Jean-Baptiste Boyer
email: jeaboyer@math.cnrs.fr

The rate of convergence for the renewal theorem in R d

Keywords: Mathematics Subject Classification. Primary : 60K05 Renewal theorem, perturbation of operators, diophantine properties

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Abstract. Let ρ be a borelian probability measure on SL d (R). Consider the random walk (Xn) on R d \ {0} defined by ρ : for any x ∈ R d \ {0}, we set X0 = x and Xn+1 = gn+1Xn where (gn) is an iid sequence of SL d (R)-valued random variables of law ρ. Guivarc'h and Raugi proved that under an assumption on the subgroup generated by the support of ρ (strong irreducibility and proximality), this walk is transient.

In particular, this proves that if f is a compactly supported continuous function on R d , then the function Gf (x) := Ex +∞ n=0 f (Xn) is well defined for any x ∈ R d \ {0}. Guivarc'h and Le Page proved the renewal theorem in this situation : they study the possible limits of Gf at 0 and in this article, we study the rate of convergence in their renewal theorem.

To do so, we consider the family of operators (P (it)) t∈R defined for any continuous function f on the sphere S d-1 and any x ∈ S d-1 by

P (it)f (x) = SL d (R) e -it ln
X 0 = x X n+1 = g n+1 X n
where (g n ) ∈ SL d (R) N is an iid sequence of SL d (R)-valued random variables of law ρ.

In the sequel, we will say that a closed subgroup of SL d (R) acts strongly irreducibly and proximally on R d if it doesn't fix any finite union of non trivial subspaces of R d and if it contains an element which has an eigenvalue that is strictly larger than the other ones and whose eigenspace has dimension 1. We will also that a borelian probability measure on SL d (R) is strongly irreducible and proximal it it's support generates a group that has these two properties.

If ρ is strongly irreducible, proximal and has a moment of order 1 1 , then a result by Furstenberg and Kesten (see [Fur63] and [START_REF] Guivarc | Frontière de Furstenberg, propriétés de contraction et théorèmes de convergence[END_REF]) shows that, if . is a norm on R d , then, for any x ∈ R d \ {0}, (1.1) 1 n ln g n . . . g 1 x -→ λ ρ :=

G P(R d )
ln gx dρ(g)dν(x) > 0 ρ ⊗N -a.e.

where ν is a stationary measure on P(R d ) (which is unique as we will see in proposition 2.5).

In particular, this implies that the walk on R d \ {0} is transient.

Given a continuous function f on R d , such that for some α ∈ R * + , sup

x∈R d |f (x)|
x α < +∞ and sup

x∈R d x α |f (x)| < +∞,
we study the function

(1.2) x → Gf (x) := +∞ n=0 E x f (X n ) 1 i.e. G | ln g |dρ(g) is finite.
Since the walk is transient and our assumptions on f , this function is well defined and even continuous on R d \ {0} (the series converges uniformly on every compact subset of R d \ {0}) and we would like to study it's behaviour at 0. This is what we call the renewal theorem in R d by analogy to the situation in R (see for instance [Bla48]).

Guivarc'h and Le Page proved in [START_REF] Guivarc | Spectral gap properties for linear random walks and Pareto's asymptotics for affine stochastic recursions[END_REF] that if T ρ , the sub-semigroup generated by the support of ρ, fixes a non trivial convex cone in R d then, there are two stationary borelian probability measures ν 1 and ν 2 on the sphere S d-1 and the space of P -invariant continuous functions on the sphere has dimension 2, a basis being given by two non negative functions p 1 , p 2 such that p 1 + p 2 = 1 and p i | supp ν j = δ i,j where δ is Kronecker's symbol; on the other hand, if T ρ doesn't fix any non trivial convex cone in R d , then, there is a unique stationary borelian probability measure ν 1 on S d-1 and we note p 1 the constant function taking the value 1 on the sphere.

In both cases, we define an operator on the space of continuous functions on R d that decay with a polynomial rate at infinity 2 setting, for such a function f and x ∈ R d \ {0},

(1.3) Π 0 f (x) = r i=1 p i x x S d-1 +∞ x f (uy) du u dν i (y)
where, r ∈ {1, 2} is the number of T ρ -invariant closed subsets on the sphere.

They also proved the following Théorème 1.1 (Guivarc'h -Le Page in [START_REF] Guivarc | Spectral gap properties for linear random walks and Pareto's asymptotics for affine stochastic recursions[END_REF]). Let ρ be a strongly irreducible and proximal borelian probability measure on SL d (R).

lim x→0 G - 1 λ ρ Π 0 f (x) = 0
Where λ ρ , G and Π 0 are defined in equations (1.1), (1.2) and (1.3).

In particular, this theorem shows that if f is a compactly supported hölder-continuous function on R d such that f (0) = 0, then the function (G -1 λρ Π 0 )f can be extended at 0 to a continuous function on R d .

So, the continuity of Gf at 0 is equivalent to the one of Π 0 f . Thus, in the case of a unique invariant closed subset on the sphere, we have that

lim x→0 +∞ n=0 P n f (x) = 1 λ ρ +∞ 0 S d-1
f (uy)dν(y) du u and in the other case, we only have a "directional limit" : for any x ∈ R d \ {0},

lim t→0 + +∞ n=0 P n f (tx) = 1 λ ρ 2 i=1 p i x x +∞ 0 S d-1 f (uy)dν i (y) du u 2 There is α ∈ R * + such that sup x∈R d x α |f (x)| is finite.
And, in particular, the function Gf cannot be extended to a continuous function at 0 in general.

Example 1.1. If T ρ contains only matrices having positive coefficients, then it fixes the cone C of the vectors having only positive coefficients and it's opposite. Therefore, taking a positive regular function f supported in C, we see that Gf = 0 on -C whereas Gf (x) will eventually be non negative on C. Thus, we won't be able to extend it to a continuous function at 0.

We would like to compute the modulus of continuity of Gf at 0 and, to do so, we want to study the rate of convergence in Guivarc'h and Le Page's result. To simplify our study, we will only consider (G -1 λρ Π 0 )f and that will allow us to make no distinction between the number of closed invariant subsets on the sphere (and we will see in proposition 4.14 that it is more than a computational trick). Thus, we will only have to study the modulus of continuity of Π 0 f to get the one of Gf and, as we have an easy formula for Π 0 f , it will be easy to find conditions that guarantee that Gf can be extended to a continuous function at 0 and to get it's modulus of continuity.

In [START_REF] Buraczewski | On the rate of convergence in the Kesten renewal theorem[END_REF], Buraczewski, Damek and Przebinda considered the case where T ρ is (conjuguated to) a subgroup of R * + × O(d) and a diophantine condition is satisfied by the projection of ρ on R * + . They prove their result by going back to the 1-dimensional case (this is why they need this diophantine condition that is necessary in this case (see for instance [START_REF] Carlsson | Remainder term estimates of the renewal function[END_REF]) ; this hypothesis will always be satisfied in our case as wee will see in section 3).

Our study (and the one of Guivarc'h and Le Page) takes place in the opposite case where the subgroup generated by the support of ρ contains an element having a strictly dominant eigenvalue (this is our proximality assumption).

More specifically, we will prove the following Théorème 1.2. Let ρ be a strongly irreducible and proximal borelian probability measure on SL d (R) having an exponential moment 3 .

Then, for any γ > 0 small enough and any compact subset K of R d , there are C, α ∈ R such that for any continuous function f ∈ C 0,γ (R d ) supported in K and such that f (0) = 0 and for any

x ∈ R d , G - 1 λ ρ Π 0 f (x) C 1 + | ln x | α f γ
Where λ ρ , G and Π 0 are defined in equations (1.1), (1.2) and (1.3).

If one studies the linear random walk on the torus T d := R d /Z d defined by a probability measure on SL d (Z) (see for instance [BFLM11]), it appears that there are finite invariant subsets (e.g. the set {0}). If A is one of them that is also minimal, then one can identify a neighbourhood of A in the torus to a neighbourhood of {0} × A in R d × A. This is why, from now on, noting Γ ρ the subgroup of SL d (R) generated by the support of ρ, we study the renewal theorem on the product of R d and a finite Γ ρ -set A on which the walk defined by ρ is irreducible and aperiodic and we consider hölder continuous functions f on R d × A.

Remark that if Gf (x, a) = +∞ n=0 P n f (x, a) has a limit g(a) when x goes to 0, then (I d -P )g(a) = f (0, a) and so g is a solution to Poisson's equation for f restricted to A (in particular, this implies that a∈A f (0, a) = 0).

Remark also that for any f ∈ C 0 (R d × A) such that a∈A f (0, a) = 0 and any a ∈ A, +∞ n=0 P n f (0, a) is well defined since the random walk on A is irreducible and aperiodic. We modify our operator Π 0 to account of the dependence in A and we note, for any continuous function f on R d × A that decays at polynomial rate at infinity 4 , any x ∈ R d \ {0} and any a ∈ A,

(1.4) Π 0 f (x, a) = 1 |A| a ′ ∈A r i=1 p i x x S d-1 +∞ x f (uy, a ′ ) du u dν i (y)
Remark that if a∈A f (0, a) = 0 and for any a ∈ A, the function f (., a) is höldercontinuous at 0 then the limit of Π 0 f (x, a) at 0 is well defined (only radially if r = 2).

The main result of this article is the following Théorème 1.3. Let ρ be a strongly irreducible and proximal borelian probability measure on SL d (R) having an exponential moment.

Let A be a finite Γ ρ -set such that the random walk on A defined by ρ is irreducible and aperiodic.

Then, for any γ > 0 small enough, there are constants C ∈ R and α ∈ R * + such that for any function f on R d × A such that We have that for any a ∈ A,

lim x→0 G - 1 λ ρ Π 0 f (x, a) = +∞ n=0 P n f (0, a)
Moreover, for any x, y ∈ R d \ {0} and any a ∈ A,

G - 1 λ ρ Π 0 f (x, a) -G - 1 λ ρ Π 0 f (y, a) Cω 0 (x, y) α f γ 4
There is some α ∈ R * + such that sup (x,a)∈R d ×A x α |f (x)| is finite.

where Π 0 is the operator defined in equation (1.4) and where we noted, for any x, y ∈ R d \ {0},

ω 0 (x, y) = |ln x -ln y | 2 + x x -y y 2 (1 + | ln x |)(1 + | ln y |)
Remarque 1.4. The definition of the function ω 0 may seem complicate but we will see that it is a kind of conical distance on R d . We will give more details about this function in section 4.4.

Remarque 1.5. The assumption on f guarantees that lim ∞ f (., a) = 0 and that there is a constant C such that for any x, y ∈ R d and any a ∈ A,

|f (x, a) -f (y, a)| C x -y (1 + x )(1 + y ) γ
In particular, compactly supported hölder-continuous functions on R d × A satisfy this assumption. Moreover, letting y go to infinity, the equation shows that for any

x ∈ R d , |f (x)| C (1 + x ) γ
So these functions vanish at polynomial speed at infinity.

We will not only consider compactly supported functions because our assumption will become very natural after identifying R d \ {0} and S d-1 × R in chapter 4.

Remarque 1.6. As we already said, it is the continuity of Gf that interests us but it is very easy to have the one of Π 0 f . To prove this theorem, we will study an analytic family of operators (see section 4) defined on C 0,γ (S d-1 × A) for z ∈ C with |ℜ(z)| small enough, a function f ∈ C 0,γ (S d-1 × A) and some point (x, a) of S d-1 × A by P (z)f (x, a) = G e -z ln gx x f (gx, ga)dρ(g) Indeed, we will prove in section 4 that the rate of convergence in the renewal theorem is linked to the growth of the norm of (I d -P (z)) -1 along the imaginary axis.

To get a control of (I d -P (it)) -1 C 0,γ (S d-1 ×A) for large values of t, we will adapt in 2 the arguments developed by Dolgopyat in [START_REF] Dolgopyat | Prevalence of rapid mixing in hyperbolic flows[END_REF] for Ruelle operators and we will prove proposition 2.22 which links the norm of (I d -P (it)) -1 to the diophantine properties of the logarithms of the spectral radii of elements of Γ ρ .

Then, we will prove that in a strongly irreducible and proximal subgroup of SL d (R), we can construct elements for which the logarithm of the spectral radius is very well controlled. This is what we will do in section 3 and more specifically in proposition 3.19.

Proofs.

In this paragraph, we prove the results that we stated in this introduction from the ones we will prove in more general settings in the following ones.

Proof of theorem 1.2 from theorem 1.3. Let γ ∈]0, 1] and K a compact subset of R d . Then, there is a constant C 0 such that for any γ-hölder continuous function

f on R d such that supp f ⊂ K, sup x,y∈R d \{0} x =y (1 + x ) γ (1 + y ) γ |f (x) -f (y)| x -y γ C 0 f γ
We can now apply theorem 1.3 to find constants C, α such that for any γ-hölder-

continuous function f with supp f ⊂ K and any x, y ∈ R d \ {0}, G - 1 λ ρ Π 0 f (x) -G - 1 λ ρ Π 0 f (y) C f γ ω 0 (x, y) α and lim y→0 G - 1 λ ρ Π 0 f (y) = 0
But, we also have that

lim y→0 ω 0 (x, y) = 1 1 + | ln x |
and this proves theorem 1.2.

Proof of theorem 1.3.

This is a direct application of our theorem 4.1. Indeed, noting X = S d-1 × A and H = {I d , ϑ} where ϑ is the antipodal application on the sphere and identity on A, we have that H acts by isometries on X × A and (X × A)/H, that we identify with the product of the projective space and A is well (ρ, γ, M, N )-contracted over A (see example 2.4 and lemma 3.7). Moreover, in section 3, we saw that the cocycle σ defined on G × P(R d ) by σ(g, X) = ln gx x for x ∈ X \ {0} also belong to Z M (P(R d )) and the result by Furstenberg that we already gave in this introduction proves that σ ρ > 0.

Moreover, we saw in theorem 3.1 that for any t 0 ∈ R * + there are constants C, L such that for any t ∈ R with |t| t 0 ,

(I d -P (it)) -1
C|t| L

This proves that we can actually apply theorem 4.1 to any function f that satisfies the assumption of theorem 1.3 since such a function can be identified to a function f in

C γ ω (X × R) such that a∈A lim x→-∞ f (x, a) = 0 and lim x→+∞ f (x, a) = 0 by the map (x, t) → e t x from S d-1 × R to R d \ {0}.
1.3. Notations and conventions. For any continuous function f on a topological space X, we note supp f the support of f . In the same way, if ν is a borelian measure on X, we note supp ν it's support.

Moreover, we note

f ∞ = sup x∈X |f (x)|
For any complex-valued function f on a metric space (X, d) and any γ ∈ ]0, 1], we note

m γ (f ) = sup x =y |f (x) -f (y)| d(x, y) γ and f γ = f ∞ + m γ (f )
Moreover, we note C 0,γ (X) the space of γ-hölder-continuous functions on X that we endow with the norm . γ . For η ∈ R * + , we note

C η = {z ∈ C||ℜ(z)| < η} and C η = {z ∈ C||ℜ(z)| η} For any A, B ⊂ R, we note A ⊕ iB = {a + ib|a ∈ A, b ∈ B} and in particular, if A ⊂ R, then A ⊕ iR = {z ∈ C|ℜ(z) ∈ A} For f ∈ L 1 (R)
, we note f the Fourier-Laplace transform of f that is defined for any z ∈ C such that the integral is absolutely convergent by

f (z) = R f (x)e -zx dx

Unitary perturbations of Markov operators

In this section, we study the perturbation of Markov operators coming from group actions by kernels of modulus one given by cocycles. The aim is to prove proposition 2.22 that shows that if the perturbated operator has an eigenvalue close to 1, then the cocycle is close to a coboundary.

Let ρ be a borelian probability measure on R having an exponential moment and a drift λ = R ydρ(y) > 0.

In [START_REF] Carlsson | Remainder term estimates of the renewal function[END_REF], Carlsson proved that to obtain the rate of convergence for the renewal theorem, we have to find some constant l ∈ R + such that lim inf

t→±∞ |t| l 1 - R e ity dρ(y) > 0.
This condition is linked to the diophantine properties of the ρ-generic elements (see for instance [Bre05] where a slightly different but similar condition is studied).

More specifically, is such a parameter l exists, then the rate of convergence in the renewal theorem is polynomial and if we can even take l = 0 (which is always the case if ρ is spread-out as proved by Riemann-Lebesgue's lemma) then we can obtain an exponential rate of convergence (see [BG07]).

In this section, G will be a second countable locally compact group acting continuously on a compact metric space (X, d). We will fix a function σ : G × X → R (that will be a cocycle) and we will study the family of operators (P (it)) t∈R defined for any continuous function f on X and any x ∈ X by

P (it)f (x) = G e -itσ(g,x) f (gx)dρ(g)
To simplify notations, we simply note P (or sometimes P ρ to insist on the measure ρ) the operator P (0). It is clear that if G acts continuously on X, then P preserves the space of continuous functions on X.

What corresponds to the diophantine condition for measures on R will be the existence of a constant l ∈ R + such that lim sup

t→±∞ 1 |t| l (I d -P (it)) -1 is finite.
Where the norm is taken in some Banach space (the space of hölder-continuous functions in our study).

To obtain this kind of control, we adapt a theorem proved for Ruelle operators by Dolgopyat in [START_REF] Dolgopyat | Prevalence of rapid mixing in hyperbolic flows[END_REF] : this will be our proposition 2.22 which is the aim of this section.

Preliminaries.

Before we state proposition 2.22 properly, we introduce in this section a few technical notions.

2.1.1. Contracting actions. From now on, we assume that X fibers G-equivariently over a finite G-set A. This means that we have a continuous map π A : X → A that is G-equivariant : for any x in X and any g in G,

π A (gx) = gπ A (x)
Definition 2.1 (Contracting action). Let G be a second countable locally compact group, N : G → [1, +∞[ a submultiplicative function on G and let (X, d) be a compact metric space endowed with a continuous action of G.

We assume that X fibers G-equivariantly over a finite G-set A.

Let ρ be a borelian probability measure on G and γ, M ∈ R ⋆ + . We say that X is (ρ, γ, M, N )-contracted over A if

(1) For any g ∈ G and any x, y ∈ X,

d(gx, gy) M N (g) M d(x, y) (2)
(2.1)

G N (g) M γ dρ(g) is finite
(3) For some n 0 ∈ N ⋆ we have that sup

x,y∈X x =y π A (x)=π A (y) G d(gx, gy) γ d(x, y) γ dρ ⋆n 0 (g) < 1
where π A : X → A is the G-equivariant projection.

Remark 2.2. If X is (ρ, γ, M, N )-contracted over A, then P preserves the space C 0,γ (X) of γ-Holder-continuous functions on X.

Remark 2.3. This notion is used for instance by Bougerol and Lacroix in [START_REF] Bougerol | Products of random matrices with applications to Schrödinger operators[END_REF] to study random walks on the projective space but the definition with such a generality is given in [START_REF]Random walks on reductive groups[END_REF] where the reader will find more details. We could have defined N (g) as the maximum d(gx, gy)/d(x, y) (assuming that it is finite) since this is a submultiplicative function on G ; however, in our applications, there will be a natural function N associated to G.

Example 2.4. Our main example will be the case where G is a strongly irreducible and proximal subgroup of SL d (R), ρ is a borelian probability measure on G having an exponential moment and whose support generates G and X will be the product of the projective space P d (R) (which is contracted according to the theorem 2.3 in chapter V in [START_REF] Bougerol | Products of random matrices with applications to Schrödinger operators[END_REF]) and of a finite G-set A endowed with the discrete distance (for any s, s ′ ∈ A, d(s, s ′ ) = 0 if s = s ′ and 1 otherwise).

Remark that the sequence (u n ) defined for any n ∈ N by

u n = sup x,y∈X x =y π A (x)=π A (y) G d(gx, gy) γ d(x, y) γ dρ ⋆n (g) is submultiplicative. Therefore, if X is (ρ, γ, M, N )-contracted over A, then there are constants C 1 , δ ∈ R ⋆ + such that for any n ∈ N and any x, y ∈ X such that π A (x) = π A (y), (2.2)
G d(gx, gy) γ dρ ⋆n (g) C 1 e -δn d(x, y) γ
Remark also that if γ ′ ∈ ]0, γ] then the function t → t γ ′ /γ is concave on [0, Diam(X)] so if the space X is (ρ, γ, M, N )-contracted, it is also (ρ, γ ′ , M, N )-contracted.

