogously, if x ∈ U 0 and x ′ ∈ B(x, M -3r), we get, writing now (1) with z = γ(2M ):

d(x ′ , γ(-2M )) -d(x ′ , γ(0)) + 2M ≥ min{d(x ′ , γ(-2M )) -d(x ′ , γ(2M )) + 4M, 4M } -2δ.
Since the RHS is smaller than 4M -2r, it cannot exceed 4M -2δ, and we get

d(x ′ , γ(0)) ≤ d(x ′ , γ(2M )) -2M + 2δ < d(x ′ , γ(2M )).
Lemma 3.2 implies that any trajectory of the random walk going from

U - 0 to G \ U 1 has to cross successively U 0 \ U - 0 , U - 1 \ U 0 and U 1 \ U - 1 . For V 1 , V 2 subsets of G, denote A V 2 V 1 the (infinite) matrix such that the row vectors indexed by v ∈ V 1 are the α V 2 v (w), w ∈ V 2 . In particular, if V 2 = {y}, set ω y V 1 for the (column) vector ω y V 1 = A {y} V 1 = (α {y} v (y)) v∈V 1 = (u(v, y)) v∈V 1 .
Fix t > T . By Propositions 2.4 and 2.5,

ω y V 1 is a vector in ℓ t (V 1 ) and α V 0 x ∈ ℓ s (V 0 ), with 1/s + 1/t = 1. With this notation, the strong Markov property yields, if U - 0 ⊂ U 0 ⊂ U - 1 ⊂ U 1 is an obstacle and x ∈ U - 0 , y ∈ U 1 : u(x, y) = v 0 ,v 1 α V 0 x (v 0 )A V 1 V 0 (v 0 , v 1 )u(v 1 , y) = α V 0 x , A V 1 V 0 ω y V 1 ,
with the natural summation rules for matrices and for the (ℓ s , ℓ t ) coupling. All series are bounded series with nonnegative terms and we set

V i = U i \ U - i .
Observe that an obstacle is completely determined by the directing geodesic segment [γ(-2M ), • • • , γ(2M + 4r)], so that there is a finite number of possible obstacles and therefore a finite number of spaces ℓ t (V ), of (infinite) matrices

A V 1 V 0 , of vectors ω z V 1 and α V 0
x if the distances d(z, γ(2M + 4r + 1)) and d(x, γ(-2M )) are bounded.

Properties of the matrix

A V 1 V 0 .

Recall that the general entry of the matrix

A = A V 1 V 0 is A(v 0 , v 1 )
, the probability that starting from v 0 ∈ V 0 , the first visit in V 1 occurs at v 1 . In particular, assume A(v 0 , v 1 ) = 0. Then, all paths from v 0 to v 1 with steps in F have to enter V 1 elsewhere before reaching v 1 . Since the support F of p contains the generators of the group, A(v, v 1 ) = 0 for all v's in the connected component of v 0 in U - 1 . By Lemma 3.1, all paths from γ(0) to v 1 with steps in F have to enter V 1 before reaching v 1 . Therefore this property depends neither on v 0 ∈ U - 1 nor on p ∈ P(F ). We say that v 1 is active if A(v 0 , v 1 ) = 0. In the sequel we will call V 1 the set of active elements of U 1 \ U - 1 . We have:

Proposition 3.3. Let γ be a geodesic, U - 0 ⊂ U 0 ⊂ U - 1 ⊂ U 1 an obstacle, V 0 = U 0 \ U - 0 , V 1 the active part of U 1 \ U - 1 . There exists a neighborhood O p of p in P(F ) and a constant c 1 such that, for all p ∈ O p , all v 0 ∈ V 0 , v 1 ∈ V 1 : (10) c -1 1 u G\U - 1 (v 0 , γ(0))α V 1 γ(0) (v 1 ) ≤ A(v 0 , v 1 ) ≤ c 1 u G\U - 1 (v 0 , γ(0))α V 1 γ(0) (v 1 ). Proof. Introduce the set U -- 1 , U -- 1 = {x ∈ G : d(x, γ(0)) < d(x, γ(2M -4r))}. By a variant of Lemma 3.2, we may write, for v 0 ∈ V 0 , v 1 ∈ V 1 : A(v 0 , v 1 ) = w∈U - 1 \U -- 1 u G\U - 1 (v 0 , w)α V 1 w (v 1 ). Using that α V 1 γ(0) (v 1 ) = w∈U - 1 \U -- 1 u G\U - 1 (γ(0), w)α V 1 w (v 1 ), we see that it suffices to prove that, for all p ∈ O p , all v 0 ∈ V 0 , w ∈ U -- 1 \ U - 1 : c -1 1 u G\U - 1 (v 0 , γ(0))u G\U - 1 (γ(0), w) ≤ u G\U - 1 (v 0 , w) ≤ c 1 u G\U - 1 (v 0 , γ(0))u G\U - 1 (γ(0), w).
This will follow from a variant of (9) once we will have located the point γ(0) with respect to the geodesic [v 0 , w].