Let X be a compact metric space and P a positive operator5 on C 0 (X). We say that the operator P is equicontinuous if it is power-bounded and if for any f ∈ C 0 (X), the sequence (P n f ) n∈N is equicontinuous. We refer to [START_REF] Raugi | Théorie spectrale d'un opérateur de transition sur un espace métrique compact[END_REF] for the properties of equicontinuous operators.

Proposition 2.5. Let G be a second countable locally compact group, N : G → [1, +∞[ a submultiplicative function on G and ρ a borelian probability measure on G.

Let (X, d) be a compact metric space endowed with a continuous action of G and which is (ρ, γ, M, N )-contracted over a finite G-set A.

Then, the operator P associated to ρ is equicontinuous on C 0 (X). Moreover, if the random walk defined by ρ on A is irreducible and aperiodic then there is a unique probability measure ν on X which is P ρ -invariant.

Finally 1 is the unique eigenvalue of P having modulus 1 and the associated eigenspace has dimension 1.

Before we prove this result, we state a lemma about Markov chains defined by group actions on finite sets.

Lemma 2.6. Let G be a second countable locally compact group acting on a finite set A and let ρ be a borelian probability measure on G such that the random walk on A defined by ρ is irreducible and aperiodic.

Then, ν A , the uniform measure on A, is the unique P ρ -stationary probability measure on A and P ρ has a spectral radius strictly smaller than 1 in the orthogonal of constant functions in L 2 (A, ν A ).

Proof. According to the theory of Markov chains on finite state spaces (or more specifically Perron-Frobénius's theorem), we only have to remark that the measure ν A is stationary.

Proof of proposition 2.5. The equicontinuity of P in the space C 0 (X) can be proved as in the case of a subgroup of SL d (R) acting on P(R d ) given in [START_REF] Benoist | Random walks on projective spaces[END_REF]. We will give more details in the proof of proposition 2.12 where the space is only locally contracted.

Let f ∈ C 0 (X) and λ a complex number of modulus 1. Assume that P f = λf . For any x, y ∈ X such that π A (x) = π A (y), we have that

λ n (f (x) -f (y)) = P n f (x) -P n f (y) = G f (gx) -f (gy)dρ ⋆n (g)
But, the space is contracted over A and |λ| = 1 so, we get that for any x, y ∈ X with

π A (x) = π A (y), f (x) = f (y).
Thus, eigenvectors of P in C 0 (X) associated to eigenvalues of modulus 1 can be identified to functions on A. As we assumed that the Markov chain defined by ρ on A is irreducible and aperiodic, we have that the only eigenvectors of P associated to eigenvalues of modulus 1 are constants (cf lemma 2.6). Using proposition 3.2 and 3.3 in [START_REF] Raugi | Théorie spectrale d'un opérateur de transition sur un espace métrique compact[END_REF], this proves that the measure ν is unique, that 1 is a simple eigenvalue and that there is no other eigenvalue of modulus 1.

We can now extend to our context the theorem 2.5 of chapter V in [START_REF] Bougerol | Products of random matrices with applications to Schrödinger operators[END_REF] that proves that, when the space is contracted, the operator P has a spectral gap in the space of hölder-continuous functions. This is the following Proposition 2.7. Let G be a second countable locally compact group, N : G → [1, +∞[ a submultiplicative function on G and ρ a borelian probability measure on G.

Let (X, d) be a compact metric space endowed with an action of G and which is (ρ, γ, M, N )-contracted over a finite G-set A on which the random walk defined by ρ is irreducible and aperiodic. Note e -κ A ∈ ]0, 1[ and C A ∈ [1, +∞[ such that for any function f on A and any n ∈ N,

P n f -f dν A ∞ C A e -κ A n f ∞
where ν A is the uniform measure on A (see lemma 2.6 for the existence of κ A , C A ).

Let ν be the unique P ρ -invariant borelian probability measure on X (given by proposition 2.5).

Then, there are constants κ, C 0 ∈ R ⋆ + that don't depend on C A and such that for any n ∈ N,

P n ρ -Π ν C 0,γ (X) C 0 C A e -κn
where we noted Π ν the operator of integration against the measure ν.

Remark 2.8. We quantify the spectral gap assumption in L ∞ (A, ν A ) since this will allow us to take a family (A i , ν i ) of G-finite G-sets on which P has a uniform spectral gap.

Remark 2.9. This proposition can be seen has a corollary of the quasicompacity of P in C 0,γ (X) that we will prove in proposition 2.12 and of the fact that, in C 0 (X), 1 is the only eigenvalue of modulus 1 and it's associated eigenspace has dimension 1. However, what interests us is the dependence between the spectral gap in L ∞ (A, ν A ) and the one in C 0,γ (X).

Proof. Let f ∈ C 0,γ (X), x, y ∈ X such that π A (x) = π A (y) and n ∈ N. Then, for any n ∈ N, we can compute

|P n f (x) -P n f (y)| m γ (f ) G d(gx, gy) γ dρ ⋆n (g) m γ (f )C 1 e -δn d(x, y) γ
where we noted C 1 , δ the constants given by equation (2.2). This proves that for any n ∈ N,

m γ (P n f ) C 1 e -δn m γ (f )
We recall that we noted ν the unique P -invariant borelian probability measure on X (given by proposition 2.5). Moreover, for any x ∈ X and any non zero integer n, we note ν x the measure defined by

ϕ(y)dν x (y) = |A| X 1 π A (x)=π A (y) ϕ(y)dν(y)
Then, for any function f ∈ C 0,γ (X), we note

f n 1 (x) = X P n f (y)dν x (y) and f n 2 (x) = P n f (x) -f n 1 (x)
Thus, for any x, y ∈ X, we have that

f n 2 (y) -m γ (f n 2 )Diam(X) f n 2 (x) f n 2 (y) + m γ (f n
2 )Diam(X) where we noted Diam(X) the diameter of X.

Therefore, integrating in the y variable and using the fact that X f n 2 (y)dν x (y) = 0, we get that

f n 2 ∞ Diam(X) γ m γ (f n 2 ) = Diam(X) γ m γ (P n f ) And as, P 2n f (x) = P n (P n f )(x) = P n (f n 2 + f n 1 )(x) = P n f n 2 (x) + P n f n 1 (x)
we also get that

P 2n f (x) - A f n 1 (a)dν A (a) f n 2 ∞ + P n f n 1 (x) - A f n 1 (a)dν A (a) Diam(X) γ C 1 e -δn m γ (f ) + C A e -κ A n P n f n 1 ∞ Diam(X) γ C 1 e -δn + C A e -κ A n f γ
Finally, using Fubini's lemma, we obtain that

A f n 1 (a)dν A (a) = X f (y)dν(y)
This last equality ends the proof of the lemma since we also have that

m γ (P n f ) C 1 e -δn m γ (f )
And so,

P 2n f -f dν γ CC 1 e -δn + C 1 e -2δn + C A e -κ A n f γ So we note κ = 1 2 min(δ, κ A ) and C 0 = (1 + C)C 1 + 1.

Fibered contracting actions.

In this paragraph, we study the case where the space is only locally contracted and we recover some results of the previous paragraph.

To study the action of SL d (R) on the sphere and not only on the projective space, the notion of contractivity of the action is not enough (since the sphere isn't contracted as x and -x stays at fixed distance). However, this is the only obstruction since if we note θ the application on the sphere that sends any point x onto -x, then it commutes to the action of G and so, noting H = {I d , θ}, we have the identification S d-1 /H ∼ P d and the projective space is (ρ, γ, M, N )-contracted (if ρ has an exponential moment and is strongly irreducible and proximal) as we already noted in example 2.4. This is why, from now on, we will consider compact metric G-spaces (X, d) endowed with an action of a finite group H that commutes to the action of G and such that the quotient (endowed with it's quotient metric) is contracted. To simplify the lecture, the reader may keep G = SL d (R), X = S d-1 , H = {I d , θ} and X/H = P d . Our first step is then to recover an equivalent of lemma 2.5 and proposition 2.7. To do so, we will use the following Theorem 2.10 (Ionescu-Tulcea and Marinescu in [START_REF] Ionescu Tulcea | Théorie ergodique pour des classes d'opérations non complètement continues[END_REF]). Let (B, . B ) be a Banach space and P a continuous operator on B.

Assume there is a norm . on B such that the identity map between the spaces (B, . B ) and (B, . ) is compact and that there are two constants r, R ∈ R + such that for any f ∈ B,

P f B r f B + R f
Then, the essential spectral radius of P in (B, . B ) is bounded by r.

Example 2.11. In our examples, (B, . B ) will be a space of hölder-continuous functions endowed with it's Banach-space norm and . will be the uniform norm.

Proposition 2.12. Let G be a second countable locally compact group, N : G → [1, +∞[ a submultiplicative function on G and ρ a borelian probability measure on G. Let (X, d) be a compact metric G-space endowed with an action of a finite group H that commutes to the one of G and such that X/H is (ρ, γ, M, N )-contracted over a finite G-set A.

Then, there are C ′ , δ ′ ∈ R ⋆ + such that for any f ∈ C 0,γ (X) and any n ∈ N,

m γ (P n f ) C ′ e -δ ′ n m γ (f ) + f ∞
In particular, P est is equicontinuous on C 0 (X) and it's spectral radius in C 0,γ (X) is strictly smaller than 1.

Proof. We do not prove this result here but later, in lemma 4.6 when the operator is perturbated by a cocycle.

Finally, we study the eigenvalues of P in C 0 (X) having modulus 1. To do so, we begin by studying the P -invariant borelian probability measures and then, we will see that, contrary to what happened when the space was contracted, there can be eigenvalues of modulus 1 and different from 1 and even non constant P -invariant functions.

This study will allow us to understand why we have to make an assumption about a cone being fixed or not in the renewal theorem.

Lemma 2.13. Let G be a second countable locally compact group, N : G → [1, +∞[ a submultiplicative function on G and ρ a borelian probability measure on G.

Let (X, d) be a compact metric G-space endowed with an action of a finite group H that commutes to the G-action and such that X/H is (ρ, γ, M, N )-contracted over a finite G-set A on which the random walk defined by ρ is irreducible and aperiodic.

Then, there are at most |H| minimal closed invariant subsets (for the action of T ρ the subsemigroup generated by the support of ρ) that we note Λ 1 , . . . , Λ r . Each one is associated to a P -invariant P -ergodic borelian probability measure ν i with supp ν i = Λ i .

Moreover, for any x ∈ X and ρ ⊗N -a.e. (g n ) ∈ G N , the sequence

1 n n-1 k=0 δ g k ...g 1 x
converges to one of the ν i and if we note, for i ∈ [1, r],

p i (x) = ρ ⊗N (g n ) 1 n n-1 k=0 δ g k ...g 1 x * ⇀ ν i
we have that the function p i is continuous, P -invariant, i p i = 1, p i = δ i,j on Λ j (where δ i,j Kronecker's symbol).

Finally, for any continuous function f on X and any x ∈ X,

1 n n-1 k=0 P k f (x) -----→ n→+∞ r i=1 p i (x) X f dν i Proof.
Let Λ be a minimal closed invariant subset (there is at least one since X is compact) and let h ∈ H. Then, hΛ is still a closed invariant subset since the actions of G and H commute. Moreover, it is also minimal since h is invertible. This proves that, HΛ is again a closed P -invariant subset. But this time, it is also H-invariant and so π H (HΛ) is an invariant subset of P seen as an operator on C 0,γ (X/H). But this closed invariant subset is unique since P is contracting on X/H and the random walk on A defined by ρ is irreducible and aperiodic (see proposition 2.5). This proves that HΛ is unique and that there are at most |H| minimal closed invariant subsets and that H acts transitively on them. We note them Λ 1 , . . . , Λ r and we note Λ their union.

We saw in proposition 2.12 that P is equicontinuous and using the propositions 3.2 and 3.3 of [START_REF] Raugi | Théorie spectrale d'un opérateur de transition sur un espace métrique compact[END_REF], we get that there are at most r continuous P -invariant functions p 1 , . . . p r forming a free family, and as each one is constant on Λ i , we can always assume that p j = δ i,j on Λ i . Thus, noting ν i the P -invariant measure on Λ i , we have that for any continuous function f ∈ C 0 (X),

lim n→+∞ 1 n n-1 k=0 P k f (x) = r i=1 p i (x) f dν i
To conclude, we only need to check that p i really is the function we defined.

First of all, the fact that for any x ∈ X, 1 n n-1 k=0 δ g k ...g 1 x converges a.e. to an ergodic measure depending continuously on x is a consequence of the equicontinuity of P and of the propositions of Raugi that we already used.

The fact that the function p i that we defined is P -invariant also comes from these propositions (see also equality 2.11 in [START_REF] Benoist | Random walks on projective spaces[END_REF]). So we can conclude using the unicity of these functions p 1 , . . . , p r . 2.1.3. Lazy random walk. Let G be a topological group. If ρ is a borelian probability measure on G, we will have to introduce the lazy random walk associated to ρ : this is the walk associated to the measure

(2.3) ρ e = 1 2 δ e + 1 2 ρ
The main interest of this measure is that the sequence (supp ρ ⋆n e ) n∈N is non decreasing. Moreover, for any λ ∈ C,

λI d -P ρe = 1 2 ((2λ -1)I d -P ρ )
and so the spectral values of P ρe and the ones of P ρ are linked (in particular, for λ = 1, we get that I d -P ρe = 1 2 (I d -P ρ )). The following lemma proves that this measure keeps other properties of ρ.

Lemma 2.14. Let G be a second countable locally compact group and ρ a borelian probability measure on G.

Let (X, d) be a compact metric G-space endowed with an action of a finite group H that commutes to the G-action and such that X/H is (ρ, γ, M, N )-contracted over a finite G-set A.

Then, X/H is also (ρ e , γ, M, N )-contracted over A.

Proof. It is clear that the first two properties are satisfied by ρ e . Moreover, for any n ∈ N, we have that

ρ ⋆n e = 1 2 n n k=0 n k ρ ⋆k
And so, for any x, y ∈ X such that x = y and π A (x) = π A (y) and any n ∈ N, we have that

G d(
G → GL(B) a representation of G such that G × B → B (g, b) → r(g)b
is continuous and G r(g) dρ(g) is finite.

We note P ρ the operator b → G r(g)(b)dρ(g).

We assume that there is an operator N 0 on B and C, κ ∈ R such that for any n ∈ N,

P n ρ -N 0 B Ce -κn .
Then, for any n ∈ N,

P n ρe -N 0 B C 1 + e -κ 2
n where P ρe is the operator associated to ρ e = 1 2 δ e + 1 2 ρ. 2.1.4. Perturbations of Markov operators by cocycles. In this paragraph, G still is a second countable locally compact group acting on a compact metric G-space (X, d) that fibers G-equivariantly over a finite G-set A and ρ a borelian probability measure on G.

We are going to study a kind of perturbation of the Markov operator associated to ρ. To do so, we make the following Definition 2.16 (Cocycles). Let G be a topological group and X a topological space endowed with a continuous action of G.

We say that a continuous function σ : G × X → R is a cocycle if for any g 1 , g 2 ∈ G and any x ∈ X, σ(g 2 g 1 , x) = σ(g 2 , g 1 x) + σ(g 1 , x) Among the cocycles, we call coboundaries the ones given by σ(g, x) = ϕ(gx) -ϕ(x) where ϕ : X → R is a continuous function.

Remark 2.17. Let σ be a cocycle. Then, the operator defined for any f ∈ C 0 (X) and any x ∈ X by

P (it)f (x) = G e -itσ(g,x) f (gx)dρ(g)
is continuous on C 0 (X) and for any f ∈ C 0 (X), any x ∈ X and any n ∈ N, we have that

P n (it)f (x) = G e -itσ(g,x) f (gx)dρ ⋆n (g) and P (it) n f ∞ f ∞
It is to have this equation that we only study cocycles and not more general functions on G × X.

As wa are going to study contracting actions (and thus holder-continuous functions) we want conditions that guarantee that P (it) preserves the space of hölder-continuous functions on X.

For a cocycle σ and g ∈ G, we note

σ sup (g) = sup x∈X |σ(g, x)
| and σ Lip (g) = sup

x,y∈X π A (x)=π A (y)

x =y |σ(g, x) -σ(g, y)| d(x, y)

Then, for any x, y ∈ X with x = y and π A (x) = π A (y), 2 γ-1 e -itσ(g,x) -e -itσ(g,y) e -itσ(g,x) -e -itσ(g,y)

γ |t| γ |σ(g, x) -σ(g, y)| γ |t| γ σ γ Lip (g)d(x, y) γ
So, for any g ∈ G, if σ Lip (g) is finite, then the function (x → e -itσ(g,x) ) is höldercontinuous. We note, for The following proposition is an extension to our context of corollary 3.21 of Guivarc'h and Le Page in [START_REF] Guivarc | Spectral gap properties for linear random walks and Pareto's asymptotics for affine stochastic recursions[END_REF].

M ∈ R + , Z M N (X) = σ is a cocycle sup g∈G σ Lip (g) N (
Proposition 2.18. Let G be a second countable locally compact group, N : G → [1, +∞[ a submultiplicative function on G and ρ a borelian probability measure on G.

Let (X, d) be a compact metric G-space endowed with an action of a finite group H that commutes to the one of G and such that X/H is (ρ, γ 0 , M, N )-contracted over a finite G-set A on which the random walk defined by ρ is irreducible and aperiodic.

Let

σ ∈ Z M N (X/H). Then there are C 2 , δ 2 ∈ R ⋆ + such that for any t ∈ R, any n ∈ N and any function f ∈ C 0,γ (X), we have that m γ (P n (it)f ) C 2 f ∞ (1 + |t|) + e -δ 2 n m γ (f )
In particular, the essential spectral radius of P (it) is smaller than e -δ 2 .

Proof. Let f ∈ C 0,γ (X) and x, y ∈ X such that x = y and π

A • π H (x) = π A • π H (y).
For any n ∈ N ⋆ , we have that

|P n (it)f (x) -P n (it)f (y)| = G e -itσ(g,x) f (gx) -e -itσ(g,y) f (gy)dρ ⋆n (g) G |f (gx) -f (gy)| dρ ⋆n (g) + f ∞ G e -itσ(g,x) -e -itσ(g,y) dρ ⋆n (g) d(x, y) γ m γ (f ) G d(gx, gy) γ d(x, y) γ dρ ⋆n (g) + f ∞ 2 1-γ |t| γ [σ] M d(x, y) γ G N γM (g)dρ ⋆n (g)
First of all, we note that as N is submultiplicative, we have that

G N (g) γM dρ ⋆n (g) G N γM (g)dρ(g) n Moreover, since H is a finite group, there is d 0 ∈ R ⋆ + such that for any x, y ∈ X, is d(x, y) d 0 , then d(x, y) = d(π H (x), π H (y)).
Thus, for any ε ∈ ]0, 1] and any x, y ∈ X such that 0 < d(x, y) εd 0 and

π A •π H (x) = π A • π H (y), we have I n (x, y) : = G d(gx, gy) γ dρ ⋆n (g) = G 1 d(gx,gy) d 0 d(gx, gy) γ + 1 d(gx,gy)>d 0 d(gx, gy) γ dρ ⋆n (g) = G 1 d(gx,gy) d 0 d(gπ H x, gπ H y) γ + 1 d(gx,gy)>d 0 d(gx, gy) γ dρ ⋆n (g) C 1 e -δn d(x, y) γ + d(x, y) γ G 1 d(gx,gy)>d 0 M γ N (g) M γ dρ ⋆n (g) C 1 e -δn + M γ G 1 M N (g) M 1/ε N (g) γM dρ ⋆n (g) d(x, y) γ Thus, if n 0 is such that C 1 e -δn 0 1/4, as G N (g) γM dρ ⋆n 0 (g) is finite, we can choose ε such that G 1 N (g) M 1/ε M γ N (g) γM dρ ⋆n 0 (g) 1/4
And so, for this choice of ε and n 0 , we have that for any x, y ∈ X such that 0 < d(x, y) εd 0 and

π A • π H (x) = π A • π H (y), G d(gx, gy) γ dρ ⋆n 0 (g) 1 2 d(x, y) γ
This proves that for any x, y ∈ X with π A • π H (x) = π A • π H (y) and d(x, y) εd 0 and any function f ∈ C 0,γ (X),

|P n 0 (it)f (x) -P n 0 (it)f (y)| d(x, y) γ 1 2 m γ (f ) + f ∞ 2 1-γ |t| γ [σ] M G N γ 0 M (g)dρ(g) n
But, as we also have, for x, y such that

π A • π H (x) = π A • π H (y) and d(x, y) εd 0 , that |P n 0 (it)f (x) -P n 0 (it)f (y)| d(x, y) γ 2 f ∞ (εd 0 ) γ
we get that for any function f ∈ C 0,γ (X),

m γ (P n 0 (it)f ) 1 2 m γ (f ) + 2 (εd 0 ) γ + 2 1-γ |t| γ [σ] M G N γ 0 M (g)dρ(g) n 0 f ∞
If we simplify the notations, what we just proves is that there is n 0 ∈ N ⋆ and a constant C ∈ R + (depending on n 0 ) such that for any f ∈ C 0,γ (X),

m γ (P n 0 (it)f ) 1 2 m γ (f ) + C(1 + |t|) f ∞
Iterating this inequality, we find that there are constants C 2 , δ 2 ∈ R ⋆ + such that for any n ∈ N and any f ∈ C 0,γ (X),

m γ (P n (it)f ) C 2 e -δ 2 n m γ (f ) + (1 + |t|) f ∞
This proves, using theorem 2.10 that P (it) has an essential spectral radius smaller than e -δ 2 and that it is equicontinuous.

2.1.5. Lower regularity of measures on metric spaces. Guivarc'h proved (cf. [START_REF] Bougerol | Products of random matrices with applications to Schrödinger operators[END_REF]) that if ρ is a strongly irreducible and proximal borelian probability measure on SL d (R) having an exponential moment, then there is a unique P -invariant probability measure ν on P(R d ). Moreover, there are ∆ + , C ∈ R ⋆ + such that for any x ∈ P(R d ) and any r ∈ R + , ν(B(x, r)) Cr ∆ + (we refer to the chapter 12 in [START_REF]Random walks on reductive groups[END_REF] for a proof of this result). This property of upper regularity of the measure means that ν is not two much concentrated at neighbourhood of points in the projective space : if, for instance, it had an atom x 0 we would have that for any ∆ + ∈ R ⋆ + , lim r→0 + ν(B(x 0 , r))/r ∆ + = +∞. Here, we will have to use the lower regularity of the measure ν : at many steps we will have to use the fact that the measure of a ball of radius r is polynomial in r. To study this, we make the following Definition 2.19. Let (X, d) be a compact metric space and ν a borelian probability measure on X.

Let ∆ ∈ R + and t, r ∈ R ⋆ + . We say that a point x ∈ X is (∆, t) -ν-regular at scale r if ν(B(x, r)) tr ∆
In the same way, we will say that a point is (∆, t) -ν-regular at any scale if

inf r∈]0,1] ν(B(x, r)) r ∆ t
Finally, we say that a point of X is ∆ -ν-regular at scale r if it is (∆, 1) -ν-regular at scale r.

Remark 2.20. If X has an Hausdorff dimension smaller than ∆ then we have (cf. [START_REF] Rudin | Real and complex analysis[END_REF]) that

ν   t∈R ⋆ + {x ∈ X|x is (∆, t) -ν -regular at any scale}   = 1
Sometimes, if π 0 : X → X 0 is a covering and ν is a probability measure on X 0 , we will say that x ∈ X is ∆ -ν-regular at scale r if π 0 (x) is.

Isotypic decomposition.

In this paragraph, we recall the isotypic decomposition that generalizes the decomposition of function on R between even and odd parts.

Let H be a finite group. For an irreducible unitary representation ξ = (ρ, V) of H, we endow End(V) of the Hilbert-Schmidt inner product defined for any A, B ∈ End(V) by

A, B HS := trA ⋆ B We note | . | HS the corresponding norm.
Let (X, d) be a compact metric space endowed with an action by isometries of H (this implies in particular that H preserves the space of hölder-continuous functions on X).

The action of H on X gives a representation of H in C 0 (X) defined for any

h ∈ H, f ∈ C 0 (X) and x ∈ X by ρ 0 (h)f (x) = f (h -1 x)
We note H a set of representatives of unitary irreducible representations of H up to isomorphism. This is a finite set. For ξ = (ρ, V) ∈ H, f ∈ C 0 (X) and x ∈ X, we note

(2.5) f (x, ξ) = dim V |H| h∈H f (h -1 x)ρ(h) ⋆
Then, we have (see theorem 8 in [START_REF] Serre | Représentations linéaires des groupes finis[END_REF]) that for any x ∈ X,

(2.6) f (x) = ξ∈ H tr f (x, ξ)
However, wee will need the following relation : for any x ∈ X and any h ∈ H,

f (hx, ξ) = dim V |H| h ′ ∈H f ((h -1 h ′ ) -1 x)ρ(h ′ ) ⋆ = f (x, ξ)ρ(h) ⋆
Thus, for any f ∈ C 0 (X), any x ∈ X, any ξ ∈ H and any h ∈ H, we have that

| f (hx, ξ)| HS = | f (x, ξ)| HS and the function (x → | f (x, ξ)| HS
) can be identified to a continuous function on X/H. The norm | . | HS allows us to define a uniform norm on bounded functions on X taking their values in End(V) : we note, for such a function f ,

f ∞ = sup x∈X |f (x)| HS
In the same way, we can define hölder-continuous functions from X to End(V) and note

C 0,γ ξ (X, End(V)) = f ∈ C 0,γ (X, End(V ξ )) ∀x ∈ X ∀h ∈ H f (hx) = f (x)ρ(h) ⋆ (
To simplify notations, we will simply note this space C 0,γ ξ (X) and sometimes C 0,γ ξ ). We now have the following Lemma 2.21. Let H be a group acting by isometries on a compact metric space (X, d).

Then, the space C 0,γ (X) injects into ξ∈ H C 0,γ ξ (X, End(V)). Moreover, for any ξ ∈ H, the projection onto C 0,γ ξ (X) is given by equation (2.5) and the reciproqual application is given by equation (2.6).

2.2. Control of the resolvant of the perturbated operator.

Statement of the proposition.

We are now ready to state proposition 2.22, aim of this section.

We keep the notations of the previous part. Let σ : G × X → R be an H-invariant cocycle. We note, for t ∈ R, P (it) (or sometimes P ρ (it) to insist on the measure ρ) the operator defined for any continuous function f on X and any x in X, by

P (it)f (x) = G e -itσ(g,x) f (gx)dρ(g) If ρ e = 1
2 δ e + 1 2 ρ, we simply note P e (it) = P ρe (it) the operator associated to the lazy random walk.

Finally, we define

[σ] M = sup g∈G sup x,y∈X x =y π A (x)=π A (y) |σ(g, x) -σ(g, y)| N (g) M d(x, y) We saw in paragraph 2.1.4 that if [σ] M is finite and if (g → N (g) M γ
) is ρ-integrable then P (it) preserves the space of γ-hölder-continuous functions on X.

The aim of this section is to study the properties of P (it) and to connect them to the ones of P . A first example is when e -itσ is a coboundary : e -itσ(g,x) = ϕ(gx)ϕ(x) -1 where ϕ is a γ-Holder continuous function taking it's values in U, the set of complex numbers of modulus 1. Indeed, in this case, P (it) is conjugated to P by the operator of multiplication by ϕ and so, these two operators have the same spectral properties. In particular, the operator P (it) has the eigenvalue 1 and an associated eigenvector is ϕ -1 .

We are going to see that we can have a partial reciprocal statement : the proposition proves that if I d -P (it) is not well invertible (the norm of it's inverse is large) then e -itσ is close to a coboundary. Proposition 2.22. Let G be a second countable locally compact group, N : G → [1, +∞[ a submultiplicative function on G and ρ a borelian probability measure on G.

Let (X, d) be a compact metric G-space endowed with an action by isometries of a finite group H that commutes to the G-action and such that the space X/H is (ρ, γ, M, N )-contracted over a finite G-set A on which the random walk defined by ρ is irreducible and aperiodic. We fix a class of representatives of unitary irreducible representations of H up to isomorphism.

Let ν be the unique P -invariant borelian probability measure on X/H (see proposition 2.5).

Then, for any γ > 0 small enough, any α 1 , β ∈ R ⋆ + , there is α 2 ∈ R + , such that for any ∆ ∈ R + such that there is a point x ∈ X/H that is ∆ -ν-regular at scale 2 -α 2 (see definition 2.19) we have that there are L, C ∈ R + such that for any t ∈ R with |t| 2, we have that if

(I d -P (it)) -1 C 0,γ (X) C|t| L then there is ξ ∈ H and a function f ∈ C 0,γ ξ (X) with f ∞ 1 and m γ (f ) C|t| such that for any point x in X whose projection on X/H is ∆ -ν-regular at scale |t| -α 2 , we have that |f (x)| HS 1 2 and G e -itσ(g,x) f (gx) -f (x) 2 HS dρ ⋆n(β,t) e (g) 1 |t| α 1
where we noted n(β, t) = ⌊β ln |t|⌋ and ρ e is the measure associated to the lazy random walk (see paragraph 2.1.3).

Remark 2.23. Note that in the conclusion of the proposition, it really is the measure ρ e that is used and not the measure ρ itself. This won't change anything in our study study since their supports generate the same subgroup of G Remark 2.24. We would want to take ϕ

(x) = 1 √ dim V
trf (x) since this will give the following inequality :

G e -itσ(g,x) ϕ(gx) -ϕ(x) 2 dρ ⋆n(β,t) e (g) 1 |t| α 1
And so, if ϕ didn't vanish on X, the theorem would imply that e -itσ were close to the coboundary ϕ(x)ϕ(gx) -1 . However, our control on |f (x)| HS doesn't give any control on |ϕ(x)|.

Proof of the proposition.

The proof relies on lemmas that are adapted from the ones of Dolgopyat for Ruelle operators.

The first difficulty comes from the fact that the space is only locally contracted over A.

To solve it, we are going to use the isotypic decomposition that we saw in lemma 2.21 and the fact that G preserves this decomposition since it's action and the H-one commutes. Moreover, we also assumed that σ is H-invariant and so, we can study the operator P (it) in each C 0,γ ξ (X). One also has to remark that there is an equivalent to proposition 2.18 in the space C 0,γ ξ (X). Proposition 2.18 suggests that we should renormalise the norm in C 0,γ (X) to study P (it). This is why, we make the following Notation. Under the assumption of proposition 2.22, we note C 2 the constant given by proposition 2.18 (we can assume without any loss of generality that C 2 1).

Let t ∈ R with |t| 2. Then, for any f ∈ C 0,γ (X), we note

. (t) = max f ∞ , m γ (f ) 2C 2 |t|
In the same way, we define the norm . (t) for functions in C 0,γ ξ (X).

Remark that for any f ∈ C 0,γ (X),

f (t) f γ (1 + 2C 2 |t|) f (t)
So (C 0,γ (X), . γ ) and (C Remark 2.26. This lemma still holds in C 0,γ ξ (X).

Proof. Let f ∈ C 0,γ such that f (t) 1. According to proposition 2.18, for any n ∈ N, we have that

m γ (P n (it)f ) 2C 2 |t| 1 2|t| 1 + |t| + 2C 2 |t|e -δ 2 n 1 + C 2 e -2δn 2C 2
Moreover, we still have that

P (it) n f ∞ f ∞ 1
and so,

P n (it) σ 2C 2
Which is what we intended to prove.

For α 2 , ∆, σ and ξ fixed, we are going to study the assumption

H(α 1 , β, ξ) : For any f ∈ C 0,γ ξ (X) with f (t) 1, there are x 0 ∈ X which is ∆ -ν- regular at scale |t| -α 2 and n ∈ [0, ⌊β ln |t|⌋] such that |P e (it) n f (x 0 )| HS 1 - 1 |t| α 1
Lemma 2.27. Under the assumptions of proposition 2.22, for any

α 1 , β ∈ R ⋆ + there is α 2 ∈ R ⋆
+ such that for any ∆ ∈ R ⋆ + , there are α ′ 1 , C, such that for any t ∈ R with |t| 2 and any ξ ∈ H, we have that if the assumption H(α 1 , β, ξ) is true, then

(I d -P (it)) -1 C 0,γ (X) C|t| α ′ 1 Proof. Let ξ ∈ H and f ∈ C 0,γ ξ (X) with f (t) 1.
By assumption, there are n ∈ [0, n(β, |t|)] and a point x 0 ∈ X whose projection on X/H is ∆ -ν-regular at scale |t| -α 2 such that

|P n e (it)f (x 0 )| HS 1 - 1 |t| α 1
We are going to prove in a first step that this control at some point x 0 can be extended to a control of the uniform norm of P m e (it)f for some m and then that this implies the expected result.

First of all, using the triangular inequality, we have that for any m, n ∈ N with m n and any x ∈ X,

|P m e (it)f (x)| HS = G e itσ(g,x) P e (it) n f (gx)dρ ⋆m-n e (g) HS G |P n e (it)f (gx)| HS dρ ⋆m-n e (g) = P m-n e |P n e (it)f | HS (x)
Moreover, since σ is H-invariant and the action of H and G commutes, we also have, by definition of C 0,γ ξ (X), that for any m ∈ N, the function |P m e (it)f | HS is H-invariant (cf. lemma 2.21). So, it passes to a function on X/H and we get, using proposition 2.7, that

P m-n e |P n e (it)f (gx)| HS (x) X/H |P n e (it)f (y)| HS dν(y) + C 0 C A e -κ(m-n) P n e (it)f γ
Moreover, using lemma 2.25, the assumption on f (t) and the fact that C 2 1, we can compute And so, we get that for any m ∈ N larger than β ln |t| and any x ∈ X,

P n e (it)f γ = P n e (it)f ∞ + m γ (P n e (it)f ) 1 + C 2 (1 + |t|) f ∞ + e -δn m γ (f ) 1 + C 2 (2|t| + 2e -δ 2 n C 2 |t|)
|P e (it) m f (x)| HS X/H |P n e (it)f (y)| HS dν(y) + 5C 0 C A C 2 2 e -κm b 1+βκ Moreover, if Z is a borelian subset of X/H and M Z = sup x∈Z |P n e (it)f (x)| HS , then, X/H |P n e (it)f (y)| HS dν(y) M Z ν(Z) + ν(Z c ) 1 + (M Z -1)ν(Z) Taking Z = B(π H (x 0 ), r) where we noted r = (1/(10C 2 2 |t| α 1 +1 )) 1/γ , we get that sup x∈Z |P n e (it)f (x)| HS |P n e (it)f (π H (x 0 ))| HS + P n e (it)f γ d(x, π H (x 0 )) γ 1 - 1 |t| α 1 + 5C 2 2 |t|r γ = 1 - 1 2|t| α 1
Taking now α 2 large enough so that |t| -α 2 (1/(5C 2 2 |t| α 1 +1 )) 1/γ , we get that |t| -α 2 r and so, as

x 0 is ∆ -ν-regular at scale |t| -α 2 , ν(Z) |t| -α 2 ∆
and so,

X/H |P n e (it)f (y)| HS dν(y) 1 - 1 2|t| α 1 +α 2 ∆
To sum-up, we found that for any m ∈ N larger than β ln |t|,

(2.7) P e (it) m f ∞ 1 - 1 2|t| α 1 +α 2 ∆ + 5C 0 C A C 2 2 e -κm |t| 1+βκ
To simplify notations, let

α 3 = α 1 + α 2 ∆. Let m = Kn(β, |t|)
for some K that we will choose later, then

P m e (it) ∞ 1 - 1 2|t| α 3 + 4C 0 C 2 2 C A |t| 1+βκ-Kκβ
and so, for K large enough (recall that |t| 2), we get that

P m e (it)f ∞ 1 - 1 4|t| α 3
Moreover, for l ∈ N larger than m, using proposition 2.18, we find that

1 2C 2 |t| m γ (P l e (it)f ) P m e (it)f ∞ + 1 2|t| e -δ(l-m) m γ (P m e (it)f ) 1 - 1 4|t| α 3 + 1 2|t| e -δl |t| Kβδ 4C 2 2 |t| 1 - 1 4b α 3 + 2C 2 2 e -δl |t| Kβδ
So, taking l = Lm = KLn(β, |t|), where L ∈ N is large enough, we get that 1 2C 2 |t| m γ (P l e (it)f ) 1 -

1 8|t| α 3
But, as we also have that

P l e (it)f ∞ P m e (it)f ∞ 1 - 1 4|t| α 3
what we proved is that under the assumptions of the theorem, is H(α 1 , β, ξ) is true, then for any f ∈ C 0,γ ξ (X) with f (t) 1,

P l e (it)f (t) 1 - 1 8|t| α 3 And so, in C 0,γ ξ (X), (I d -P l e (it)) -1 (t)
8|t| α 3 Moreover, as

(I d -P e (it)) -1 = l-1 k=0 P e (it) k (I d -P e (it) l ) -1 and 1 2 (I d -P (it)) -1 = 2(I d -P e (it)) -1 ,
we can compute

(I d -P (it)) -1 (C 0,γ ξ (X), . (t) ) 2 l-1 k=0 P k e (it) (t) (I d -P e (it) l ) -1 (t) 2C 2 l8b α 3
Finally, we recall that for any f ∈ C 0,γ ξ (X),

f (t) f γ (1 + 2C 2 |t|) f (t)
and this proves that, in C 0,γ ξ (X),

(I d -P (it)) -1 γ (1 + 2C 2 |t|)16C 2 |t| 1+α 3 l
And as l is bounded by the product of ln |t| and some constant, we get the expected result for α ′ > 1 + α 3 .

We saw in lemma 2.27 what happens if, under the assumptions of proposition 2.22, for any hölder-continuous function f on X there is a point x 0 and an integer n such that |P n e (it)f (x 0 )| HS is far from 1. We are now going to study the other alternative : the case where there is a function f such that for any x ∈ X, |P n e (it)f (x)| HS stays close to 1 for many values of n.

To explain how this works, we begin with the extreme case where we have that |f (x)| and |P e (it)f (x)| are not only close to 1 but we even have that for any x ∈ X, |P e (it)f (x)| = |f (x)| = 1. Then, for any x ∈ X, P e (it)f (x) is an average of complex numbers of modulus 1 and so, for any x ∈ X and ρ e -a.e. g ∈ G,

(2.8) e -itσ(g,x) f (gx) = P e (it)f (x)
In particular, since by definition ρ e (e) > 0, we get that for any x ∈ X,

f (x) = e -itσ(e,x) f (x) = P e (it)f (x)
This proves that for any x ∈ X and ρ e -a.e. g ∈ G,

e -itσ(g,x) = f (x) f (gx)
and so, e -itσ is a coboundary.

We quantify this argument in next

Lemma 2.28. Under the assumptions of proposition 2.22.

Let ξ ∈ H, f ∈ C 0,γ ξ (X) with f ∞ 1, x ∈ X and L ∈ N ⋆ . Let s be such that 1 -s |f (x)| HS , |P L e (it)f (x)| HS , then we have that G e -itσ(g,x) f (gx) -f (x) 2 HS dρ ⋆L e (g) 2 L+3 s
Proof. Expending the following expression, we can compute

I(x) : = G e -itσ(g,x) f (gx) -P L e (it)f (x) 2 HS dρ ⋆L e (g) = P L e |f | 2 HS (x) + |P L e (it)f (x)| 2 HS -2|P L e (it)f (x)| 2 HS 1 -|P L e (it)f (x)| 2 HS 2s Thus, ρ * L e (e) f (x) -P L e (it)f (x) 2 HS G e -itσ(g,x) f (gx) -P L e (it)f (x) 2 HS dρ ⋆L e (g) 2s
This proves, using the fact that ρ e (e) 1/2, that

f (x) -P L e (it)f (x) HS √ 2 L+1 s
And so, the triangular inequality proves that,

G e -itσ(g,x) f (gx) -f (x) 2 HS
dρ ⋆L e (g)

1/2 I(x) + f (x) -P L e (it)f (x) HS √ 2s + √ 2 L+1 s √ 2 L+3 s
And this finishes the proof of the lemma.

Lemma 2.29. Under the assumptions of proposition 2.22. For any α 1 , β there is α 2 such that for any ∆ there are α ′ 1 , β ′ such that for any σ ∈ Z M , and any ξ ∈ H, is the assumption

H(α ′ 1 , β ′ , ξ) is false then there is f ∈ C 0,γ ξ (X) with f (t) 1 such that for any x ∈ X whose projection on X/H is ∆ -ν-regular at scale |t| -α 2 , we have that |f (x)| HS 1 - 1 |t| α 1 and G e iσ(g,x) f (gx) -f (x) 2 HS dρ n(β,|t|) e (g) 1 |t| α 1
Proof. Fix α 1 , α 2 , β, ∆ and take α ′ 1 ∈ R ⋆ + large (we will precise this later in the proof). If the assumption

H(α ′ 1 , β, ξ) is not satisfied, then there is f ∈ C 0,γ ξ (X) with f (t) 1 such that for any n ∈ [0, n(β, t)] and any x ∈ X that is ∆ -ν-regular at scale |t| -α 2 , |P e (it) n f (x)| HS 1 - 1 |t| α ′ 1 Using the previous lemma, we get that G e -itσ(g,x) f (gx) -f (x) 2 HS dρ ⋆n(β,t) e 2 n(β,t)+3 |t| α ′ 1 8 |t| α ′ 1 -β ln 2
And this proves the lemma if we take α ′ 1 > β ln 2 + α 1 + 3 since we have by assumption that |t| 2. Lemmas 2.27 and 2.29 prove proposition 2.22.