Observe that if v 0 ∈ U 0 , then d(v 0 , γ(0)) ≥ M -3r. Indeed, writing that

(γ(-2M ), γ(4r)) v 0 ≥ min{(γ(-2M ), γ(-M + 2r)) v 0 , (γ(4r), γ(-M + 2r)) v 0 } -δ = (γ(-2M ), γ(-M + 2r)) v 0 -δ,
we get that d(v 0 , γ(4r)) ≥ M +2r-δ ≥ M -3r and the claim follows. Since, by Lemma 3.1, the whole geodesic [v 0 , γ(-M + 2r)] lies in U 0 , we have

d(γ(0), [v 0 , γ(-M + 2r)]) ≥ M -3r. But we know that γ(0) ∈ N ([v 0 , γ(M )] ∪ [v 0 , γ(-M + 2r)], 4δ). It follows that there is a point z 1 ∈ [v 0 , γ(M )] with d(γ(0), z 1 ) ≤ 4δ. In the same way, since w ∈ G \ U -- 1 , d(γ(0), [w, γ(M -2r)]) ≥ M -3r and therefore d(z 1 , [w, γ(M -2r)]) ≥ M -3r -4δ. It follows that there is a point z ∈ [v 0 , w] such that d(z, γ(0)) ≤ d(z, z 1 ) + d(z 1 , γ(0)) ≤ 8δ. Let y 0 be the point in [v 0 , w] at distance R from w. Then G \ U - 1 contains N ([v 0 , y 0 ], R)
and the point z belongs to [v 0 , y 0 ] 2 . So we may apply (9) to the points v 0 , z, y 0 and the domain

∆ = G \ U - 1 to obtain, for all p ∈ O p , all v 0 ∈ V 0 , v 1 ∈ V 1 : c -1 0 u G\U - 1 (v 0 , z)u G\U - 1 (z, y 0 ) ≤ u G\U - 1 (v 0 , y 0 ) ≤ c 0 u G\U - 1 (v 0 , z)u G\U - 1 (z, y 0 ).
By changing the constant, we can replace y 0 by w (since d(w, y 0 ) = R) and z by γ(0) (since d(z, γ(0)) ≤ 8δ). We obtain the desired inequality.

For V a subset of G, t > 0, denote C t V the convex cone of nonnegative sequences in ℓ t (V ) and define on C t V the projective distance between half lines as ϑ(f, g)

:= ln[τ (f, g)τ (g, f )],
where τ (f, g) := inf{s, s > 0, sf -g ∈ C t V }. Represent the space of directions as the sector of the unit sphere D t V = C t V ∩ S t V ; then, ϑ defines a distance on D t V for which D t V is a complete space (Lemma 2.6). We fix t > T such that the sequences α {y} v (y) ∈ ℓ t (V ) and we consider the matrix A V 1 V 0 as an operator from ℓ t (V 1 ) into the space of sequences indexed on V 0 . We have: Proposition 3.4. Choose t > T and s such that 1/s + 1/t = 1. For any obstacle

U - 0 ⊂ U 0 ⊂ U - 1 ⊂ U 1 , all p ∈ O p , the operator A V 1 V 0 sends C t V 1 into C t V 0 , the adjoint operator (A V 1 V 0 ) * sends C s V 0 into C s V 1 and Diam C t V 0 A V 1 V 0 (C t V 1 ) ≤ 4 ln c 1 , Diam C s V 1 (A V 1 V 0 ) * (C s V 0 ) ≤ 4 ln c 1 ,
where c 1 and O p are the ones in (10).

2 Since w ∈ U -- 1 , we have d(w, z) ≥ d(w, γ(0)) -8δ ≥ M -3r -8δ ≥ M -11r > R.
Proof. By definition,

v 1 ∈V 1 α V 1 γ(0) (v 1 ) ≤ 1 so that α V 1 γ(0) (v 1 ) ∈ ℓ s (V 1 )
. By (10), for any

v 0 ∈ V 0 , any f ∈ ℓ t (V 1 ), A V 1 V 0 f (v 0 ) ≤ c 1 u G\U 1 (v 0 , γ(0)) α V 1 γ(0) (v 1 ) s f t .
By the same argument as in the proof of Proposition 2.5, we see that

v 0 → u G\U - 1 (v 0 , γ(0)) ∈ ℓ t (V 0 ). It follows that for any f ∈ ℓ t (V 1 ), A V 1
V 0 f belongs to ℓ t (V 0 ). By (10), we know that for any

f ∈ C t V 1 , (11) c -1 1 u G\U 1 (v 0 , γ(0)) ≤ A V 1 V 0 f (v 0 ) < α V 1 γ(0) (.), f (.) > ≤ c 1 u G\U 1 (v 0 , γ(0)).
It follows that Diam

C V t 0 A V 1 V 0 (C t V 1 ) ≤ 4 ln c 1 .
The same argument works for the adjoint operator

(A V 1 V 0 ) * , since we know that v 0 → u G\U 1 (v 0 , γ(0)) ∈ ℓ t (V 0 ) and v 1 → α V 1 γ(0) (v 1 ) ∈ ℓ s (V 1 ). Proposition 3.5. Choose t > T + 1, s such that 1/s + 1/t = 1. The mapping p → A V 1 V 0 (re- spectively p → (A V 1 V 0 ) * ) is C ∞ from P(F ) into L(ℓ t (V 1 ), ℓ t (V 0 (respectively L(ℓ s (V 0 ), ℓ s (V 1 ))).
Proof. We follow the scheme of the proofs of Propositions 2.4 and 2.5. By Proposition 2.2, there is a neighbourhood O p of p in P(F ) and numbers C, ζ, 0 < ζ < 1, such that for q ∈ O p and for all y ∈ G, q (n) (y) ≤ Cζ n .