Diophantine properties in linear groups

In this section, we prove that the logarithms of the spectral radii of elements of strongly irreducible and proximal subgroups of SL d (R) have good diophantine properties. We also prove a property of lower regularity of the stationary measure on the projective space by showing that it is lower regular at fixed points of proximal elements. This will allow us, in theorem 3.1, to control the operator P (it) defined in section 2 and that appears in the study of the renewal theorem in section 4.

In SL d (R), the application mapping a matrix to it's spectral radius is not a group morphism (for d 2).

In a Zariski-dense subgroup Γ of SL d (R), we can construct sequences of elements (g n ) and (h n ) for which we have a good control of the difference between the logarithm of the spectral radius of g n h n and the sum of the ones of g n and h n (see [START_REF] Quint | Groupes de Schottky et comptage[END_REF]). In particular, this proves that the logarithms of the spectral radii of proximal elements of a Zariski-dense subgroup of SL d (R) generate a dense subgroup of R.

In this section, we quantify this construction to prove a technical result that will allow us to check the assumption of proposition 2.22 and to prove theorem 3.1 that gives a control of the resolvent of the pertubated operator.

More specifically, studying the renewal in R (see [START_REF] Carlsson | Remainder term estimates of the renewal function[END_REF] and the beginning of section 2), we see that the rate of convergence depends on a diophantine condition on the measure and we are about to prove that it's equivalent in SL d (R) is always satisfied for measures having an exponential moment and whose support generate a strongly irreducible and proximal subgroup. This will be the Proposition (3.19). Let ρ be a strongly irreducible and proximal borelian probability measure on G := SL d (R) having an exponential moment.

Then, there are α, β ∈ R * + and p ∈ N * such that

lim inf b→±∞ |b| α G e ibλ 1 (g) -1 2 dρ * pn(β,b) (g) > 0
where we noted λ 1 (g) the logarithm of the spectral radius of g and

n(β, b) = ⌊β ln |b|⌋
We recall that an element g of SL d (R) is said to be proximal if it has a locally attractive fixed point in P(R d ).

We will prove the genericness of the lower regular points for the stationary measure on the projective space, other condition that we used in the study of the perturbated operator, in the Corollary (3.11). Let ρ be a strongly irreducible and proximal borelian probability measure on SL d (R) having an exponential moment.

Let ν be the unique borelian probability measure on P(R d ) (existence and uniqueness of ν are proved in proposition 2.5).

Then, for any M ∈ R * + , there are n 0 ∈ N, ∆ ∈ R and t ∈ R * + such that for any n ∈ N with n n 0 ,

ρ * n g ∈ G g is proximal and ν B V + g , e -M n e -∆M n 1 -e -tn
where V + g is the locally attractive fixed point of g in P(R d ) and we endowed P(R d ) with the usual distance (see equation (3.1)).

These two results will allow us to prove the next Théorème 3.1. Let ρ be a borelian probability measure on SL d (R) having an exponential moment and whose support generates a strongly irreducible and proximal subgroup Γ.

Let (A, ν A ) be a finite Γ-set endowed with the uniform probability measure and assume that the random walk on A defined by ρ is irreducible and aperiodic.

Let σ : Γ × S d-1 × A → R be the cocycle defined for any g ∈ Γ, x ∈ S d-1 and a ∈ A, by σ(g, (x, a)) = ln gx x and let, for any t ∈ R, P (it) be the operator defined on C 0 (S d-1 × A) by

P (it)f (x, a) = G e -itσ(g,(x,a)) f (gx, ga)dρ(g)
Then, for any γ > 0 small enough and any t 0 ∈ R * + , there are C, L ∈ R + such that for any t ∈ R with |t| t 0 ,

(I d -P (it)) -1 C 0,γ (S d-1 ×A) C|t| L
Proof. This is a direct use of lemma 3.12 and of proposition 3.19 that we can apply (with the measure ρ e = 1 2 δ e + 1 2 ρ) thanks to lemma 3.1 and lemma 3.7 (remark that we can assume without any loss of generality that p = 1 since it just changes P (it) into P p (it) and a control of (I d -P p (it)) -1 gives a control of (I d -P (it)) -1 as we already saw in the proof of lemma 2.27).

Notations and preliminaries.

We first fix the notations that we will use in this section.

3.1.1. Proximal elements of SL d (R). Let (V, . ) be a finite dimensional R-vector space endowed with an euclidian norm and an orthonormal basis (e 1 , . . . , e d ).

We define a distance on P(V) setting, for any X = Rx, Y = Ry ∈ P(V),

(3.1) d(X, Y ) = x ∧ y x y
where we extended the scalar product of V to ∧ 2 V asking the basis (e i ∧ e j ) 1 i<j l to be orthonormal. We also define a pairing between P(V) and P(V * ), setting, for

X = Rx ∈ V and Y = Rϕ ∈ V * , δ(X, Y ) := |ϕ(x)| ϕ x = inf Y ′ ∈Y ⊥ d(X, Y ′ )
where

Y ⊥ = {Y ′ = Ry ′ ∈ Y |ϕ(y ′ ) = 0}.
We refer to the chapter 9 in [START_REF]Random walks on reductive groups[END_REF] for a proof of the next lemma that shows that we really are in the context developed in section 2. Lemma 3.1. For any g ∈ G and any X, Y ∈ P(R d ),

d(gX, gY ) g 2d d(X, Y )
Moreover, there is C ∈ R such that for any X, Y ∈ P(R d ) and any g ∈ G,

|σ(g, X) -σ(g, Y )| C g C d(X, Y )
where we noted, for X = Rx ∈ P(R d ) and g ∈ G, σ(g, X) = ln gx x

So, with the notations of the previous section, we have that σ ∈ Z C N (P(R d )) where we noted N (g) = g . An element g of SL(V) is said to be proximal if it has a locally attractive fixed point in P(V). Equivalently, a proximal element is one having a unique eigenvalue of maximal modulus and whose eigenspace is a line. In this case, this eigenvalue is real. We note that g is proximal if and only if t g is proximal in V * .

If g is proximal, we note V + g ∈ P(V) the space associated to it's eigenvalue of maximal modulus and V < g ∈ P(V * ) the class defined by the g-invariant supplementary subspace of V + g (or equivalently the locally attractive fixed point of t g in V * ). In the sequel, we will always note v + g a representative of V + g in V and ϕ < g a representative of V < g in V * and we will use these representatives in a way such that our formulas will not depend on their choices.

For any g ∈ G, we note λ 1 (g), . . . , λ d (g) the logarithms of the eigenvalues of g sorted in decreasing order and counted with multiplicities. So, g is proximal if and only if λ 1 (g) > λ 2 (g). Moreover, if g is proximal, we have by definition that gv + g = ε 1 (g)e λ 1 (g) v + g for some ε 1 (g) ∈ {±1}.

For an element g ∈ SL(V), we choose a Cartan decomposition g = k g a g l g . This means that k g , l g ∈ O(V) and a g is a diagonal matrix :

a g =    κ 1 (g) 0 . . . 0 κ d (g)   
where the κ i (g) are the singular values of g, and satisfy κ 1 (g) • • • κ d (g) and are given by κ i (g) = ∧ i g ∧ i-1 g where we noted ∧ i g the endomorphism defined by the action of g in i V endowed with the scalar product induced by the one of V : this means that

∧ i g(v 1 ∧ • • • ∧ v i ) = (gv 1 ) ∧ • • • ∧ (gv i ) Moreover, we note κ 1,2 (g) = κ 2 (g) κ 1 (g) = ∧ 2 g g 2
Finally, for an element g ∈ G and a choice of a Cartan decomposition g = k g a g l g , we note

x M g = k g e 1 , X M g = Rx M g , y m g = t l g e * 1 and Y m g = Ry m g Lemma 3.2. Let V = R d endowed with an euclidian norm, g an element of SL(V), X = Rx ∈ P(V) and Y = Rϕ ∈ P(V * ).

Then, (1)

δ(X, Y m g ) gx g x δ(X, Y m g ) + κ 1,2 (g) (2) δ(X M g , Y ) t gϕ g ϕ δ(X M g , Y ) + κ 1,2 (g) (3) d(gX, X M g )δ(X, Y m g ) κ 1,2 (g) Proof.
As the norm is supposed to be euclidian, we can assume without any loss of generality that g is the diagonal matrix (κ 1 (g), . . . , κ d (g)). We write x = x 1 + x 2 with x 1 ∈ Vect(e 1 ) and x 2 ∈ Vect(e 2 , . . . , e d ).

Then,

gx 1 = κ 1 (g)x 1 and gx 2 κ 2 (g) x 2
And so, using the fact that κ 1 (g) = g , we get that

x 1 x gx g x x 1 x + κ 1,2 (g) x 2 x Finally, x 1 x = d(X, Vect(e 2 , . . . , e d )) = δ(X, Y m g )
and this proves the first inequalities.

The second ones can be proved in the same way if we work in the dual space.

Finally, the last one comes from the fact that

d(gX, X M g )δ(X, Y m g ) = gx 2 gx x 1 x κ 2 (g) κ 1 (g) = κ 1,2 (g)
In the sequel, we will have to control the Cartan decomposition of products of elements of G. To do so, we will use the following

Lemma 3.3. For any p ∈ N, p 2, any ε ∈ ]0, 1/4], any g 1 , . . . , g p ∈ G with κ 1,2 (g i ) ε 3 , δ(X M g i+1 , Y m g i ) 2ε and δ(X M g i , Y m g i+1 ) 2ε
, we have that

κ 1 (g p . . . g 1 ) ε p-1 κ 1 (g p ) . . . κ 1 (g 1 ), κ 1,2 (g p . . . g 1 ) κ 1,2 (g 1 ) . . . κ 1,2 (g p ) ε 2(p-1) and, d(X M gp...g 1 , X M gp ) κ 1,2 (g p ) ε , d(Y m gp...g 1 , Y m g 1 ) κ 1,2 (g 1 ) ε
Proof. According to lemma 3.2, we have that for any (g i ) ∈ G p and any x ∈ R d \ {0}, noting X = Rx, g p . . . g 1 x g p g p-1 . . . g 1 x δ(g p-1 . . . g 1 X, Y m gp ) g p . . . g 1 x δ(g p-1 . . . g 1 X, Y m gp ) . . . δ(X, Y m g 1 ) Moreover, taking x orthogonal to Y m g 1 , we have that for any l ∈ [1, p],

d(g l-1 . . . g 1 X, Y m g l ) l + 1 l ε
Indeed, this is true for l = 1 by assumption and by induction, we have that for any

l ∈ [1, p -1], d(g l . . . g 1 X, X M g l ) κ 1,2 (g l ) d(g l-1 . . . g 1 X, Y m g l ) l l + 1 ε 2 l l + 1 ε and so, δ(g l . . . g 1 X, Y m g l+1 ) ε 2 - l l + 1 = l + 2 l + 1 ε This proves that g p . . . g 1 x x p 2 g p . . . g 1 ε p-1 and therefore, κ 1 (g p . . . g 1 ) p 2 ε p-1 κ 1 (g p ) . . . κ 1 (g 1 )
Moreover, using the sub-multiplicativity of the function κ 1 κ 2 and that κ 1,2 (g i ) ε 3 , we find that κ 1,2 (g p . . . g 1 ) = κ 2 (g p . . . g 1 )κ 1 (g p . . . g 1 ) κ 1 (g p . . . g 1 ) 2 4 p 2 ε 2(p-1) κ 1,2 (g 1 ) . . . κ 1,2 (g p ) 4 p 2 ε p-1 κ 1,2 (g p ) Finally, still using lemma 3.2, we have that δ(X, Y m gp...g 1 ) + κ 1,2 (g p . . . g 1 )

p 2 ε p-1
and so, δ(X, Y m gp...g 1 )

p 2 ε p-1 - 4 p 2 ε p+2 = p 2 ε p-1 1 - 8 p 3 ε 3 This proves that d(g p . . . g 1 X, X M gp...g 1 ) κ 1,2 (g p . . . g 1 ) δ(X, Y m gp...g 1 ) 2 p κ 1,2 (g p ) 1 -8ε 3 /p 3 And so, d(X M gp...g 1 , X M gp ) p p + 1 κ 1,2 (g p ) ε + 2 p κ 1,2 (g p ) (1 -8ε 3 /p 3 ) κ 1,2 (g) ε p p + 1 + 2ε p(1 -8ε 3 /p 3 )
This proves the third inequality since for any ε ∈ ]0, 1/4] and p 2,

2ε p(1-8ε 3 /p 3 ) 1 p+1 .
To get the control of d(Y m gp...g 1 , Y m g 1 ), we do the same computations in the dual space.

Next lemma will allow us to, knowing the Cartan decomposition of an element g of G, prove that it is proximal and have a control on V + g and V < g .

Lemma 3.4. For any ε ∈ ]0, 1/4] and any element g in G, if κ 1,2 (g) ε 3 and δ(X M g , Y m g ) 2ε then g is proximal and

d(V + g , X M g ) κ 1,2 (g) ε , d(V < g , Y m g ) κ 1,2 (g) ε
Moreover,

e λ 1 (g) κ 1 (g)δ(X M g , Y m g ) et g | V < g 2κ 2 (g) ε
Proof. The first three inequalities come from lemma 13.14 in [START_REF]Random walks on reductive groups[END_REF].

To prove that the norm of g restricted to V < g is controlled by κ 2 (g), we remark that, according to lemma 3.2, for any x ∈ R d \ {0}, noting X = Rx, we have that

gx x κ 1 (g)δ(X, Y m g ) + κ 2 (g) But, for any X ∈ V < g , we have that δ(X, Y m g ) d(V < g , Y m g ) κ 1,2 (g) ε
This proves that for any x ∈ V < g \ {0},

gx x κ 2 (g) 1 + 1 ε 2κ 2 (g) ε
And thus we get the last expected inequality.

From now on, we note, for any g ∈ SL d (R) and X = Rx ∈ P(R d ),

(3.2) σ(g, X) = ln gx x Lemma 3.5. For any ε ∈ ]0, 1/4], and any element g in G, if κ 1,2 (g) ε 4 and d(X M g , Y m g ) 2ε, then we have that for any

X ∈ P(R d ) with δ(X, V < g ) 2ε, σ(g, X) -λ 1 (g) -ln δ(X, V < g ) δ(V + g , V < g ) 2 κ 1,2 (g) ε 3
Moreover, for any X, Y ∈ P(R d ) with δ(X, V < g ), δ(Y, V < g ) 2ε, we have that

d(gX, gY ) κ 1,2 (g) 4ε 4
Proof. Note v + g , ϕ < g such that V + g = Rv + g and V < g = Rϕ < g . Then, according to the previous lemma,

δ(V + g , V < g ) 2ε -2 κ 1,2 (g) ε 2ε(1 -ε 2 ) 3 2 ε
For any x ∈ R d , we can write

x = ϕ < g (x) ϕ < g (v + g ) v + g + x - ϕ < g (x) ϕ < g (v + g ) v +
g and so, as gv + g = ε 1 (g)e λ 1 (g) v + g , we get

gx = ε 1 (g)e λ 1 (g) ϕ < g (x) ϕ < g (v + g ) v + g + g x - ϕ < g (x) ϕ < g (v + g ) v + g But, x - ϕ < g (x) ϕ < g (v + g ) v + g ∈ V <
g and so, according to lemma 3.4,

g x - ϕ < g (x) ϕ < g (v + g ) v + g κ 2 (g) ε x - ϕ < g (x) ϕ < g (v + g ) v + g κ 2 (g) ε 2 x δ(V + g , V < g ) Thus, if x = 0, e λ 1 (g) δ(X, V < g ) δ(V + g , V < g ) (1 -u ) gx x e λ 1 (g) δ(X, V < g ) δ(V + g , V < g ) (1 + u ) with u = δ(V + g , V < g ) e λ 1 (g) δ(X, V < g ) x g x - ϕ < g (x) ϕ < g (v + g ) v + g and u 1 e λ 1 (g) δ(X, V < g ) 4κ 2 (g) ε
But, still using lemma 3.4, we have that e λ 1 (g) 2 g ε and so,

4κ 2 (g) e λ 1 (g) δ(X, V < g )ε 2κ 1,2 (g) ε 2 δ(X, V < g )
So, for X with δ(X, V < g ) 2ε, we have that

u κ 1,2 (g) ε 3 ε and ln(1 -u ) σ(g, X) -λ 1 (g) -ln δ(X, V < g ) δ(V + g , V < g ) ln(1 + u )
and we get the expected result if we use that

u min κ 1,2 (g) ε 3 , 1 2 Finally, for any X, Y ∈ P(R d ) with δ(X, V < g ), δ(Y, V < g ) 2ε, d(gX, gY ) = ∧ 2 g(x ∧ y) gx gy κ 1 (g)κ 2 (g) x y gx gy κ 1 (g)κ 2 (g) e -2λ 1 (g) 4δ(V + g , V < g ) 2 δ(X, V < g )δ(Y, V < g ) κ 1,2 (g) 4ε 4
3.1.2. Genericness of proximal elements. First, recall that if ρ is a borelian probability measure on SL d (R) having a moment of order 1 6 , then, there are λ 1 , . . . ,

λ d ∈ R, called Lyapunov exponents of ρ, such that λ 1 + • • • + λ d = 0 and for any i ∈ [1, d], 1 n ln ∧ i g n . . . g 1 -→ λ 1 + • • • + λ i ρ ⊗N -a.e.
Moreover, if the support of ρ generates a strongly irreducible and proximal subgroup, then λ 1 > λ 2 (see [START_REF] Guivarc | Frontière de Furstenberg, propriétés de contraction et théorèmes de convergence[END_REF]).

In the sequel, we will have to produce elements in the support of ρ * n having good properties. To do so, we will use the following result which is proved in the chapter 12 of [START_REF]Random walks on reductive groups[END_REF].

Lemma 3.6. Let ρ be a borelian probability measure on SL d (R) having an exponential moment and whose support generate a strongly irreducible and proximal subgroup. Then, for any ε ∈ R * + , there are t ∈ R * + and n 0 ∈ N such that for any n ∈ N with n n 0 , we have that for any x ∈ P(R d ) and any y ∈ P((R d ) * ),

ρ * n g ∈ G ∀i ∈ [1, d], 1 n κ i (g) -λ i ε 1 -e -tn ρ * n g ∈ G δ(x, y m g ) 2e -εn 1 -e -tn ρ * n g ∈ G d(gx, x M g ) e -(λ 1 -λ 2 -ε)n 1 -e -tn ρ * n g ∈ G δ(x M g , y) 2e -εn 1 -e -tn ρ * n g ∈ G δ(gx, y) 2e -εn 1 -e -tn ρ * n g ∈ G δ(x M g , y m g ) 2e -εn 1 -e -tn
Moreover, we have, with lemma 3.1, the Lemma 3.7. Let ρ be a borelian probability measure on G having an exponential moment and whose support generates a strongly irreducible and proximal subgroup of G. Note, for g ∈ G, N (g) := g . Then, there is

γ ∈ R * + such that P(R d ) is (ρ, γ, 2d, N )-contracted.
Proof. We refer to [START_REF] Bougerol | Products of random matrices with applications to Schrödinger operators[END_REF] for a proof of this result.

6

G ln g dρ(g) is finite.

A regularity lemma for convolution powers of measures.

In this paragraph, we prove a technical lemma about regularity of powers of convolution of measures on SL d (R).

With the language we used in the previous section, this means that for a measure on SL d (R) having an exponential moment, there is a parameter ∆ such that for any n large enough, any point of supp ρ * n is ∆-regular at scale e -t 2 n except on a set of exponentially small measure.

Lemma 3.8. Let ρ be a borelian probability measure on SL d (R) having an exponential moment.