We write α V 1 v 0 (v 1 ) as the sum of the probabilities

α n,V 1 v 0 (v 1 ) of entering V 1 at v 1 in exactly n steps. We have, for all v 0 ∈ V 0 , v 1 ∈ V 1 : α n,V 1 v 0 (v 1 ) ≤ q (n) (v -1 0 v 1 ) ≤ Cζ n . As before, the function p → α n,V 1 v 0 (v 1
) is a homogeneous polynomial of degree n on P(F ) and for all α

= {n 1 , n 2 , • • • , n |B| , n i ∈ N ∪ {0}}, all v 0 ∈ V 0 , v 1 ∈ V 1 , | ∂ α ∂p α α n,V 1 v 0 (v 1 )| ≤ Cn |α| (inf i∈F p i ) |α| α n,V 1 v 0 (v 1 ) ≤ Cn |α| (inf i∈F p i ) |α| ζ n . Let f ∈ ℓ t (V 1 ). Then, v 1 | ∂ α ∂p α α n,V 1 v 0 (v 1 )||f (v 1 )| ≤ Cn |α| (inf i∈F p i ) |α| α n,V 1 v 0 (v 1 ) s f t ≤ Cn |α| ζ s-1 s n (u G\U - 1 (v 0 , γ(0))) 1 s f t .
To obtain the last inequality, we use that

α n,V 1 v 0 (v 1 ) s ≤ C ζ (s-1)n v 1 α n,V 1 v 0 (v 1 ) 1/s , ( 10
)
and v 1 α V 1 γ(0) (v 1 ) ≤ 1. Therefore, v 1 | ∂ α ∂p α α n,V 1 v 0 (v 1 )||f (v 1 )| ℓ t (V 0 ) ≤ Cn |α| ζ s-1 s n v 0 (u G\U 1 (v 0 , γ(0))) t s 1 t f t .
Since t > T + 1, t/s > T , the series v 0 (u G\U 1 (v 0 , γ(0)))

t s converges and the operator

f → v 1 ∂ α ∂p α α n,V 1 v 0 (v 1 )f (v 1 ) has norm smaller than Cn |α| ζ s-1 s n in L(ℓ t (V 1 ), ℓ t (V 0 )
). The series of operators which defines

∂ α ∂p α A V 1
V 0 is converging. The proof is the same for the adjoint operator. We estimate, for g ∈ ℓ s (V 0 ),

v 0 | ∂ α ∂p α α n,V 1 v 0 (v 1 )||g(v 0 )| ≤ Cn |α| (inf i∈F p i ) |α| α n,V 1 v 0 (v 1 ) t g s ≤ Cn |α| ζ n t α V 1 γ(0) (v 1 ) t-1 t g s .
As before, we find that the operator

g → v 0 ∂ α ∂p α α n,V 1 v 0 (v 1 )g(v 0 )
has norm smaller than

Cn |α| ζ n t v 1 α V 1 γ(0) (v 1 ) s t-1 t ≤ Cn |α| ζ n t in L(ℓ s (V 0 ), ℓ s (V 1 )) (recall that s t-1 t = 1
). The series of operators which defines

∂ α ∂p α (A V 1 V 0 ) * is converging as well.

Hölder regularity of the Martin kernel.

Fix x ∈ G and a geodesic γ with γ(0) = e. Consider the family

U - 0 ⊂ U 0 ⊂ • • • ⊂ U - n ⊂ U n such that for all j = 1, • • • n -1, U - j ⊂ U j ⊂ U - j+1 ⊂ U j+1 is an obstacle for γ • σ 2jM +K .
The integer K is chosen so that x, e ∈ U - 0 , e.g. K = 4M + |x|. With that choice, γ(n) ∈ U k as soon as n > K + 2kM + 4r. Iterating the strong Markov property, we get, for z ∈ U k :

u(x, z) u(e, z) = α V 0 x , A V 1 V 0 • • • A V k V k-1 ω z V k α V 0 e , A V 1 V 0 • • • A V k V k-1 ω z V k . Choose t > T + 1 and s such that 1/s + 1/t = 1. Set f k (z) := ω z V k ω z V k t , α := α V 0 e , β := α V 0 x . For all z ∈ U k , f k (z) ∈ D t V k and α, β ∈ C s V 0 -{0}. By Proposition 2.4, if z, z ′ ∈ U k , ϑ C t (A V k V k-1 f k (z), A V k V k-1 f k (z ′ )) ≤ 4 ln c 1 . Set τ = c 2 1 -1 c 2 1 +1
. By repeated application of (8), we have, as soon as z, z ′ ∈ U k , (12)

ϑ C t A V 1 V 0 • • • A V k V k-1 f k (z), A V 1 V 0 • • • A V k V k-1 f k (z ′ ) ≤ τ k-1 ϑ C t (A V k V k-1 f k (z), A V k V k-1 f k (z ′ )) ≤ 4 ln c 1 τ k-1 .
We are interested in the function Φ : ∂G → R,

Φ(ξ) = -ln K x (ξ) = -ln lim xn→ξ u(x, x n ) u(e, x n ) .
If we choose the reference geodesic γ converging towards ξ, then setting Φ n (ξ) = u(x,γ(n)) u(e,γ(n)) , we have lim n Φ n (ξ) = Φ(ξ). More precisely, as soon as n, m > K + 2kM + 4r, we may write:

Φ n (ξ) -Φ m (ξ) = ln α, T j 0 • • • T j k-1 f k (γ(n)) β, T j 0 • • • T j k-1 f k (γ(n)) β, T j 0 • • • T j k-1 f k (γ(m)) α, T j 0 • • • T j k-1 f k (γ(m)) ,
where T js is the projective action of A

V s+1
Vs . By ( 12) and Lemma 2.6, we have, as soon as

n, m > K + 2kM + 4r, |Φ n (ξ) -Φ m (ξ)| ≤ Cτ k . For the same reason, for any fixed family of f k ∈ D t V k , the sequence T j 0 • • • T j k-1 f k converge in D t
V 0 towards some f ∞ , independent the choice of f k and a priori depending on the geodesic γ converging towards ξ. In any case, we have:

(13) T j 0 • • • T j k-1 f k -f ∞ t ≤ Cτ k and Φ(ξ) = ln α, f ∞ α 1 , f ∞ .
Consider now two points ξ, η ∈ ∂G such that ρ(ξ, η) < a -n-C . Then there is a geodesic γ converging to ξ and a sequence {y ℓ } ℓ≥1 going to η such that for ℓ, m large enough, (γ(m), y ℓ ) > n. For fixed x and K = 4M + |x|, consider the same family

U - 0 ⊂ U 0 ⊂ • • • ⊂ U - k ⊂ U k such that for all j = 1, • • • k -1, U - j ⊂ U j ⊂ U - j+1 ⊂ U j+1
is an obstacle for γ • σ 2jM +K . We have Lemma 3.6. Assume 2kM < n -K -4r -22δ and ℓ large enough. Then, y ℓ ∈ U k .

Proof. Choose ℓ large enough that lim m→∞ (γ(m), y ℓ ) > n and we choose a geodesic [y ℓ , ξ] such that (y ℓ , ξ) > n. By definition of U j , we have to show that d(y ℓ , γ(2(j -1)M + K)) ≥ d(y ℓ , γ(2jM + K + 4r)) for 2jM + K + r + 22δ < n. By continuity, there is a point s 0 where the function s → d(y ℓ , γ(s)) attains its minimum. We are going to show that s 0 ≥ n -12δ. By 8δ convexity of s → d(y ℓ , γ(s))([GH], Proposition 25, page 45), this prove the claim 3 .

By continuity, there is a point s 1 such that d(γ(s 1 ), [γ(0),

y ℓ ]) = d(γ(s 1 ), [y ℓ , ξ]) ≤ 4δ. On the one hand, s 1 ≥ d(γ(0), [y ℓ , ξ]) -4δ ≥ n -4δ (recall that (ξ, y ℓ ) > n).
On the other hand, we know that

d(y ℓ , γ(s 1 )) ≤ (γ(0), ξ) y ℓ + 8δ ≤ d(y ℓ , γ(s 0 )) + 8δ
(see the proof of Lemma 22.4 in [W]). It follows that s 0 ≥ s 1 -8δ ≥ n -12δ.

3 Indeed, since 2jM + K + r + 22δ < n, γ(2jM + K + 4r + 10δ) lies between γ(2(j -1)M + K) and s0 and thus, by 8δ convexity of the distance, d(y ℓ , γ(2jM + K + 4r + 10δ)) ≤ d(y ℓ , γ(2(j -1)M + K)) + 8δ (recall that s0 achieves the minimum of d(y ℓ , γ(s))). The inequality follows by writing the δ-hyperbolicity relation (1) with x = y ℓ , y = γ(2jM + K + 4r), z = γ(2jM + K + 4r + 10δ) and w = γ(2(j -1)M + K).

We have that Φ(ξ) -Φ(η) = lim xm→ξ,y ℓ →η ln u(e,xm) u(x,xm) u(x,y ℓ ) u(e,y ℓ ) . With the above notation, assume that k is such that 2kM < n -K -4r -22δ. If ℓ and m are large enough,

y ℓ , γ(m) ∈ U k and Φ(ξ) -Φ(η) = lim xm→ξ,y ℓ →η ln β, T j 0 • • • T j k-1 f k (γ(m)) α, T j 0 • • • T j k-1 f k (γ(m)) α, T j 0 • • • T j k-1 f k (y ℓ ) β, T j 0 • • • T j k-1 f k (y ℓ ) .
Since as above, we have ϑ(T

j 0 • • • T j k-1 f k (γ(m)), T j 0 • • • T j k-1 f k (y ℓ )) < Cτ k
, and α, β take a finite number of values, we have:

|Φ(ξ) -Φ(η)| ≤ Cτ k ≤ Cρ n 0
for a new constant C and ρ 0 = τ 1/2M . This shows that for all x ∈ G, the function ξ → -ln K x (ξ) is Hölder continuous on ∂G. Moreover, the Hölder exponent | ln ρ 0 |/ ln a and the Hölder constant C are uniform on a neighbourhood of p in P(F ).