Then, for any t 1 , t 2 ∈ R * + , there are n 0 ∈ N and t 3 ∈ R * + such that for any n ∈ N with n n 0 , we have that

ρ * n g ∈ G ρ * n B(g, e -t 2 n ) e -t 3 n 1 -e -t 1 n
Proof. Let ε ∈ R * + be such that G g ε dρ(g) is finite and fix a n ∈ N. Using Markov's inequality we have that for any M ∈ R + ,

ρ * n g ∈ G g e M n e -εM n G g ε dρ * n (g) e -εM G g ε dρ(g) n So, noting Ω n = g ∈ G g e M n
, we have that

ρ * n Ω c n e -εM G g ε dρ(g) n
Moreover, there is a constant C(d) depending only on the dimension d and g 1 , . . . g L ∈ Ω n such that

Ω n ⊂ L i=1 B(g i , e -t 2 n /2)
and L C(d)e (M +t 2 )d 2 n . Now, we note, for K ∈ R * + ,

G n = g ∈ {g 1 , . . . , g L } ρ * n (B(g, e -t 2 n /2)) e -Kn and Ω n = g∈Gn B(g, e -t 2 n /2)

Then, for h ∈ Ω n , there is g ∈ G n such that d(g, h) e -t 2 n /2 and so, B(h, e -t 2 n ) ⊃ B(g, e -t 2 n /2) and so,

ρ * n (B(h, e -t 2 n )) ρ * n (B(g, e -t 2 n /2)) e -Kn
Finally, as ρ is a probability measure we have that

1 = ρ * n Ω c n + ρ * n (Ω n ) + ρ * n Ω n \ Ω n
But, by definition,

ρ * n Ω n \ Ω n Le -Kn
And so, for any n ∈ N,

ρ * n (Ω n ) 1 -e -εM G g ε dρ(g) n -Le -Kn 1 -e -εM G g ε dρ(g) n -C(d)e -Kn e (M +t 2 )d 2 n
And this proves the lemma since we can choose K and M as large as we want.

3.2. Regular points in the projective space.

In this paragraph, we study the lower regularity of the stationary measure on the projective space at fixed points of generic proximal elements.

In our study of the perturbation of Markov operators on compact spaces (section 2), we used an assumption of lower regularity on the points we considered and it is time to prove that this assumption holds for the walk on the projective space.

We remind that given a compact metric space (X, d) endowed with a borelian probability measure ν, we say that a point x ∈ X is ∆ -ν-regular at scale r where r ∈ R * + and ∆ ∈ R + , if ν(B(x, r)) r ∆ We are going to prove that for a strongly irreducible and proximal subgroup of SL d (R), the fixed points of proximal elements are regular.

First of all, we have the following Lemma 3.9. Let ρ be a borelian probability measure on G whose support generates a strongly irreducible and proximal subsemigroup T ρ .

Then, there is a unique borelian P -invariant probability measure ν on P(R d ).

Moreover, for any proximal g ∈ T ρ , we have that V + g ∈ supp ν. Proof. Existence and unicity of ν come from [START_REF] Guivarc | Frontière de Furstenberg, propriétés de contraction et théorèmes de convergence[END_REF].

To prove the end of the lemma, note that for any proximal g ∈ T ρ , there is X ∈ supp ν such that X ∈ V < g . Indeed, if not, we would have some proximal g ∈ T ρ such that supp ν ⊂ V < g but this is impossible since we assumed that T ρ is strongly irreducible. Moreover, for any proximal g ∈ T ρ and any X ∈ V < g , we have that g n X -→ V + g and, as supp ν is closed and T ρ -invariant, this proves that V + g ∈ supp ν. We are now ready to prove the regularity of the stationary measure in the Proposition 3.10. Let ρ be a strongly irreducible and proximal borelian probability measure on SL d (R), having an exponential moment.

Let ν be the unique stationary borelian probability measure on P(R d ).

Then, for any m ∈ N * and any ε ∈ R * + small enough we have that for any g ∈ G with κ 1,2 (g) ε 4 and δ(X M g , Y m g ) 2ε, g is proximal and

ν(B(V + g , (κ 1,2 (g)/ε 4 ) m ) 1 2 (ρ * n (B(g, r))) m
where we noted r = κ 1 (g) -(1+d)m

Proof. Let ε ∈ ]0, 1/4] and g ∈ G such that κ 1,2 (g) ε 3 and δ(X M g , Y m g ) 2ε. According to lemma 3.4, g is proximal. Moreover, using the ρ-stationarity of the measure ν we have that for any m ∈ N,

ν(B(V + g , κ 1,2 (g) m )) = X 1 B(V + g ,κ 1,2 (g) m ) (X)dν(X) = X G 1 B(V + g ,κ 1,2 (g) m ) (hX)dρ * mn (h)dν(X) = X ρ * mn h hX ∈ B(V + g , κ 1,2 (g) m ) dν(X) X 1 δ(X,Y m g ) 2ε ρ * mn h hX ∈ B(V + g , κ 1,2 (g) m ) dν(X)
But, we saw in lemma 3.5 that if

X ∈ P(R d ) is such that δ(X, Y m g ) 2ε, then d(g m X, V + g ) = d(g m X, g m V + g ) κ 1,2 (g) 4ε 4 d(g m-1 X, V + g ) κ 1,2 (g) 4ε 4
m where we used that gV + g = V + g . Moreover, if r ∈ ]0, 1] then for any h 1 , . . . , h m ∈ B(g, r) we have that

g m -h 1 . . . h m g -h 1 g m-1 + h 1 g m-1 -h 2 . . . h m m(2 g ) m-1 r
and so,

d(h 1 . . . h m X, V + g ) g m -h 1 . . . h m + d(g m X, V + g ) m(2 g ) m-1 r + κ 1,2 (g) 4ε 4
m Therefore, taking r = κ 1 (g) -(1+d)m and using that κ 1 (g) = g , κ 2 (g) κ 1 (g) 1-d (since κ 1 (g) . . . κ d (g) = 1 and κ 1 (g) κ 2 (g)

• • • κ d (g)), κ 1,2 (g) = κ 2 (g)/κ 1 (g), we get that 1 m κ 1,2 (g) 8 g ε 4 m κ 1 (g) -(1+d)m 1 m 1 8ε 4 m
and we can assume that ε is small enough so that for any m ∈ N * , 1 m (8ε 4 ) -m 1. In this way, we get that for any h 1 , . . . , h m ∈ B(g, r),

d(h m . . . h 1 X, V + g ) 2 κ 1,2 (g) 4ε 4 m κ 1,2 (g) ε 4 m Thus, for any X ∈ P(R d ) such that δ(X, Y m g ) 2ε, ρ * mn h hX ∈ B(V + g , (κ 1,2 (g)/ε 4 ) m ) (ρ * n (B (g, r))) m
This proves that for any g ∈ G with δ(X M g , Y m g ) 2ε and κ 1,2 (g) ε 3 , we have that

ν(B(V + g , (κ 1,2 (g)/ε 4 ) m )) (ρ * n B(g, r)) m ν X ∈ P(R d ) δ(X, Y m g ) 2ε (ρ * n B(g, r)) m (1 -(2ε) c )
where we used the upper regularity of the measure ν (see chapter 12 in [START_REF]Random walks on reductive groups[END_REF]) to have that for some constant c and ε small enough, ν(X|δ(X, Y m g ) 2ε) 1 -(2ε) c . Corollary 3.11. Let ρ be a strongly irreducible and proximal borelian probability measure on SL d (R) having an exponential.

Let ν be the unique P -invariant borelian probability measure on P(R d ).

Then, for any M ∈ N * , there are n 0 ∈ N, ∆ ∈ R and t ∈ R * + such that for any n ∈ N with n n 0 ,

ρ * n g ∈ G g is proximal and ν B V + g , e -M n e -∆M n 1 -e -tn
Proof. Let m ∈ N, 0 < ε < (λ 1 -λ 2 )/4 and λ = λ 1 -λ 2 -3ε. According to proposition 3.10, we have that for any n large enough and any g ∈ G such that κ 1,2 (g) e -λn and δ(X M g , Y m g ) 2e -εn , ν(B(g, e -mεn )) 1 2 ρ * n (B(g, e -(λ 1 +ε)(1+d)mn ))

m
In particular, taking m M/ε, we have that

ν(B(g, e -M n )) 1 2 ρ * n (B(g, e -(λ 1 +ε)(1+d)mn )) m
To conclude, we use the fact that such elements g with δ(X M g , Y m g ) 2e -εn and κ 1,2 (g) e -λn are generic according to lemma 3.6 and so are the elements for which we have a good lower bound on the measure of B(g, e -(λ 1 +ε)(1+d)mn ) according to lemma 3.8. Now, we use this regularity property to pass from a condition in proposition 2.22 where the action of G on X plays a role to a condition on G. Lemma 3.12. Let ρ be a borelian probability measure on G having an exponential moment and whose support generates a strongly irreducible and proximal subgroup.

Let X = S d-1 × A and σ : G × P(R d ) → R the cocycle defined for any g ∈ G and X = Rx ∈ P(R d ) by σ(g, X) = ln gx x that we extend to X noting for any (x, a) ∈ X, σ(g, (x, a)) = σ(g, Rx).

Let ν be the unique P ρ -invariant borelian probability measure on X/H where we noted H = {I d , ϑ} where ϑ is the involution map on X that is the antipodal application on S d-1 and the identity on A (existence and unicity of ν are given in proposition 2.5).

For any α, t 0 , t 2 , t 3 , β, ∆ ∈ R * + , there are α 2 , C, L ∈ R * + such that for any t ∈ R with |t| t 0 , if

(I d -P (it)) -1 C 0,γ (S d-1 ×A) C|t| L Then, G 1 g is proximal and V + g is ∆-ν-regular at scale |t| -α 2 e -2i|A|tλ 1 (g) -1 2 dρ * n(β,t) (g) 1 |t| α
where we noted n(β, t) = ⌊β ln |t|⌋.

Proof. We can assume without any loss of generality that t 0 = 2 (otherwise we set σ ′ = t 0 σ/2 and t ′ = 2t/t 0 ).

Note that as H is isomorphic to Z/2Z, it's irreducible unitary representations are 1-dimensional and the decomposition of lemma 2.21 is just the decomposition between even and odd parts.

Applying proposition 2.22, for any α 1 , β, ∆ ∈ R * + , there are α 2 , C, L such that if

(I d -P (it)) -1 C 0,γ (S d-1 ×A) C|t| L
then there are ξ ∈ H and a function f ∈ C 0,γ ξ (X) with f ∞ C and m γ (f ) C|t| such that for any point x in X whose projection on X/H is ∆ -ν-regular at scale |t| -α 2 , we have

|f (x)| 1 2 and G e -itσ(g,x) f (gx) -f (x) 2 dρ ⋆n(β,t) e (g) 1 |t| α 1
where we set n(β, t) = ⌊β ln |t|⌋ and ρ e is the measure associated to the lazy random walk (see paragraph 2.1.3).

To prove the proposition, we are going to make the point x in the integral depend on the point g chosen with the measure ρ * n(β,t) e

to take x = V + g since this in this way we will have that gx = x and σ(g, x) = λ 1 (g).

To do so, we will first get a control for any point x and any g except on a set of exponentially small measure, choose the point x we want and then integrate the result. The cost of this operation is passing from α to α 1 .

We note n = n(β, t) and for t 2 , t 3 ∈ R * + whose value will be fixed later, we note

G n = g ∈ G g is proximal, V + g is ∆ -ν-regular at scale |t| -α 2 and ρ * n (B(g, e -t 2 n )) e -t 3 n
Then, for any g ∈ G n and any x that is ∆ -ν-regular at scale

|t| -α 2 , G 1 h∈B(g,e -t 2 n ) e -itσ(h,x) f (hx) -f (x) 2 dρ * n (h) 1 |t| α 1
Using the triangular inequality, we get that

I b (x) : = ρ * n (B(g, e -t 2 n )) e -itσ(g,x) f (gx) -f (x) B(g,e -t 2 n ) e -itσ(h,x) f (hx) -f (x) 2 dρ * n (h) 1/2 + B(g,e -t 2 n ) e -itσ(g,x) f (gx) -e -itσ(h,x) f (hx) 2 dρ * n (h) 1/2 1 |t| α 1 /2 + C ′ |t|e -t 2 γn ρ * n (B(g, e -t 2 n ))
where we use that for any h ∈ B(g, e -t 2 n ), we have that e -itσ(g,x) f (gx) -e -itσ(h,x) f (hx) e -itσ(h,x) -e -itσ(g,x) |f (gx

)| + |f (hx) -f (gx)| 2|t| γ C |σ(g, x) -σ(h, x)| γ + m γ (f )d(hx, gx) γ C ′ |t|e -t 2 γn
For some constant C ′ depending on σ (for more details, we refer to the computations in the proof of proposition 2.18). Thus, we get that for any g ∈ G n and any x that is ∆ -ν-regular at scale |t| -α 2 ,

e -itσ(g,x) f (gx) -f (x) e t 3 n/2 |t| α 1 /2 + C ′ |t|e -t 2 γn
Note now, for any x = (u, a) ∈ S d-1 × A,

ϕ(x) = a∈A f (u, a)f (-u, a)
This function is H-invariant and moreover, we have that for any x whose projection on the projective space is ∆ -ν-regular at scale |t| -α 2 , e -2it|A|σ(g,x) ϕ(gx) -ϕ(x)

a∈A h∈H e -itσ(g,x) ϕ(ghx) -ϕ(hx)

2|A| e t 3 n/2 |t| α 1 /2 + C ′ |t|e -t 2 γn
But, we can now use that, by definition of G n , V + g is ∆ -ν-regular at scale |t| -α 2 and so, we get that for any g ∈ G n ,

e -2it|A|λ 1 (g) -1 |ϕ(V + g )| = e -2it|A|λ 1 (g) ϕ 1 (V + g ) -ϕ 1 (V + g ) 2|A| e t 3 n/2 |t| α 1 /2 + C ′ |t|e -t 2 γn
where we identified ϕ with a function on the projective space. This proves, using that |ϕ(V + g )| 2 -2|A| , that for any g ∈ G n ,

e -2it|A|λ 1 (g) -1 2 1+2|A| |A| e t 3 n/2 |t| α 1 /2 + C ′ |t|e -t 2 γn
Thus, we get that

G 1 Gn (g) e -2i|A|tλ 1 (g) -1 2 dρ * n(β,t) e (g) 2 1+2|A| |A| e t 3 n/2 |t| α 1 /2 + C ′ |t|e -t 2 γn
Moreover, we note C A = 2 1+2|A| |A| and

G 1 n := g ∈ G g is proximal, V + g is ∆ -ν-regular at scale |t| -α 2 (g) 1 |t| α
and this is what we intended to prove.

Diophantine properties of the lengths of translations.

In this paragraph, we prove that for strongly irreducible and proximal measures on SL d (R), the logarithms of the spectral radii of generic elements satisfy a diophantine condition of the same kind of the one used by Carlsson in [START_REF] Carlsson | Remainder term estimates of the renewal function[END_REF] in the study of the walk on R.

Let ρ be a borelian probability measure on G = SL d (R). For g ∈ G, we note λ 1 (g) the logarithm of the spectral radius of g. As in the study of the renewal theorem on R, the study of the renewal theorem on R d requires a diophantine assumption. However, we are going to prove that it is always satisfied for measures whose support generate a strongly irreducible and proximal subgroup.

First of all, we are going to get an estimation of the difference between λ 1 (gh) and λ 1 (g) + λ 1 (h) when g and h are proximal elements in generic position. Lemma 3.13. There are c 1 , c 2 ∈ R * + such that for any ε ∈ ]0, c 1 ] and any g, h ∈ G with κ 1,2 (g), κ 1,2 (h)

ε 4 , δ(X M g , Y m g ) 2ε, δ(X M h , Y m h ) 2ε, δ(X M g , Y m h ) 2ε and δ(X M h , Y m g ) 2ε, we have that λ 1 (g) + λ 1 (h) -λ 1 (gh) -ln δ(V + h , V < h )δ(V + g , V < g ) δ(V + g , V < h )δ(V + h , V < g ) c 2 κ 1,2 (g) ε 3 + κ 1,2 (h) ε 3
Remark 3.14. According to lemma 3.6, there are many pairs of elements g and h that satisfy the assumption of the lemma in the support of ρ * n .

Proof. We take at first c 1 = 1/4 and c 2 = 1. First, according to lemma 3.3, we have that

κ 1,2 (gh) κ 1,2 (g)κ 1,2 (h) ε 2 , d(X M gh , X M g ) κ 1,2 (g) ε and d(Y m gh , Y m h ) κ 1,2 (h) ε and so, κ 1,2 (gh) ε 4 and δ(X M gh , Y m gh ) 2ε(1 -ε) 3 2 ε
We note ε ′ = 3 4 ε and so we have that d(X M gh , Y m gh ) 2ε ′ and κ 1,2 (gh)

1 4 4 3 3 ε ′3 ε ′3 .
Thus, using lemma 3.4, gh is proximal and

d(V + gh , V + g ) d(V + gh , X M gh ) + d(X M gh , X M g ) + d(X M g , V + g ) κ 1,2 (gh) ε ′ + 2κ 1,2 (g) ε
In the same way we get

d(V < gh , V < h ) d(V < gh , Y m gh ) + d(Y m gh , Y m h ) + d(Y m h , V < h ) κ 1,2 (gh) ε ′ + 2 κ 1,2 (h) ε 4 3 ε 3 + 2ε 2 3ε 2
and so,

δ(V + h , V < gh ) 2ε -3ε 2 = 2ε(1 -3ε/2) ε 2 So we note ε ′′ = ε
4 and then, δ(V + h , V < gh ) 2ε ′′ and κ 1,2 (gh) 64c 1 ε ′′3 so, assuming that c 1 1 64 , according to lemma 3.5, we have that

σ(gh, V + h ) -λ 1 (gh) -ln δ(V + h , V < gh ) δ(V + gh , V < gh ) 2 κ 1,2 (gh) ε ′′3 2 κ 1,2 (h)ε ε ′′3
Moreover, using the cocycle relation and the fact that, by definition of

V + h , σ(h, V + h ) = λ 1 (h), we get that σ(gh, V + h ) = σ(g, hV + h ) + σ(h, V + h ) = σ(g, V + h ) + λ 1 (h)
and finally, we also have, still with lemma 3.5, that

σ(g, V + h ) -λ 1 (g) -ln δ(V + h , V < g ) δ(V + g , V < g ) 2 κ 1,2 (g) ε 3
This leads to

λ 1 (g) + λ 1 (h) -λ 1 (gh) -ln δ(V + h , V < gh )δ(V + g , V < g ) δ(V + gh , V < gh )δ(V + h , V < g ) 2 κ 1,2 (g) ε 3 + 2 7 κ 1,2 (h) ε 2
To conclude, remark that

δ(V + h , V < gh ) -δ(V + h , V < h ) d(V < h , V < gh ) 3ε 2 and δ(V + h , V < gh )
ε 2 so for ε small enough, we have that for some constant c 3 ,

ln δ(V + h , V < gh ) δ(V + h , V < h ) c 3 ε d(V < h , V < gh ) c 3 ε κ 1,2 (gh) ε ′ + 2 κ 1,2 (h) ε We can control | ln δ(V + gh , V < gh ) -ln δ(V + g , V < h )|
with the same kind of ideas and get the expected result.

In the sequel, we will need a result about the continuity of the Cartan decomposition. This will be the following Lemma 3.15. Let g ∈ G.

Then, for any h ∈ G,

d X M h , X M g (2 g -h + κ 1,2 (g)) , and d Y m h , Y m g (2 g -h + κ 1,2 (g))
Moreover, there are constants c 1 , c 2 ∈ R * + such that for any p ∈ N * , any ε ∈ ]0, c 1 ], any g in G with κ 1,2 (g) ε 3 and δ(X M g , Y m g ) 2ε, any r ∈ R * + with r ε 2 κ 1,2 (g)

ε 2 p and any element f ∈ B(g p , r), δ(X M f , Y m f ) ε, d(X M f , X M g ) 2 κ 1,2 (g) ε , d(Y m f , Y m g ) 2 κ 1,2 (g) ε κ 1 (f ) 1 2 ε p-1 κ 1 (g) p , κ 1,2 (f ) 16 ε 2(p-1) κ 1,2 (g) p d(V + f , V + g ) c 2 κ 1,2 (g) p ε 2p-1 et d(V < f , V < g ) c 2 κ 1,2 (g) p ε 2p-1
Proof. We are going to evaluate g t l h e 1 in two ways. First,

κ 1 (h)x M h = h t l h e 1 = g t l h e 1 + (h -g) t l h e 1 = κ 1 (g) t l h e 1 , t l g e 1 x M g + u + (h -g) t l h e 1 for some u such that u κ 2 (g). So, κ 1 (h) x M h -t l h e 1 , t l g e 1 x M g |κ 1 (g) -κ 1 (h)| + u + g -h κ 2 (g) + 2 g -h and as x M h = 1 = x M g , we can deduce that d(X M g , X M h ) = x M g ∧ x M h = x M h -t l h e 1 , t l g e 1 x M g ∧ x M g κ 1,2 (g) + 2 g -h g
This proves the first part of the lemma since for any g ∈ G, g = κ 1 (g) 1.