Let us choose κ < 1, κ < -ln ρ 0 2 ln a , and consider the space Γ κ of functions φ on ∂G such that there is a constant C κ with the property that |φ(ξ)-φ(η)| ≤ C κ (d(ξ, η)) κ . For φ ∈ Γ κ , denote φ κ the best constant C κ in this definition. The space Γ κ is a Banach space for the norm φ := φ κ + max ∂G |φ|. In this subsection, we showed that for p ∈ P(F ), x ∈ G, there exist κ > 0 and a neighborhood O p of p in P(F ) such that for p ′ ∈ O p , the function Φ p ′ (ξ) = -ln K ξ (x) belongs to Γ κ and that the mapping

p ′ → Φ p ′ is bounded from O p into Γ κ .
4. The Martin kernel depends regularly on p Proposition 4.1. Fix x ∈ G. For all p ∈ P(F ), there exist κ > 0 and a neighborhood O p of p in P(F ) such that the mapping

p → Φ(ξ) = -ln K ξ (x) is Lipschitz continuous from O p into Γ κ .
Proof. Let p ∈ P(F ) and choose κ = κ(p) given by Section 3.3. We have to find a neighborhood O of p in P(F ) and a constant C such that, for p ′ ∈ O,

Φ p -Φ p ′ = max ξ |Φ p (ξ) -Φ p ′ (ξ)| + Φ p -Φ p ′ κ ≤ Cϑ(p, p ′ ),
where, for convenience, we use on P(F ) the already defined projective distance on R F . We treat the two terms separately.

Claim 1: max ξ |Φ p (ξ) -Φ p ′ (ξ)| ≤ Cϑ(p, p ′ ).
Choose the geodesic γ converging to ξ. Applying Section 3.3 and (13), there are vectors

f ∞ (p), f ∞ (p ′ ) ∈ ℓ t (V 0 ) such that |Φ p (ξ) -Φ p ′ (ξ)| = ln α(p), f ∞ (p) α(p ′ ), f ∞ (p ′ ) β(p ′ ), f ∞ (p ′ ) β(p), f ∞ (p) .
By Proposition 2.4, we make an error of order Cϑ(p, p ′ ) when replacing β(p ′ ) by β(p) and α(p ′ ) by α(p). The remaining term is

(14) lim k ln α, T j 0 • • • T j k-1 f k α, T ′ j 0 • • • T ′ j k-1 f k β, T ′ j 0 • • • T ′ j k-1 f k β, T j 0 • • • T j k-1 f k ,
where T js is the projective action of A V s+1

Vs (p), T ′ js the projective action of A V s+1

Vs (p ′ ) and we have chosen once for all f k ∈ ℓ t (V k ), independent of p ∈ O.

We have:

ϑ T j 0 • • • T j k-1 f k , T ′ j 0 • • • T ′ j k-1 f k ≤ k-1 i=1 ϑ T j 0 • • • T j i-1 T ′ j i • • • T ′ j k-1 f k , T j 0 • • • T j i T ′ j i+1 • • • T ′ j k-1 f k ≤ k-1 i=1 τ i-1 ϑ T ′ j i • • • T ′ j k-1 f k , T j i T ′ j i+1 • • • T ′ j k-1 f k ,
where we used (12) to write the last line. If the neighborhood O is relatively compact in

P(F ), all points T ′ j i+1 • • • T ′ j k-1 f k are in a common bounded subset of D t V j i+1
. By Proposition 3.5, there is a constant C and a neighborhood O such that for

p ′ ∈ O, i = 1, • • • , k -1, ϑ T ′ j i T ′ j i+1 • • • T ′ j k-1 f k , T j i T ′ j i+1 • • • T ′ j k-1 f k ≤ Cϑ(p, p ′ ).
Finally, we get that for all k, ϑ T

j 0 • • • T j k-1 f k , T ′ j 0 • • • T ′ j k-1 f k ≤ C 1 -τ ϑ(p, p ′ ). Reporting in (14) proves Claim 1. Claim 2: Φ p -Φ p ′ κ ≤ Cϑ(p, p ′ ).
Let ξ, η ∈ ∂G be such that ρ(ξ, η) < a -n-C . We want to show that there is a constant C and a neighbourhood O, independent on n such that, for p ′ ∈ O:

|Φ p (ξ) -Φ p ′ (ξ) -Φ p (η) + Φ p ′ (η)| ≤ Ca -κn ϑ(p, p ′ ).
Choose as before a geodesic γ converging to ξ and a sequence {y ℓ } ℓ≥1 going to η such that for ℓ, m large enough, (γ(m), y ℓ ) > n. For fixed x and K = 4M + |x|, consider the same family

U - 0 ⊂ U 0 ⊂ • • • ⊂ U - k ⊂ U k such that for all j = 1, • • • k -1, U - j ⊂ U j ⊂ U - j+1 ⊂ U j+1 is an obstacle for γ • σ 2jM +K . By Lemma 3.6, for ℓ large enough, y ℓ ∈ U k and we may write Φ p (ξ) -Φ p ′ (ξ) -Φ p (η) + Φ p ′ (η) as: (15) lim xm→ξ,y ℓ →η ln β, T j 0 • • • T j k-1 g k α, T j 0 • • • T j k-1 g k α ′ , T ′ j 0 • • • T ′ j k-1 g ′ k β ′ , T ′ j 0 • • • T ′ j k-1 g ′ k α, T j 0 • • • T j k-1 h k β, T j 0 • • • T j k-1 h k β ′ , T ′ j 0 • • • T ′ j k-1 h ′ k α ′ , T ′ j 0 • • • T ′ j k-1 h ′ k , where α = α(p), α ′ = α(p ′ ), β = β(p), β ′ = β(p ′ ), T js is the projective action of A V s+1 Vs (p), T ′ js the projective action of A V s+1
Vs (p ′ ) and g k , g ′ k are f k (γ(m)) calculated with p and p ′ respectively, h k , h ′ k are f k (y ℓ ) calculated with p and p ′ .

Recall that

g k = f k (γ(m)) is the direction of ω γ(m) v in ℓ t (V k ).
It can obtained by a series of obstacles along γ between U k and γ(m). Let us show that we can choose m large enough (depending on p ′ ) such that we have ϑ(g k , g ′ k ) ≤ Cϑ(p, p ′ ). Indeed,

ϑ(g k , g ′ k ) = ϑ(f k (γ(m)), f ′ k (γ(m))) = ϑ T j k • • • T j m-1 f m , T ′ j k • • • T ′ j m-1 f ′ m .
We have ϑ(f m , f ′ m ) < C and for m large enough,

ϑ T ′ j k • • • T ′ j m-1 f m , T ′ j k • • • T ′ j m-1 f ′ m < τ m-k C ≤ ϑ(p, p ′ ).
By the same computation as in Claim 1, we then have

ϑ T j k • • • T j m-1 f m , T ′ j k • • • T ′ j m-1 f m ≤ Cϑ(p, p ′ ).
Since ϑ(g k , g ′ k ) ≤ Cϑ(p, p ′ ), using the contraction of the T j , we can replace g ′ k by g k in (15) with an error less than Cτ k ϑ(p, p ′ ) < Cρ n 0 ϑ(p, p ′ ). In the same way, following obstacles along the geodesic between γ(n) and y ℓ , we have, for ℓ large enough, ϑ(h k , h ′ k ) ≤ ϑ(p, p ′ ) and we can replace h ′ k by h k in ( 15) with an error less than Cρ n 0 ϑ(p, p ′ ). Observe also that all terms α = α/ α s , β = β/ β s belong to D s V j 0 . We may write, considering for instance α ′ , T

′ j 0 • • • T ′ j k-1 g ′ k : α ′ , T ′ j 0 • • • T ′ j k-1 g ′ k α, T ′ j 0 • • • T ′ j k-1 g ′ k = α ′ , A ′ j 0 • • • A ′ j k-1 g ′ k α, A ′ j 0 • • • A ′ j k-1 g ′ k = (A ′ j k-1 ) * • • • (A ′ j 0 ) * α ′ , g ′ k (A ′ j k-1 ) * • • • (A ′ j 0 ) * α, g ′ k = α ′ s α s (T ′ j k-1 ) * • • • (T ′ j 0 ) * α′ , g ′ k (T ′ j k-1 ) * • • • (T ′ j 0 ) * α, g ′ k ,
where (T ′ j ) * denotes the projective action of (A ′ j ) * on D s V j . Observe that if we replace α ′ by α, β ′ by β in (15) and use the above equation and its analogs, the ratios α ′ s α s , β ′ s β s cancel one another and using the contraction of the (T ′ j ) * , we make an other error of size at most Cρ n 0 ϑ(p, p ′ ). We find that, up to an error of size at most Cρ n 0 ϑ(p, p ′ ), the difference Φ p (ξ) -Φ p ′ (ξ) -Φ p (η) + Φ p ′ (η) is given by: lim

xm→ξ,y ℓ →η ln β, A j 0 • • • A j k-1 g k β, A ′ j 0 • • • A ′ j k-1 g k α, A ′ j 0 • • • A ′ j k-1 g k α, A j 0 • • • A j k-1 g k α, A j 0 • • • A j k-1 h k α, A ′ j 0 • • • A ′ j k-1 h k β, A ′ j 0 • • • A ′ j k-1 h k β, A j 0 • • • A j k-1 h k ,
where we reordered the denominators to get a sum of four terms of the form

± ln α, A j 0 • • • A j k-1 g α, A ′ j 0 • • • A ′ j k-1 g ,
Claim 2 is proven (recall that κ < -ln ρ 0 2 ln a so that ρ 1/2 < a -κ ).

Markov coding and regularity of p ∞

In this section, we discuss the regularity of the mapping p → p ∞ from P(F ) into the space Γ * κ of continuous linear forms on Γ κ . By Theorem 2.1, p ∞ is the only p-stationary measure for the action of ∂G, and thus depends continuously on p. In the case of the free group, p ∞ appears as the eigenform for an isolated maximal eigenvalue of an operator on Γ κ (see [L1], Chapter 4c) and therefore depends real analytically on p. This argument does not seem to work in all the generality of a hyperbolic group and we are going to use the Markov representation of the boundary which was described by M. Coornaert and A. Papadopoulos in [CP].

5.1. Markov coding. Following [CP], we call horofunctions any integer valued function on G such that, for all λ ≤ h(x), the distance of a point x to the set h -1 (λ) is given by h(x) -λ. Two horofunctions are said to be equivalent if they differ by a constant. Let Φ 0 be the set of classes of horofunctions. Equipped with the topology of uniform convergence on finite subsets of G, the space Φ 0 is a compact metric space. G acts naturally on Φ 0 . The Busemann boundary

∂ B G is a G-invariant subset of Φ 0 . For each horofunction h, sequences {x n } n≥1 such that d(x n , x n+1 ) = h(x n ) -h(x n+1 ) = 1
converge to a common point in ∂G, the point at infinity of h. Two equivalent horofunctions have the same point at infinity. The mapping π : Φ 0 → ∂G which associates to a class of horofunctions its point at infinity is continuous, surjective, G-equivariant and uniformly finite-to-one. Fix an arbitrary total order relation on the set of generators S. Define a map α : Φ 0 → Φ 0 by setting, for a class ϕ = [h] ∈ Φ 0 , α(ϕ) = a -1 ϕ, where a = a(ϕ) is the smallest element in S satisfying h(e) -h(a) = 1. In [CP] is proven:

Theorem 5.1 ( [CP]). The dynamical system (Φ 0 , α) is topologically conjugate to a subshift of finite type.