To get the control of d(Y m g , Y m h ), we do the same computations in the dual space. To prove the end of the lemma, we note that according to lemma 3.3, we have that

κ 1 (g p ) ε p-1 κ 1 (g) p , κ 1,2 (g p ) 1 ε 2(p-1) κ 1,2 (g) p and d(X M g p , X M g ) κ 1,2 (g) ε , d(Y m g p , Y m g ) κ 1,2 (g) ε Thus, for f ∈ B(g p , r), δ(X M f , Y m f ) 2ε -d(X M g p , X M g ) -d(Y m g p , Y m g ) -d(X M f , X M g p ) -d(Y m f , Y m g p ) 2ε -2 κ 1,2 (g) ε -2 (2r + κ 1,2 (g p )) ε(2 -6ε)
and so, for maybe some smaller c 1 , we have that

δ(X M f , Y m f ) ε Moreover, κ 1 (f ) κ 1 (g p ) -r ε p-1 κ 1 (g) p -ε p+2 ε p-1 κ 1 (g) p 1 - ε 3 κ 1 (g) p
And using that κ 1 (g) 1 and ε c 1 , we get that if c 1 is small enough,

κ 1 (f ) ε p-1 κ 1 (g) p
Moreover, using the equality κ 1 (g)κ 2 (g) = κ 1 (∧ 2 g), we have that

κ 1 (f )κ 2 (f ) κ 1 (g p )κ 2 (g p ) + ∧ 2 g p -∧ 2 f κ 1 (g p )κ 2 (g p ) + (κ 1 (g p ) + κ 1 (f )) f -g p so, κ 1,2 (f ) κ 1 (g p )κ 2 (g p ) κ 1 (f ) 2 + κ 1 (g p ) + κ 1 (f ) κ 1 (f ) 2 g p -f 16 ε 2(p-1) κ 1,2 (g) p 16ε p+2
Therefore, is c 1 is small enough, we get that κ 1,2 (f ) (ε/2) 3 , and so, using to lemma 3.4 we find that for any f ∈ B(g p , r), f is proximal and

d(V + f , X M f ) 2 κ 1,2 (f ) ε et d(V < f , Y m f ) 2 κ 1,2 (f ) ε
Thus, using that V + g p = V + g , we find that

d(V + f , V + g ) d(V + f , X M f ) + d(X M f , X M g p ) + d(X M g p , V + g ) c 2 κ 1,2 (g) p ε 2p-1 for some universal constant c 2 .
Working in the same way in the dual space, we obtain the inequalities for d(V < f , V < g ) and finish the proof of the lemma.

We are now ready to evaluate the difference between the logarithms of the spectral radii of well chosen elements of G. We would like to take two proximal elements g and h and study the elements g p gh on side and g p and gh on the other side (as in [START_REF] Quint | Groupes de Schottky et comptage[END_REF]). However, as we don't want to study only purely atomic measures on SL d (R), we have to take, not g p but an element f in a small neighbourhood. This is what we do in next Lemma 3.16. There are constants c 1 , c 2 , c 3 such that for any p ∈ N * , any ε ∈ ]0, c 1 ], any g ∈ G with κ 1,2 (g) ε 5 and δ(X M g , Y m g ) 2ε we have that for any h ∈ G with

κ 1,2 (h) ε 3 , δ(X M h , Y m h ) 2ε, δ(X M h , Y m g ) 2ε, δ(X M g , Y m h ) 2ε and any f ∈ G such that g p -f ε 2 κ 1,2 (g) ε p
, we have that

λ 1 (f gh) -λ 1 (f ) -λ 1 (gh) -ln δ(V + g , V < g )δ(gV + h , V < h ) δ(V + g , V < h )δ(gV + h , V < g ) c 2 κ 1,2 (g) p ε 2p + κ 1,2 (h) ε
Moreover, we note π g the projection onto V < g parallel to V + g and we have that if

X M h = V + g , d(gπ g X M h , Y m h ) 2ε, d(X M g , X M h ) 2ε and κ 1,2 (h)κ 1 (g) 1 2 ε 3 , then ε 3 c 3 κ 1 (g) d ln δ(V + g , V < g )δ(gV + h , V < h ) δ(V + g , V < h )δ(gV + h , V < g ) c 3 κ 1,2 (g) ε 5
Proof. We want to apply lemma 3.13 tof and gh. To do so, we are going to prove at first that gh is proximal.

According to lemma 3.3, we have that

κ 1,2 (gh) κ 1,2 (g)κ 1,2 (h) ε 2 ε 4 and d(X M gh , X M g ) κ 1,2 (g) ε and d(Y m gh , Y m h ) κ 1,2 (h) ε This proves that δ(X M gh , Y m gh ) δ(X M g , Y m h ) -d(X M gh , X M g ) -d(Y m h , Y m gh ) 2ε(1 -ε) 3 2 ε
So, for c 1 small enough, gh satisfy the assumptions of lemma 3.4 and 3.5 with ε ′ = ε/2. So, gh is proximal and

d(V + gh , gV + h ) = d(ghV + gh , ghV + h ) κ 1,2 (gh) 4ε ′4 c 2 κ 1,2 (h) ε and d(V < gh , V < h ) d(V < gh , Y m gh ) + d(Y m gh , Y m h ) + d(V < h , Y m h ) 2 κ 1,2 (h) ε + κ 1,2 (gh) ε Moreover, according to lemma 3.15, for f ∈ G such that g p -f ε 2 κ 1,2 (g) ε p , we have that δ(X M f , Y m f ) ε, d(X M f , X M g ) 2 κ 1,2 (g) ε , d(Y m f , Y m g ) 2 κ 1,2 (g) ε Moreover, κ 1 (f ) 1 2 ε p-1 κ 1 (g) p , κ 1,2 (f ) 16 ε 2(p-1) κ 1,2 (g) p and d(V + f , V + g ) c 2 κ 1,2 (g) p ε 2p-1 et d(V < f , V < g ) c 2 κ 1,2 (g) p ε 2p-1 Finally, δ(X M f , Y m gh ) δ(X M g , Y m h ) -d(X M f , X M g ) -d(Y m h , Y m gh ) ε 2 -2 κ 1,2 (g) ε - κ 1,2 (h) ε ε 2 -3ε 2 So, is c 1 is small enough, we have that δ(X M f , Y m gh ) ε and also, δ(X m gh , Y m g ) ε.
Thus, according to lemma 3.13,

λ 1 (f ) + λ 1 (gh) -λ 1 (f gh) -ln δ(V + gh , V < gh )δ(V + f , V < f ) δ(V + f , V < gh )δ(V + gh , V < f ) c 2 κ 1,2 (f ) ε 2 + κ 1,2 (gh) ε 2
This finishes the proof of the first part of the lemma since we also have seen controls on

d(V + gh , gV + h ), d(V + f , V + g ), d(V < f , V < g ) and d(V < gh , V < h ).
To prove the second part, we use the equality

δ(V + g , V < g )δ(gV + h , V < h ) δ(V + g , V < h )δ(gV + h , V < g ) = ϕ < g (v + g )ϕ < h (gv + h ) ϕ < h (v + g )ϕ < g (gv + h ) = 1 + ϕ < g (v + g )ϕ < h (gv + h ) -ϕ < h (v + g )ϕ < g (gv + h ) ϕ < h (v + g )ϕ < g (gv + h ) (3.3)
And, as

v + h = ϕ < g (v + h ) ϕ < g (v + g ) v + g + v + h - ϕ < g (v + h ) ϕ < g (v + g ) v + g = ϕ < g (v + h ) ϕ < g (v + g ) v + g + π g (v + h )
where we noted π g the projection on V < g parallel to V + g , we have that

ϕ < g (gv + h ) = ε 1 (g)e λ 1 (g) ϕ < g (v + h ) and ϕ < h (gv + h ) = ε 1 (g)e λ 1 (g) ϕ < g (v + h ) ϕ < g (v + g ) ϕ < h (v + g ) + ϕ < h gπ g (v + h )
This proves that

ϕ < g (v + g )ϕ < h (gv + h ) -ϕ < h (v + g )ϕ < g (gv + h ) = ϕ < g (v + g )ϕ < h (gπ g v + h
) And so, replacing in equation (3.3), we get that

δ(V + g , V < g )δ(gV + h , V < h ) δ(V + g , V < h )δ(gV + h , V < g ) = 1 + ε 1 (g)e -λ 1 (g) ϕ < g (v + g )ϕ < h (gπ g v + h ) ϕ < h (v + g )ϕ < g (v + h ) Moreover, we use that π g v + h ∈ V < g
, to get, with lemma 3.4 that

gπ g v + h κ 2 (g) ε π g v + h κ 2 (g) ε 2 δ(V + g , V < g ) v + h 2κ 2 (g) ε 2 v + h
and so, using that e λ 1 (g) 2κ 1 (g)ε, we find

e -λ 1 (g) ϕ < g (v + g )ϕ < h (gπ g v + h ) ϕ < h (v + g )ϕ < g (v + h ) κ 1,2 (g) ε 3 ϕ < g v + g ϕ < h v + h |ϕ < h (v + g )ϕ < g (v + h )| κ 1,2 (g) 5ε 4
We can also compute

ϕ < g (v + g )ϕ < h (gπ g v + h ) ϕ < h (v + g )ϕ < g (v + h ) = δ(V + g , V < g )δ(gπ g V + h , V < h ) ϕ < g v + g ϕ < h gπ g v + h |ϕ < h (v + g )ϕ < g (v + h )| δ(V + g , V < g )δ(gπ g V + h , V < h ) gπ g v + h v + h δ(V + g , V < g )δ(gπ g V + h , V < h ) 1 g -1 π g v + h v + h (3.4) And finally, as v + g ∧ v + h = v + g ∧ π g v + h , we get that d(V + g , V + h ) = v + g ∧ v + h v + g v + h = v + g ∧ π g v + h v + g v + h π g v + h v + h
and so, using inequality (3.4), this proves that

ϕ < g (v + g )ϕ < h (gπ g v + h ) ϕ < h (v + g )ϕ < g (v + h ) 1 g -1 δ(V + g , V < g )δ(gπ g V + h , V < h )d(V + g , V + h )
To conclude, note that κ d (g) . . . κ 1 (g) = det(g) = 1 and κ d (g) = g -1 -1 and so

g -1 -1 κ 1 (g) 1-d . Moreover, δ(gπ g V + h , V < h ) δ(gπ g x M h , y m h ) -d(gπ g V + h , gπ g x M h ) -d(V < h , y m h ) 2ε -g π g d(V + h , x M h ) -d(V < h , y m h ) 2ε -2 κ 1 (g) ε κ 1,2 (h) ε 2ε 1 - κ 1 (g)κ 1,2 (h) ε 3 ε And d(V + g , V + h ) d(X M g , X M h ) -d(V + g , X M g ) -d(V + h , X M h ) 2ε - κ 1,2 (g) ε - κ 1,2 (h) ε ε
and so, using the fact that e -λ 1 (g) κ 1 (g) -1 , we find that

e -λ 1 (g) ϕ < g (v + g )ϕ < h (gπ g v + h ) ϕ < h (v + g )ϕ < g (v + h ) ε 3 κ 1 (g) d
And this is what we intended to prove. Lemma 3.16 proved that if we make good assumptions of proximality and transversality on the elements g and h in G, then we have a good control on λ 1 (f gh)-λ 1 (f )-λ 1 (gh) for f in a small neighbourhood of g p . Using lemma 3.6 we will get that those elements g and h are generic and this leads to the following Lemma 3.17. Let ρ be a borelian probability measure on SL d (R) having an exponential moment and whose support generates a strongly irreducible and proximal subgroup.

Let ν be the unique borelian stationary measure on P(R d ).

Then, there are n 0 , p

∈ N et c 1 , c 2 ∈ R * + such that for any α 2 ∈ R * + , there are ∆, c 3 ∈ R * + such that for any n ∈ N with n n 0 , ρ * pn ⊗ ρ * pn      g, h
g, h and gh are proximal, V + g , V + h and V + gh are ∆ -ν-regular at scale e -α 2 pn and e

-c 1 n |λ 1 (gh) -λ 1 (g) -λ 1 (h)| e -c 2 n      e -c 3 n Proof. We note, for n ∈ N and η, t 2 ∈ R * + , G n = g ∈ G ρ * n (B(g, e -t 2 n )) e -t 3 n , δ(X M g , Y m g ) 2e -ηn ∀i ∈ {1, 2} 1 n κ i (g) -λ i η Then, for any g ∈ G n , κ 1,2 (g) = κ 2 (g) κ 1 (g) e -(λ 1 -λ 2 -2η)n
And so, noting ε = e -ηn , we have that if 7η < λ 1 -λ 2 , for n large enough, g satisfy the assumptions of lemma 3.16. Moreover, for g ∈ G n and p ∈ N * , we note

H p n (g) = h ∈ G pn δ(X M h , Y m g ) 2e -ηn , δ(X M g , Y m h ) 2e -ηn d(gπ g X M h , Y m h ) 2e -ηn , d(X M g , X M h ) 2e -ηn
where π g is the projection to V < g parallel to V + g . If p is such that (p -1)(λ 1 -λ 2 + η) > λ 1 + η, then for any g ∈ G n and any h ∈ H p n (g), the pair (g, h) satisfy the assumptions of lemma 3.16 and so we have that for any f ∈ B(g p , e -p(λ 1 -λ 2 -η)n ),

λ 1 (f gh) -λ 1 (f ) -λ 1 (gh) -ln d(V + g , V < g )d(gV + h , V < h ) d(V + g , V < h )d(gV + h , V < g ) 2c 2 e -p(λ 1 -λ 2 -4η)n and e -(d(λ 1 +η)+5η)n c 3 ln d(V + g , V < g )d(gV + h , V < h ) d(V + g , V < h )d(gV + h , V < g ) c 3 e -(λ 1 -λ 2 -7η)n
Moreover, according to lemma 3.3, we also have that

κ 1,2 (gh) κ 1,2 (g)κ 1,2 (h) ε 2 e -(p+1)(λ 1 -λ 2 -4η)n and δ(X M gh , Y m gh ) δ(X M g , Y m h ) -d(X M gh , X M g ) -d(Y M gh , Y M h ) 2e -ηn -2e -(λ 1 -λ 2 -3η)n
So, according to proposition 3.10, choosing m such that m(λ 1 -λ 2 -4η) α 2 , we get that

ν(B(V + gh , e -pα 2 n ) ν(B(V + gh , (κ 1,2 (gh)/ε 4 ) m ) 1 2 (ρ * pn (B(gh, r))) m
With r = e -(λ 1 +η)(d+1)pn . Moreover, if g 1 ∈ B(g, r 1 ) and h 1 ∈ B(h, r 2 ), then

gh -g 1 h 1 r 1 h + r 2 g 1
and so,

ρ * pn (B(gh, r)) ρ * n (B(g, e -2(λ 1 +η)(d+1)n ))ρ * (p-1)n (B(h, e -2(λ 1 +η)(d+1)n ))
Thus, taking t 2 = 2(λ 1 + η)(d + 1), this proves that V + gh is ∆ -ν-regular at scale e -α 2 pn for some ∆ ∈ R + .

Doing the same for V + f and V + f gh (using lemma 3.13 and lemma 3.3), we get that V + f and V + f gh are also ∆ -ν-regular at scale e -α 2 pn (maybe for some different ∆ but there is one that works for V + gh , V + f and V + f gh ).

We just proved that for p such that p(λ 1 -λ 2 -4η) > dλ 1 + 5η,

ρ * pn ⊗ ρ * pn      g, h
g, h and gh are proximal, V + g , V + h and V + gh are ∆ -ν-regular at scale e -α 2 pn and e

-c 1 n |λ 1 (gh) -λ 1 (g) -λ 1 (h)| e -c 2 n      G 1 Gn (g)ρ * (p-1)n (H p-1 n )(g)ρ * pn (B(g p , e -p(λ 1 -λ 2 +η)n ))dρ * n (g)
But, according to lemma 3.6, we have that any h ∈ H p-1 n (g) except a set of exponentially small measure (to get the lower bound on δ(gπ g X M h , Y m h ), we do as in the proof of inequality 13.38 in lemma 13.13 in [START_REF]Random walks on reductive groups[END_REF] to get that X M h and Y M h are essentially independent and then guarantee that

X M h ∈ V + g and d(V < g , Y m h ) 2e -ηn
). So, we only need to find a lower bound on

G 1 Gn (g)ρ * pn (B(g p , e -p(λ 1 -λ 2 +η)n ))dρ * n (g)
But, doing the same as previously, we find that, maybe for some other constant t 2 , there is a constant c 3 such that for any g ∈ G n , ρ * pn (B(g p , e -p(λ 1 -λ 2 +η)n )) e -c 3 n And finally, using lemma 3.8 and lemma 3.6, we have that every g belong to G n except a set of exponentially small measure and this finishes the proof of the lemma.

We can now state an "integrated version" of lemma 3.17. Lemma 3.18. Let ρ be a strongly irreducible and proximal borelian probability measure on SL d (R) having an exponential moment.

There are α, β ∈ R * + and p ∈ N * such that for any

α 2 ∈ R * + , there is ∆ ∈ R * + such that lim inf b→±∞ |b| α G ∆,2 (b -α 2 ) e ib(λ 1 (gh)-λ 1 (g)-λ 1 (h)) -1 2 dρ * pn(β,b) (g)dρ * pn(β,b) (h) > 0
Where we noted, n(β, b) = ⌊β ln |b|⌋ and, for r ∈ R * + , G ∆,2 (r) := (g, h) ∈ G 2 g, h and gh are proximal and

V + g , V + h , V + gh are ∆ -ν-regular at scale r
Proof. We note, with the same notations as in the proof of the previous lemma, for any n ∈ N,

G 2 n :=    g, h
g, h and gh are proximal, V + g , V + h and V + gh are ∆ -ν-regular at scale e -α 2 pn and e

-c 1 n |λ 1 (gh) -λ 1 (g) -λ 1 (h)| e -c 2 n    We choose α, β ∈ R *
+ and we will fix their value later. We note p the parameter given by the previous lemma. For b ∈ R, we note n = ⌊β ln |b|⌋.

Then, for b large enough and any (g,

h) ∈ G 2 n , |b| 1-c 1 β |b||λ 1 (gh) -λ 1 (g) -λ 1 (g)| |b| 1-c 2 β So, if β > 1/c 2 , we use the inequalities 0 < inf x∈[-1,1] e ix -1 |x| sup x∈[-1,1] e ix -1 |x| < +∞
to get that for large enough b and any (g, h) ∈ G n ,

e ib(λ 1 (gh)-λ 1 (g)-λ 1 (g)) -1 ≍ |b||λ 1 (gh) -λ 1 (g) -λ 1 (g)| |b| 1-c 1 β
So, for b large enough and uniformly for any (g, h) ∈ G 2 n ,

e ib(λ 1 (gh)-λ 1 (g)-λ 1 (g)) -1 |b| 1-c 1 β
Moreover, according to lemma 3.17,

ρ * pn(β,b) ⊗ ρ * pn(β,b) (G 2 n ) e -c 3 n(β,b) |b| -c 3 β So, if α is such that α -c 3 β + 2(1 -c 1 β) > 0, then lim inf b→±∞ |b| α G ∆,2 (b -α 2 ) e ib(λ 1 (gh)-λ 1 (g)-λ 1 (h)) -1 2 dρ * pn(β,b) (g)dρ * pn(β,b) (h) > 0
and this is what we intended to prove.

We can finally prove the control on the logarithm of the spectral radii of proximal elements of SL d (R).

Proposition 3.19. Let ρ be a strongly irreducible and proximal borelian probability measure on SL d (R) having an exponential moment.

Then, there are α, β ∈ R * + and p ∈ N * such that for any

α 2 ∈ R * + , there is ∆ ∈ R + such that lim inf b→±∞ |b| α G 1 g is proximal and V + g is ∆-ν-regular at scale b -α 2 e ibλ 1 (g) -1 2 dρ * pn(β,b) (g) > 0
where we noted λ 1 (g) the spectral radius of g and n(β, b) = ⌊β ln |b|⌋.

Proof. Note, for r ∈ R * + , G ∆ (r) := g ∈ G g is proximal and V + g is ∆ -ν-regular at scale r Suppose that there are no such α, β, p. Then, for any α, β, p,

lim inf b→±∞ |b| α G 1 G ∆ (b -α 2 ) (g) e ibλ 1 (g) -1 2 dρ * pn(β,b) (g) = 0
In particular, for any α, β ∈ R * + and any p ∈ N,

lim inf b→±∞ |b| α G 2 1 G ∆ (b -α 2 ) (gh) e ibλ 1 (gh) -1 2 dρ * pn(β,b) (g)dρ * pn(β,b) (h) = lim inf b→±∞ |b| α G e ibλ 1 (g) -1 2 dρ * 2pn(β,b) (g) = 0
But, using the triangular inequality and keeping the notation G ∆ 2 of the previous lemma, we obtain that

I b (β) : = G ∆,2 (b -α 2 ) e ib(λ 1 (gh)-λ 1 (g)-λ 1 (h)) -1 2 dρ * pn(β,b) (g)dρ * pn(β,b) (h) 1/2 G 1 G ∆ (b -α 2 ) (g) e ibλ 1 (g) -1 2 dρ * 2pn(β,b) (g) 1/2 + 2 G 1 G ∆ (b -α 2 ) (g) e ibλ 1 (g) -1 2 dρ * pn(β,b) (g) 1/2 So, for any α, β, lim inf b→±∞ |b| α I b (β) = 0
and this contradicts lemma 3.18.

4. The renewal theorem In this section, we link, as we said in the introduction, the rate of convergence for the renewal theorem to the control of (I d -P (it)) -1 in some Banach space. The aim is to prove theorem 4.1 that we need to establish theorem 1.3. 4.1. Preliminaries. Given a second countable locally compact group G acting continuously on a metric space X and a cocycle σ : G × X → R (see definition 2.16), we can define an action of G on X × R by setting g.(x, t) = (gx, t + σ(g, x))

If ρ is a borelian probability measure on G, this defines a Markov chain on X × R whose associated operator is the one defined for any continuous function f on X × R and any (x, t) ∈ X × R by

P f (x, t) = G f (g.(x, t))dρ(g) = G f (gx, t + σ(g, x))dρ(g)
This operator commutes to translations on R and this implies that for any f ∈ L ∞ (X×R) and any g ∈ L 1 (R), (P f ) * g = P (f * g)

where we noted, for f ∈ L ∞ (X × R), g ∈ L 1 (R) and (x, t) ∈ X × R, f * g(x, t) = R f (x, u)g(t -u)du
We call renewal kernel the operator G = +∞ n=0 P n when it is defined. Kesten studied in [START_REF] Kesten | Renewal theory for functionals of a markov chain with general state space[END_REF] the properties of G in a very general setting and Guivarc'h and Le Page used his result in [START_REF] Guivarc | Spectral gap properties for linear random walks and Pareto's asymptotics for affine stochastic recursions[END_REF] to get the renewal theorem in R d associated to a borelian probability measure on SL d (R) : this is theorem 1.1 that we stated in the introduction.

In this chapter, we would like to study the rate of convergence in Kesten's theorem but we will not do it in his general setting but directly in the case of a group contracting some compact metric space.

So, in the sequel, we fix a second countable locally compact group G and a borelian probability measure ρ on G.

Let (X, d) be a compact metric G-space endowed with an action of a finite group H that commutes to the G-action and such that X/H is (ρ, γ, M, N )-contracted over a finite G-set A on which the random walk defined by ρ is irreducible and aperiodic (see section 2). To simplify notations, we simply note π A the two projections of X to A and of X/H to A.

We recall that for us,

G = SL d (R), ρ is a probability measure on SL d (Z), X = S d-1 × A, X/H = P(R d ) × A and A is finite Γ ρ -set.
For technical reasons that we will justify in section 4.4, we introduce the function

ω : (X × R) 2 → R + defined by ω((x, t), (x ′ , t ′ )) = e -|t|+|t ′ | 2 d(x, x ′ ) 2 + e (t-t ′ )/2 -e (t ′ -t)/2 2 if π A (x) = π A (x ′ ) 1 if not
and we set, for γ ∈ ]0, 1],

C 0,γ ω =      f ∈ C 0 (X × R) f γ,ω := sup (x,t),(x ′ ,t ′ )∈X×R (x,t) =(x ′ ,t ′ ) |f (x, t) -f (x ′ , t ′ )| ω((x, t), (x ′ , t ′ )) γ is finite     
In the same way, we note,

ω 0 ((x, t), (x ′ , t ′ )) = √ |t-t ′ | 2 +d(x,x ′ ) 2 (1+|t ′ |)(1+|t|) si π A (x) = π A (x ′ ) 1
si non and we define C 0,γ ω 0 (X × R) like C 0,γ ω (X × R). We will see in section 4.4 that for any f in C 0,γ ω , there are functions p + (f ), p -(f ) on A such that for any x ∈ X,

p -(f )(π A (x)) = lim t→-∞ f (x, t) and p + (f )(π A (x)) = lim t→+∞ f (x, t)
We recall that for a cocycle σ on X and t ∈ R, we note P (it) the operator defined for any continuous function f on X and any x ∈ X by

P (it)f (x) = G e -itσ(g,x) f (gx)dρ(g)
We refer to section 2 and more specifically to paragraph 2.1.4 for more details.

The main result of this section is the following Theorem 4.1. Let G be a second countable locally compact group, N : G → [1, +∞[ a submultiplicative function on G and ρ a borelian probability measure on G. Let X be a compact metric G-space endowed with an action of a finite group H that commutes to the G action and such that X/H is (ρ, γ 0 , M, N )-contracted over a finite G-set A on which the random walk defined by ρ is irreducible and aperiodic.

Let σ ∈ Z M (X/H) and σ ρ = G X/H σ(g, x)dν(x)dρ(g) where ν is the unique P -invariant probability measure on X/H given by proposition 2.5. We assume that σ ρ > 0.

We also assume that there is γ 0 ∈ ]0, 1] such that for any γ ∈ ]0, γ 0 ] and t 0 ∈ R * + , there are C 0 , L such that for any t ∈ R with |t| t 0 ,

(I d -P (it)) -1 C 0,γ (X) C 0 |t| L We note Π 0 the operator define for any f ∈ C γ ω (X × R) such that p + (f ) = 0 par Π 0 f (x, t) = +∞ t N 0 f (x, u)du
where N 0 is the projector on the space of P -invariant function in C 0 (X) and we make the abuse of notations

N 0 f (x, u) = N 0 f u (x) with f u = f (., u).
Then, for any γ > 0 small enough, there are α, C ∈ R * + such that for any

f ∈ C γ ω (X × R) with p + (f ) = 0 = a∈A p -(f )(a) and for any x ∈ X, lim t→-∞ G - 1 σ ρ Π 0 f (x, t) = n∈N P n p -(f )(π A (x)) Moreover, (G -1 σρ Π 0 )f ∈ C α ω 0 (X × R) and G - 1 σ ρ Π 0 f α,ω 0 C f γ,ω
Remark 4.2. The function Gf is well defined under the assumptions of the theorem since the convergence in the series is uniform on any compact subset of X × R.

Proof. To prove the theorem, we use the decomposition given in lemma 4.19, the corollary 4.21 and lemma 4.22 4.2. Non-unitary perturbations by cocycles.

In this paragraph, we study the inverse of the operator I d -P (z) and we prove proposition 4.3 that shows that a control of the growth of the norm on the imaginary axis gives a control of the norm of the operator and it's derivatives on a neighbourhood with a nice shape.

Let G be a second countable locally compact group acting on a compact metric space (X, d) and ρ a borelian probability measure on G.

For a cocycle σ : G × X → R and g ∈ G, we recall that we noted We note C γ := {z ∈ C||ℜ(z)| < γ}. For z ∈ C γ and σ ∈ Z M N (X/H), we define the operator P (z) on C 0 (X) by

P (z)f (x) = G e -zσ(g,x) f (gx)dρ(g)
This is a continuous operator since for any continuous function f on X, any z ∈ C γ and any x ∈ X,

|P (z)f (x)| f ∞ G e -ℜ(z)σ(g,x) dρ(g) f ∞ G [σ] γ ∞ N (g) γM dρ(g)
and G N (g) γM dρ(g) is finite since the action is contracting.

In the sequel, we will note, for η ∈ R * + , C η := {z ∈ C||ℜ(z)| < η}.

We group the main results of this paragraph in the next Proposition 4.3. Under the assumptions and notations of theorem 4.1.

For any γ > 0 small enough, there are η, C, L, t ∈ R * + such that (z → P (z)) is an analytic function from C η to the space of continuous operators on C 0,γ (X). Moreover, for any z ∈ [0, η[⊕iR and any n ∈ N,

P (z) n γ C(1 + |z|)e -tℜ(z)n
Finally, noting

U (z) = (I d -P (z)) -1 - 1 σ ρ z N 0 ,
we have that (z → U (z)) (which is definite a priori on iR \ {0}) can be extended to an analytic function onto the space of continuous operators on C 0,γ (X) and defined on

D η,C,L := z ∈ C -1 C(1 + |ℑz|) L+1 < ℜ(z) < η
and for any n ∈ N and any z ∈ D η,C,L

U (n) (z) γ n!C n+1 (1 + |ℑz|) (L+1)(n+1)
Remark 4.4. This proposition generalises the situation in R when P (z) is the Fourier-Laplace transform of the measure ρ. In this case, the same estimations can be obtained under the "non-lattice of type p" assumption used by Carlsson in [START_REF] Carlsson | Remainder term estimates of the renewal function[END_REF].

Before we prove each of the assertions of the proposition, we draw the zone Proof. First of all, according to Jensen's inequality, for any 0 s η, G e -sσ(g,x) dρ * n (g) G e -ησ(g,x) dρ * n (g)

D η,C,L . O η -1/C z ∈ C -1 C(1+|ℑz|) L+1 < ℜ(z) < η
s/η = (P (η) n 1(x)) s/η
Moreover, as σ ρ > 0, lemma 10.17 in [START_REF]Random walks on reductive groups[END_REF] and the fact that (with their notations but reminding that their operator P ϑ is what we called P (-iϑ)) and their equality 10.32,

λ ′ (0) = X P ′ (0)1dν(x) = -σ ρ < 0
prove that there are η, t, C ∈ R * + such that

sup x P (η) n 1(x) Ce -tn
and this is what we intended to prove.

Lemma 4.6. Under the assumptions of proposition 4.3, for any γ > 0 small enough there is η ∈ R * + such that the function (z → P (z)) is analytic from C η to the space of continuous operators on C 0,γ (X) and there are t ∈ R * + and C ∈ R such that for any z ∈ C η with ℜ(z) 0, any function f ∈ C 0,γ (X) and any n ∈ N,

P (z) n f γ C e -tn m γ (f ) + (1 + |z|) f ∞ Remark 4.7.
This proof is very close to the one of proposition 2.18. The difficulty here is that the perturbation no longer have modulus 1.

Proof. To see that (P (z)) is an analytic family of operators, we refer to lemma 10.16 de [START_REF]Random walks on reductive groups[END_REF].

Let γ > 0 small enough and η ∈ R * + .

Compute, for z (g,x) f (gx) -e -zσ(g,y) f (gy) dρ * n (g) G e -ℜ(z)σ(g,x) |f (gx) -f (gy)| + f ∞ e -zσ(g,x) -e -zσ(g,y) dρ * n (g) (4.1) But, using the definition of [σ] M and [σ] ∞ (cf. equation (2.4)), we have that for any ε ∈ R * + e -zσ(g,x) -e -zσ(g,y)

∈ C η , f ∈ C 0,γ (X), x, y ∈ X with π A • π H (x) = π A • π H (y) and n ∈ N, |P (z) n f (x) -P (z) n f (y)| G e -zσ
2 1-γ |z|e ℜ(z)σsup(g) (σ sup (g))

1-γ |σ(g, x) -σ(g, y)| γ 2 1-γ |z|[σ] η ∞ N (g) M (γ+|ℜ(z)|) ln([σ] ∞ N (g) M ) 1-γ [σ] γ M d(x, y) γ C ε |z|N (g) M (γ+η+ε) [σ] ∞ [σ] M d(x, y) γ
where we noted C ε such that for any x ∈ [1, +∞[, x C ε e εx /2 1-γ .

And so, (4.2) G e -zσ(g,x) -e -zσ(g,y) dρ * n (g)

C ε |z|[σ] ∞ [σ] M d(x, y) γ G N (g) M (γ+η+ε) dρ * n (g) Moreover, (4.3) G e -ℜ(z)σ(g,x) |f (gx) -f (gy)|dρ * n (g) m γ (f )[σ] η ∞ G N (g) M η d(gx, gy) γ dρ * n (g) Let d 0 ∈ R * + be such that if d(x, y) d 0 then d(x, y) = d(π H x, π H y).
Then, for any ε ′ ∈]0, 1] and any x, y with 0 < d(x, y) ε ′ d 0 , we have that

I n (x, y) : = G N (g) M η d(gx, gy) γ dρ * n (g) G N (g) M η 1 d(gx,gy) d 0 d(gx, gy) γ dρ * n (g) + G N (g) M η 1 d(gx,gy) d 0 d(gx, gy) γ dρ * n (g) G N (g) M η d(gπ H x, gπ H y) γ dρ * n (g) + G N (g) M η 1 M N (g) M 1/ε M N (g) M γ d(x, y) γ dρ * n (g) d(x, y) γ G N (g) M η d(gπ H x, gπ H y) γ d(π H x, π H y) γ dρ * n (g) +M G N (g) M (η+γ) 1 M N (g) M 1/ε dρ * n (g) (4.4)
But, using Cauchy-Schwartz's inequality and the contraction of X/H, we have that

J n (x, y) : = G N (g) M η d(gπ H x, gπ H y) γ d(π H x, π H y) γ dρ * n (g) G N (g) 2M η dρ * n (g) G d(gπ H x, gπ H y) 2γ d(x, y) 2γ dρ * n (g) 1/2 C 2γ e -δ 2γ n/2 G N (g) 2ηM dρ(g) n/2 (4.5)
We can now choose n such that C 2γ e -δ 2γ n/4 1/4[σ] ∞ and then, for η such that

e -δ 2γ n/4 G N (g) M η dρ(g) 1,
we find, with equation (4.5), that

J n (x, y) = G N (g) M η d(gπ H x, gπ H y) γ d(π H x, π H y) γ dρ * n (g) 1 4[σ] ∞ Moreover, for this fixed n we can choose ε ′ ∈ R * + such that M G N (g) M (η+γ) 1 M N (g) M 1/ε ′ dρ * n (g) 1/4[σ] ∞
and this proves, with equations (4.1), (4.2), (4.3) and (4.4), that for any f ∈ C 0,γ (X), any x, y ∈ X with

π A • π H (x) = π A • π H (y) and d(x, y) ε ′ d 0 , |P (z) n f (x) -P (z) n f (y)| d(x, y) γ 1 2 m γ (f ) + 2 f ∞ |z|[σ] ∞ [σ] M G N (g) M (γ+η+ε) dρ * n (g) Moreover, if d(x, y) ε ′ d 0 , then |P n (z)f (x) -P n (z)f (y)| 2 d(x, y) γ (ε ′ d 0 ) γ f ∞ [σ] ∞ G N (g) M η dρ * n (g)
So, what we proved is that for any γ, η > 0 small enough there are n ∈ N * and a constant C (depending on n, σ and ρ) such that for any z ∈ C η and any f ∈ C 0,γ (X),

m γ (P (z) n f ) 1 2 m γ (f ) + Ce Cn (1 + |z|) f ∞
But, we also have, according to lemma 4.5, that for ℜ(z) 0,

P (z) n f ∞ f ∞ sup x∈X G e -ℜ(z)σ(g,x) dρ * n (g) C f ∞ e -ℜ(z)tn
So, we obtain the expected inequalities by iterating this relations and we refer to [START_REF] Ionescu Tulcea | Théorie ergodique pour des classes d'opérations non complètement continues[END_REF] for a proof that we can choose a constant C that doesn't depend on n nor on z.

An analytic family of operators (P (z)) is said to be meromorphic at z 0 if there is N ∈ N such that the family ((z -z 0 ) N P (z)) is analytic on a neighbourhood of z 0 .

We are now going to use a version of the analytic Fredholm theorem that holds for quasi-compacts operators and that we state in next Theorem 4.8. Let (B, . B ) be a Banach space.

Let . be a norm on B such that the unit ball in B for . B is relatively compact for . . Let U be a connected open subset of C. Let (P (z)) z∈U be an analytic family of operators defined on U such that there are r ∈ [0, 1[ and a real valued function (z → R(z)) such that for any f ∈ B and any z ∈ U ,

P (z)f B r f B + R(z) f
Then, we have the following alternative :

• The operator

I d -P (z) is invertible for no z ∈ U . • The function (z → (I d -P (z)) -1 ) is meromorphic on U .
Proof. The proof is the same as in the classic case when we remark that, according to Ionescu-Tulcea and Marinescu's theorem (that we recalled in theorem 2.10) we have a control on the essential spectral radius of P (z) that is uniform in z.

Lemma 4.9. Under the assumptions of proposition 4.3, the family ((I d -P (z)) -1 ) z∈Cη is a meromorphic family of operators that is analytic on ]0, η[⊕iR.

Moreover, we can choose η such that there are C, t ∈ R * + such that for any z ∈ ]0, η[ ⊕ iR and any n ∈ N,

P (z) n γ C(1 + |z|)e -tℜ(z)n
Finally, if σ ρ = 0 and if η ′ is small enough, we can write, for z ∈ B(0, η ′ ) \ {0},

(I d -P (z)) -1 = 1 σ ρ z N 0 + U (z)
where N 0 is the operator of projection on to the space of P -invariant functions and (U (z)) is an analytic family of continuous operators on C 0,γ (X) and defined on B(0, η ′ ).

Proof. For any z ∈ [0, η] ⊕ iR, any n ∈ N and any f ∈ C 0 (X), we have, according to lemma 4.5, that

P (z) n f ∞ f ∞ sup x∈X G e -ℜ(z)σ(g,x) dρ * n (g) C f ∞ e -tℜ(z)n
And so, for any f ∈ C 0,γ (X) and any n ∈ N, (we can assume without any loss of generality that the constants t given in lemma 4.6 and lemma 4.5 are equal and the same thing for the constants C), according to lemma 4.6 applied to P n (z)(P n (z)f ) and then to P n (z)f , we have that

P (z) 2n f γ C e -tn P n (z)f γ + (1 + |z|) P n (z)f ∞ C e -tn C(m γ (f ) + (1 + |z|)e -tn f ∞ ) + Ce -tℜ(z)n (1 + |z|) f ∞ C 2 (1 + |z|) f γ e -tn + e -tℜ(z)n
And this is what we intended to prove. Moreover, lemma 4.6 shows that the essential spectral radius of P (z) is uniformly bounded by e -t on C η and so we can apply theorem 4.8 to show that the family (I d -P (z)) -1 is meromorphic on C η since we just proved that (I d -P (z)) -1 is well defined on ]0, η] ⊕ iR.

Moreover, this proves that (I d -P (z)) -1 has no pole in ]0, η[⊕iR.

Finally, we refer to lemma 3.2 de [START_REF] Bougerol | Products of random matrices with applications to Schrödinger operators[END_REF] or to lemma 10.17 in [START_REF]Random walks on reductive groups[END_REF] for the expansion of (I d -P (z)) -1 on the neighbourhood of 0. Indeed, adapting their argument, we find that there are two analytic families of continuous operators (N (z)), (U 1 (z)) defined on the neighbourhood of 0 and an analytic function λ such that for any non zero z in the considered neighbourhood of 0,

(I d -P (z)) -1 = 1 1 -λ(z) N (z) + U 1 (z)
This finishes the proof of the lemma since λ and N are analytic, λ(0) = 1, λ ′ (0) = -σ ρ = 0 and N (0) is the projector on ker(I d -P ).

Lemma 4.10. Under the assumptions and notations of proposition 4.3 We note

U (z) = (I d -P (z)) -1 - 1 σ ρ z N 0
Then, for any γ, η > 0 small enough, there are C, L such that for any z ∈ C with

-1 C(1 + |ℑz|) L < ℜ(z) < η
and any n ∈ N, we have that

U (n) (z) γ C n+1 n!(1 + |z|) (L+1)(n+1)
Proof. First of all, we note that, according to the previous lemma, U (z) is well defined on a neighbourhood B(0, η ′ ) of 0.