We assume, as we may, that the number R 0 used in the construction of [CP] satisfies R 0 > r. In order to fix notation, let (Σ, σ) be the subshift of finite type of Theorem 5.1. That is, there is a finite alphabet Z and a Z × Z matrix A with entries 0 or 1 such that Σ is the set of sequences z = {z n } n≥0 such that for all n, A zn,z n+1 = 1 and σ is the left shift on Σ. We can decompose Σ into transitive components. Namely, there is a partition of the alphabet Z into the disjoint union of Z j , j = 0, • • • , K in such a way that for j = 1, • • • , K, Σ j := {z, z 0 ∈ Z j } is a σ-invariant transitive subshift of finite type and ∪ K j=1 Σ j is the ωlimit set of Σ. By construction, G-invariant closed subsets of Σ are unions of Σ j for some j ∈ {1, • • • , K}. We denote such G-invariant subsets by Σ J , where J is the corresponding subset of {1, • • • , K}. In particular, the supports of stationary measures on G are subsets of ∂ B G which are identified with such Σ J .

For χ > 0 consider the space Γ χ of functions φ on Σ such that there is a constant C χ with the property that, if the points z and z ′ have the same first n coordinates, then |φ(z) -φ(z ′ )| < C χ χ n . For φ ∈ Γ χ , denote φ χ the best constant C χ in this definition. The space Γ χ is a Banach space for the norm φ := φ χ + max Σ |φ|. Identifying Σ with Φ 0 , we still write π : Σ → ∂G the mapping which associates to z ∈ Σ the point at infinity of the class of horofunctions represented by z.

Proposition 5.2. The mapping π : Σ → ∂G is Hölder continuous.

Proof. Let z and z ′ be two elements of Σ such that z i = z ′ i for 1 ≤ i ≤ n. Denote h and h ′ the corresponding horofunctions with h(e) = h ′ (e) = 0. Let {x n } n≥0 be define inductively such that x 0 = e and (x n-1 ) -1 x n is the smallest element a in S such that h(x n-1 ) -h(x n-1 a) = 1. The sequence {x n } n≥0 is a geodesic and converges to π(z). By [CP], Lemma 6.5, h and h ′ coincide on N ({x 0 , • • • , x n+L 0 }, R 0 ), where L 0 and R 0 has been chosen as in [CP], page 439. In particular, if one associates {x ′ n } n≥0 similarly to h ′ , the sequence {x ′ n } n≥0 is a geodesic which converges to π(z ′ ) and we have

x k = x ′ k for 0 ≤ k ≤ n + L 0 . It follows that for all m, m ′ > n + L 0 , (x m , x m ′ ) e = n + L 0 + (x m , x m ′ ) x n+L 0 ≥ n + L 0 ≥ n. Therefore (π(z), π(z ′ )) e ≥ lim inf m,m ′ (x m , x m ′ ) e ≥ n and ρ(π(z), π(z ′ )) ≤ e -an+c 1 .
In the same way we have: Proposition 5.3. Let x be fixed in G with |x| < R 0 . Then the mapping z → h z (x) depends only on the first coordinate in Σ, where h z is the horofunction representing z in Theorem 5.1. Proof. As above, if z 0 = z ′ 0 and h, h ′ are the corresponding horofunctions with h(e) = h ′ (e) = 0, h and h ′ coincide on N (e, R 0 ) ⊃ {x}.

Let ν be a stationary probability measure on Φ 0 . By equivariance of π, the measure π * ν is stationary on ∂G and, by Theorem 2.1, we have π * ν = p ∞ . Actually, there is a more precise result: Proposition 5.4. Let ν be a stationary measure on Φ 0 . Then, for ν-a.e. ϕ ∈ Φ 0 , all x,

(16) dx * ν dν (ϕ) = K π(ϕ) (x).
Proof. Since the mapping π : Φ 0 → ∂F is G-equivariant and finite-to-one, the measure ν can be written as

ψ(ϕ)dν(ϕ) = ϕ:π(ϕ)=ξ ψ(ϕ)a(ϕ) dp ∞ (ξ),
where a is a nonnegative measurable function on Φ 0 such that ϕ:π(ϕ)=ξ a(ϕ) = 1 for p ∞ -a.e. ξ. Moreover, since ∂G is a Poisson boundary for the random walk ([K], Theorem 7.6), the conditional measures a(ϕ) has to satisfy a(xϕ) = a(ϕ) p ∞ -a.s.( [KV], Theorem 3.2). The formula (16) for the density then follows from formula (6).

Identifying Φ 0 with Σ, we see that, for z ∈ Σ, σ -1 z is given by some az, where a is one of the generators. We can describe the restriction of a stationary measure to Σ j . More precisely, we have: Proposition 5.5. For each j = 1, • • • , K, there is a unique probability measure ν j such that any p-stationary measure on Σ has the restriction to Σ j proportional to ν j . Moreover, for all p ∈ P(F ), there exist χ > 0 and a neighborhood O p of p in P(F ) such that the mapping p → ν j is Lipschitz continuous from O p to Γ * χ (Σ j ).