Moreover, for any hölder-continuous function f , any z ∈ C η and any x ∈ X, we have that (g,x) f (gx)dρ(g) And so, doing the same kind of computations than in the proof of lemma 4.6, we find that for some constant C 1 ∈ R + and any z ∈ C η ,

P ′ (z)f (x) = G σ(g, x)e -zσ
P ′ (z) γ C 1 (1 + |z|) So, for any f ∈ C 0,γ (X), (I d -P (z))f γ (I d -P (ℑz))f γ -|ℜ(z)| sup z ′ P ′ (z) γ f γ (I d -P (ℑz))f γ -C 1 |ℜ(z)|(1 + |z|) f γ Thus, inf f ∈C 0,γ (X)\{0} (I d -P (z))f γ f γ inf f ∈C 0,γ (X)\{0} (I d -P (ℑz))f γ f γ -C 1 |ℜ(z)|(1 + |z|) But, by assumption, for z ∈ C with |z| η ′ , inf f ∈C 0,γ (X)\{0} (I d -P (ℑz))f γ f γ = 1 (I d -P (ℑz)) -1 γ 1 C 0 (1 + |z|) L So, for any z ∈ C η with |z| η ′ and |ℜ(z)| C 0 C 1 (1 + |z|) L+1 , we have that inf f ∈C 0,γ (X)\{0} (I d -P (z))f γ f γ 1 2C 0 (1 + |z|) L
This proves, using that P (z) has an essential spectral radius strictly smaller than 1, that I d -P (z) is invertible and that

(I d -P (z)) -1 γ 2C 0 (1 + |z|) L
We just proved that there is a constant C such that for any z ∈ C η with

|ℜ(z)| 1 C(1 + |z|) L+1
we have that U is analytic at z and

U (z) γ C(1 + |z|) L
Moreover, according to lemma 4.9, if z ∈ C η with ℜ(z) 1/C(1 + |z|) L+1 , then, we have that for any n ∈ N, P

C(1 + |z|)e -tℜ(z)n So, (I d -P (z)) -1 γ C(1 + |z|) 1 -e -t 0 ℜ(z) C(1 + |z|) tℜ(z) C 2 (1 + |z|) L+2 (z) n γ
t And this proves that the function U is analytic on ]0, η[⊕iR.

We just proved that for η small enough and any z ∈ C with -1

C(1 + |z|) L+1 < ℜ(z) < η we have that U (z) γ C ′ (1 + |z|) L+2 for some constant C ′ .
To conclude, we do the same kind of computations than Gelfand and Shilov in the proof of theorem 15 in [GC64] to get the control of the derivatives of U on the domain D η,C ′′ ,L+1 .

The renewal theorem for regular functions.

In this paragraph, we prove a result of representation of the renewal kernel and we deduce the rate of convergence in the renewal theorem for regular functions.

Let γ ∈ R * + . For f ∈ C 0 (X × R), we note m γ,E (f ) = sup t∈R sup x,x ′ ∈X x =x ′ π A (x)=π A (x ′ ) e γ|t| |f (x, t) -f (x ′ , t)| d(x, x ′ ) γ and f γ,∞ = sup x∈X sup t∈R e γ|t| |f (x, t)| Moreover, we note f γ,E = f γ,∞ + m γ,E (f ) And finally, we set (4.6) E γ,k (X × R) = f ∈ C k (R, C 0 (X)) ∀m ∈ [0, k] f (m)
γ,E is finite where we noted

f (k) (x, t) = ∂ k f ∂t k (x, t) And, for f ∈ E γ,k (X × R), f γ,k = max m∈[0,k] f (m) γ,E If f ∈ E γ,0 (X × R), then, we note for x ∈ X and ξ ∈ R, f (x, ξ) = R e -iξt f (x, t)dt
It is clear that, for fixed ξ the function f (., ξ) is hölder-continuous on X. Moreover,

∂ l f ∂ξ l (x, ξ) = R (-it) l e -iξt f (x, t)dt
And integrating by parts, we find that for any ξ ∈ R and any m ∈ N,

ξ m f (x, ξ) = (-i) m R e -iξt f (m) (x, t)dt So, if f ∈ E γ,k (X × R), then (1 + |ξ| k )| f (x, ξ)| R |f (x, t)|dt + R |f (k) (x, t)|dt 2 f γ,k R e -γ|t| dt
In the same way, if

π A • π H (x) = π A • π H (x ′ ), then (1 + |ξ| k )| f (x, ξ) -f (x ′ , ξ)| 2 f γ,k d(x, x ′ ) γ R e -γ|t| dt
So, as we can do the same with the functions ∂ l f ∂ξ l , we just prove the following Lemma 4.11. Let k, l ∈ N. There is a constant C such that for any f ∈ E γ,k (X × R) and any ξ ∈ R, we have that

∂ l f ∂ξ l ( . , ξ) γ C f γ,k 1 + |ξ| k
Remark 4.12. This lemma shows that we recover the classical properties of the Fourier transform that exchanges regularity of functions and decay at infinity.

We shall now prove that convolution with functions of E γ,k regularises functions of E γ,0 . As we will not need this result in full generality, we just prove it for some particular function in E γ,k . Lemma 4.13. Let, for k ∈ N, ϕ k be the function defined for t ∈ R by

ϕ k (t) = t k e -t 1 R + (t)
Then for any γ ∈ ]0, 1[ and any k ∈ N, there is a constant C k such that for any f ∈ E γ,0 (X × R),

ϕ k+1 * f ∈ E γ,k (X × R) and ϕ k+1 * f γ,k C k f γ,0
Proof. Thus usual properties of the convolution product prove that, as ϕ k+1 ∈ C k (R), the function f * ϕ k+1 is k-times differentiable and it k-th derivative is continuous (since f is continuous) and for any x ∈ X, any t ∈ R and any m ∈ [0, k]

(ϕ k+1 * f ) (m) (x, t) = ϕ (m) k+1 * f (x, t) But, ϕ (m) k+1 (t) = m l=0 m l (-1) m-l (k + 1)! (k + 1 -l)! t k+1-l e -t 1 R + (t) So, (ϕ k+1 * f ) (m) (x, t) = m l=0 m l (-1) m-l (k + 1)! (k + 1 -l)! R + u k+1-l f (x, t -u)e -u du and so, if x, x ′ ∈ X are such that π A • π H (x) = π A • π H (x ′ ), I k,m,x,x ′ ,t := (ϕ k+1 * f ) (m) (x, t) -(ϕ k+1 * f ) (m) (x ′ , t) m l=0 m l (k + 1)! (k + 1 -l)! R + u k+1-l e -γ|t-u| d(x, x ′ ) γ f γ,0 e -u du e -γ|t| d(x, x ′ ) γ f γ,0 m l=0 m l (k + 1)! (k + 1 -l)! R + e γ|u| u k+1-m e -u du
where we used that for any v, w ∈ R,

e |v|-|v+w| e |w|
In the same way, we get that

J k,m,x,t = (ϕ k+1 * f ) (m) (x, t) C k,m f γ,0 e -γ|t|
for some constant C k,m and this finishes the proof of the lemma.

We can now state our proposition about representation of the renewal kernel in next Proposition 4.14. Under the assumption of theorem 4.1, for any γ > 0 small enough, there is K ∈ N such that for any f ∈ E γ,K (X × R), any x ∈ X and any t ∈ R,

+∞ n=0 P n f (x, t) = 1 σ ρ Π 0 f (x, t) + 1 2π R e iξt U (-iξ) f (x, ξ)dξ
Where U is the operator defined in proposition 4.3 and we made the abuse of notations

U (-iξ) f (x, ξ) = U (-iξ) f ξ (x) with f ξ = f (., ξ).
Proof. For s ∈ R * + , we note P s the operator defined by

P s f (x, t) = G e -sσ(g,x) f (gx, t + σ(g, x))dρ(g)
We are first going to prove that if f is non negative,

+∞ n=0 P n f (x, t) = lim s→0 + +∞ n=0 P n s f (x, t)
And then that

+∞ n=0 P n s f (x, t) = 1 2π R e iξt (I d -P (s -iξ)) -1 f (x, ξ)dξ
To prove the first equality, note that

G e -sσ(g,x) f (g.(x, t))dρ * n (g) = G e -sσ(g,x) (1 σ(g,x) 0 + 1 σ(g,x)>0 )f (g.(x, t))dρ * n (g)
And the monotone convergence theorem proves that lim

s→0 + +∞ n=0 G e -sσ(g,x) 1 σ(g,x)>0 f (g.(x, t))dρ * n (g) = +∞ n=0 G 1 σ(g,x)>0 f (g.(x, t))dρ * n (g)
Moreover, lemma 4.5 and Bienaymé-Tchebytchev's inequality prove that for any x ∈ X,

ρ * n (g ∈ G|σ(g, x ) 0)
G e -ησ(g,x) dρ * n (g) Ce -tηn so the dominated convergence theorem proves that lim

s→0 + +∞ n=0 G e -sσ(g,x) 1 σ(g,x) 0 f (g.(x, t))dρ * n (g) = +∞ n=0 G 1 σ(g,x) 0 f (g.(x, t))dρ * n (g)
And this finishes the proof of the first equality. Moreover,

I s (t, x) : = +∞ n=0 P n s f (t, x) = +∞ n=0 G e -sσ(g,x) f (gx, t + σ(g, x))dρ * n (g) = 1 2π +∞ n=0 G e -sσ(g,x) R e iξ(t+σ(g,x) f (gx, ξ)dξdρ * n (g) = 1 2π R e iξt +∞ n=0 G e -(s-iξ)σ(g,x) f (gx, ξ)dρ * n (g)dξ = 1 2π R e iξt +∞ n=0 P (s -iξ) n f (x, ξ)dξ = 1 2π R e iξt (I d -P (s -iξ)) -1 f (x, ξ)dξ
and this computation proves the second equality.

Finally, we noted U (z) the family of operators defined by

(I d -P (z)) -1 = 1 σ ρ z N 0 + U (z)
and we saw in proposition 4.3 that (U (z)) is an analytic family of continuous operators on C 0,γ (X). And so,

R e iξt (I d -P (s -iξ)) -1 f (x, ξ)dξ = R e iξt s -iξ N 0 f (x, ξ) dξ σ ρ + R e iξt U (s -iξ) f (x, ξ)dξ
But, for any continuous function f on X, using lemma 2.13, we can write

N 0 f (x) = r i=1 p i (x) f dν i
where the p i are P -invariant functions and the ν i are stationary probability measures on X. And so,

1 2π R e iξt s -iξ N 0 f (x, ξ)dξ = 1 2π r i=1 p i (x) X R e iξt s -iξ f (y, ξ)dξdν i (y) = r i=1 p i (x) X +∞ 0 f (y, t + u)e -su dudν i (y) = +∞ 0 N 0 f (x, t + u)e -su du
where we used that 1 s -iξ

= +∞ 0 e -(s-iξ)u du So, for any f ∈ L 1 (R) such that f ∈ L 1 (R), R e iξt s -iξ f (ξ)dξ = +∞ 0 e -su R e iξ(t+u) f (ξ)dξdu = 2π +∞ 0 e -su f (t + u)du
This proves, using the definition of Π 0 , that lim

s→0 + 1 2π R e iξt s -iξ N 0 f (x, ξ)dξ = +∞ t N 0 f (x, u)du = Π 0 f (x, t) Thus Gf (x, t) = +∞ n=0 P n f (x, t) = 1 σ ρ Π 0 f (x, t) + lim s→0 + 1 2π R e iξt U (s -iξ) f (x, ξ)dξ
and, as, for fixed ξ we have, according to proposition 4.3, that

U (s -iξ) f (x, ξ) ∞ U (s -iξ) γ f (x, ξ) γ C(1 + |ξ|) L+1 f (x, ξ) γ
and as f ∈ E γ,K (X × R), we can conclude with the dominated convergence theorem 4.11 taking K = L + 3.

Corollary 4.15. Under the assumption and notations of proposition 4.3, for any γ > 0 small enough there are constants C, K such that for any f ∈ E γ,K , any x, x ′ ∈ X and any t, t ′ ∈ R we have that

G - 1 σ ρ Π 0 f (x, t) -G - 1 σ ρ Π 0 f (x ′ , t ′ ) C f γ,K ω 0 ((x, t), (x ′ , t ′ )) γ
where,

ω 0 ((x, t), (x ′ , t ′ )) = √ |t-t ′ | 2 +d(x,x ′ ) 2 (1+|t ′ |)(1+|t|) if π A (x) = π A (x ′ ) 1 otherwise
Remark 4.16. Letting t ′ go to +∞ and using Riemann-Lebesgue's lemma and proposition 4.14 to get that under the assumption of the corollary, lim

t ′ →±∞ (G - 1 σ ρ Π 0 )f (x, t ′ ) = 0 we find that for any f ∈ E γ,K (X × R), tout x ∈ X and any t ∈ R, Gf (x, t) - 1 σ ρ Π 0 f (x, t) C (1 + |t|) γ f γ,K Proof. According to proposition 4.14, G - 1 σ ρ Π 0 f (x, t) = 1 2π R e iξt U (-iξ) f (x, ξ)dξ
So, integrating by parts and noting ψ(x, ξ) = U (-iξ) f (x, ξ), we find that for any t

∈ R * , (G - 1 σ ρ Π 0 )f (x, t) = 1 2π R e iξt -t 2 ψ ′′ (x, ξ)dξ So, for any x, x ′ ∈ X such that π A • π H (x) = π A • π H (x ′ ) and any t, t ′ ∈ R * , I(x, t, x ′ , t ′ ) := (G - 1 σ ρ Π 0 )f (x, t) -(G - 1 σ ρ Π 0 )f (x ′ , t ′ ) = 1 2π R e iξt -t 2 ψ ′′ (x, ξ)dξ - 1 2π R e iξt ′ -t ′2 ψ ′′ (x ′ , ξ)dξ = R e iξt -t 2 - e iξt ′ -t ′2 ψ ′′ (x, ξ) - R e iξt ′ t ′2 ψ ′′ (x, ξ) -ψ ′′ (x ′ , ξ) dξ 2π So, I(x, t, x ′ , t ′ ) R e iξt t 2 - e iξt ′ t ′2 |ψ ′′ (x, ξ)|dξ + 1 |t ′ | 2 R ψ ′′ (x, ξ) -ψ ′′ (x ′ , ξ) dξ But, assuming that |t ′ | |t| 1, we find that e iξt t 2 - e iξt ′ t ′2 |t 2 -t ′2 | t 2 t ′2 + |ξ| |t -t ′ | t ′2 |t -t ′ | |t||t ′ | (2 + |ξ|)
And, as we also have for t ∈ R with |t| 1, that

1 |t| 2 1 + |t| , what we get is that R e iξt t 2 - e iξt ′ t ′2 |ψ ′′ (x, ξ)|dξ 4|t -t ′ | (1 + |t|)(1 + |t ′ |) R (2 + |ξ|)|ψ ′′ (x, ξ)|dξ Moreover, 1 |t ′ | 2 R ψ ′′ (x, ξ) -ψ ′′ (x ′ , ξ) dξ 4d(x, y) γ (1 + |t|)(1 + |t ′ |) R m γ (ψ ′′ (., ξ))dξ
So we only need to check the integrability of ψ ′′ ( . , ξ) γ . But,

ψ ′′ (x, ξ) = -U ′′ (-iξ) f (x, ξ) -2iU ′ (-iξ) f ′ (x, ξ) + U (-iξ) f ′′ (x, ξ)
and according to proposition 4.3 there is a constant C such that for any m ∈ {0, 1, 2},

U (m) (-iξ) γ C m+1 m!(1 + |ξ|) (L+1)(m+1)
And, as we also have, according to lemma 4.11 that for any k ∈ N, there is a constant C such that for any l ∈ {0, 1, 2},

∂ l f ∂ξ l (x, ξ) γ C f γ,k 1 + |ξ| k
we get that for some constants C, K and any function f

∈ E γ,K 0 (X × R), any x, x ′ ∈ X with π A • π H (x) = π A • π H (x ′ ) and any t, t ′ ∈ R * with |t|, |t ′ | 1, (4.7) I(x, t, x ′ , t ′ ) C f γ,K |t -t ′ | + d(x, x ′ ) γ (1 + |t|)(1 + |t ′ |)
We can now use the fact that for some constants C γ , C, we have that for any t, t ′ ∈ R and any x, x ′ ∈ X,

|t -t ′ | + d(x, x ′ ) γ (1 + |t|)(1 + |t ′ |) C γ |t -t ′ | + d(x, x ′ ) (1 + |t|)(1 + |t ′ |) γ CC γ |t -t ′ | 2 + d(x, x ′ ) 2 (1 + |t|)(1 + |t ′ |) γ
To prove that inequality (4.7) still holds for |t| 1, we are going to use the fact that the objects we consider behave well with the translations on R.

Indeed, if f ∈ E γ,k (X × R) and if we note, for s ∈ R, f s (t) = f (t -s) then for any x ∈ X and any t ∈ R, (G - 1 σ ρ Π 0 )f (x, t) = (G - 1 σ ρ Π 0 )f s (x, t + s) and f s γ,k e γ|s| f γ,k
So, if |t| 1, we take s ∈ [-10, 10] such that 1 |t + s| |t ′ + s| and we get that for some other constant C,

|I(x, t, x ′ , t ′ )| 4Ce 10γ f γ,k |t -t ′ | 2 + d(x, x ′ ) 2 (1 + |t|)(1 + |t ′ |) γ
and this is finally what we intended to prove.

4.4. Renewal theorem for hölder-continuous functions. Until now, we proved the renewal theorem only for regular functions when we are interested in functions that will just be hölder-continuous on R d .

To do so, we are going to regularize our functions by convolving them to regular ones and then use tauberian theorems to get the result.

This method is the one used for instance in [START_REF] Buraczewski | On the rate of convergence in the Kesten renewal theorem[END_REF] and we will use, as they do, a tauberian theorem by Frennemo (see [START_REF] Frennemo | On general Tauberian remainder theorems[END_REF] and appendix A).

We are going to define a new class of functions on X × R for which it is easy to work for the rate of convergence in the renewal theorem.

Example 4.17. On R d , we study functions having (a power of)

ω ′ (x, y) = x -y (1 + x )(1 + y )
as continuity modulus. Using the application Φ :

S d × R → R d \ {0} that maps (x, t) onto e t x and identifies R d \ {0} and S d × R, our function ω becomes ω ′ ((x, t), (x ′ , t ′ )) = e t x -e t ′ x ′ (1 + e t )(1 + e t ′ )
But, e t x -e t ′ x ′ 2 = e 2t + e 2t ′ -2e t+t ′ x, x ′ = e t -e t ′ 2 + e t+t ′ x -x ′ 2 = e t+t ′ e (t-t ′ )/2 -e (t ′ -t)/2 2 + x -x ′ 2 So ω ′ ((x, t), (x ′ , t ′ )) = e t/2 1 + e t e t ′ /2 1 + e t ′ d(x, x ′ ) 2 + e (t-t ′ )/2 -e (t ′ -t)/2 2

This leads to the following definition when (X, d) is some compact metric space and (x, t), (x ′ , t ′ ) ∈ X × R, ω((x, t), (x ′ , t ′ )) = e -|t|+|t ′ | 2 d(x, x ′ ) 2 + e (t-t ′ )/2 -e (t ′ -t)/2 2 si π A (x) = π A (x ′ ) 1 si non And we note

C γ ω (X × R) :=      f ∈ C 0 (X × R) sup (x,t),(x ′ ,t ′ )∈X×R (x,t) =(x ′ ,t ′ )
|f (x, t) -f (x ′ , t ′ )| ω((x, t), (x ′ , t ′ )) γ is finite

    
The following lemma proves that functions in C γ ω can be extended to continuous functions on X × R.

Lemma 4.18. Let f ∈ C γ ω (X × R). Then, there are p + , p -∈ L ∞ (A) such that for any x ∈ X and any t ∈ R, |f (x, t) -p + (f )(π A (x))| 2e γt f γ,ω et |f (x, t) -p -(f )(π A (x))| 2e -γt f γ,ω (we recall our abuse of notations : we keep the function π A for the function π A • π H ) Proof. We only do the proof for p -because it is the same idea for p + .

For any x ∈ X ans any t, t ′ ∈ R, we have that |f (x, t) -f (x, t ′ )| e -|t|+|t ′ | 2 e (t-t ′ )/2 -e (t ′ -t)/2 f γ,ω

So, according Cauchy's criterion for functions, for any x ∈ X, (t → f (x, t)) has a finite limit at -∞.

So we can note p -(f )(x) = lim t→-∞ f (