Proof. Consider a p-stationary probability measure on Σ that has a nonzero restriction to Σ j . Let ν j be this (normalized) restriction. By ( 16), for all x such that x -1 Σ j = Σ j , we have dx * ν j dν j (z) = K π(z) (x). We shall show that there is a unique probability measure on Σ j satisfying dσ * ν j dν j (z) = K π(z) (z 0 ) and that it depends Lipschitz continuously on p as an element of Γ * χ for some suitable χ. We use thermodynamical formalism on the transitive subshift of finite type Σ j . For χ < 1 and φ ∈ Γ χ with real values, we define the transfer operator L φ on Γ χ (Σ j ) by L φ ψ(ξ) := η∈σ -1 ξ e φ(η) ψ(η).

Then, L φ is a bounded operator in Γ χ . Ruelle's transfer operator theorem (see [Bo], Theorem 1.7 and [R], Proposition 5.24) applies to L φ and there exists a number P (φ) and a linear functional N φ on Γ χ such that the operator L * φ on (Γ χ ) * satisfies L * φ N φ = e P (φ) N φ . The functional N φ extends to a probability measure on Σ j and is the only eigenvector of L * φ with that property. Moreover, φ → L φ is a real analytic map from Γ χ to the space of linear operators on Γ χ ([R], page 91). Consequently, the mapping φ → N φ is real analytic from Γ χ into the dual space Γ * χ (see e.g. [Co], Corollary 4.6). For p ∈ P(F ), define φ p (z) = ln K π(z) (z 0 ). By Propositions 4.1 and 5.2, we can choose χ such that the mapping p → φ p is Lipschitz continuous from a neighborhood O p of p in P(F ) into the space Γ χ . It follows that the mapping p → N φp is Lipschitz continuous from O p into Γ * χ . From the relation dσ * ν j dν j (z) = K π(z) (z 0 ), we know that ν j is invariant under L * φp . This shows that ν j is the only probability measure satisfying this relation, that P (φ p ) = 0 and that ν j extends N φp . Let Σ J be a minimal closed G-invariant subset of Σ. We know that Σ J is a finite union of transitive subshifts of finite type. We have Corollary 5.6. For p ∈ P(F ), there is a unique p-stationary probability measure ν J (p) on Σ J . There is a χ and a neighborhood O of p such that the mapping p → ν J (p) is Lipschitz continuous from O into Γ * χ (Σ J ). Proof. Let ν J be a p stationary measure on Σ J . We know by Proposition 5.5 that the conditional measures on the transitive subsubshifts are unique and Lipschitz continuous from O into Γ * χ (Σ k ). We have to show that the ν J (Σ k ) are well determined and Lipschitz continuous in p. Write again equation ( 16), but now for elements x ∈ G that exchange the Σ k within Σ J and write that k ν J (Σ k ) = 1. We find that the ν J (Σ k ) are given by a system of linear equations. By Proposition (4.1) and (5.5), we know that the coefficients of this linear system are Lipschitz continuous on O. We know that there is a solution, and that it is unique, since otherwise there would be a whole line of solutions, in particular one which would give ν J (Σ k ) = 0 for some k and this is impossible. Then the unique solution is Lipschitz continuous. For the escape rate part, recall that the Busemann boundary ∂ B G is made of horofunctions so that it can be identified with a G-invariant subset of Σ. Stationary measures on ∂ B G are therefore convex combinations of the ν J ′ , where J ′ are such that ν J ′ (∂ B G) = 1. Formula (4) yields ℓ S p = max J ′ x∈F Σ J ′ h(x -1 )dν J ′ (h) p(x) . By Proposition 5.3, for a fixed x ∈ F the function h(x) is in Γ χ (Σ j ) for all χ. Therefore, Corollary 5.6 implies that each one of the functions Σ J ′ h(x -1 )dν J ′ (h) is Lipschitz continuous on O. This achieves the proof of Theorem 1.1 because the function p → ℓ S p is also written as the maximum of a finite number of Lipschitz continuous functions on O.

6.

  Proof of Theorem 1.1 Choose χ small enough and O a neighbourhood of p in P(F ) such that Proposition 4.1 and Corollary 5.6 apply: the mappings p → ln K π(z) (x) and p → ν J are Lipschitz continuous from O into respectively Γ χ (Σ) and Γ * χ (Σ J ). Then, by the definition (2), the function p → h p (Σ J , ν) is Lipschitz continuous on O. By (3) and (5), the function h p is the maximum of a finite number of Lipschitz continuous functions on O; this proves the entropy part of Theorem 1.1.

This work was partially supported by NSF grant DMS-0811127.

. We can arrange each such term and write:

We are reduced to estimate:

Since, g i , h i remain in a bounded part of the D V t and α i , β i in a bounded part of the D V s , using Propositions 2.4 and 2.5, one gets a constant C such that ϑ(A

) are all smaller than Cϑ(p, p ′ ). Furthermore, using the contraction of T j and (T j ) * (Proposition 3.4) we see that:

Moreover, all products in the formula are approximations of α, f ∞ and thus are uniformly bounded away from 0. It follows that for i ≤ k/2,

and for i > k/2:

so that finally the main term of ( 15) is estimated by: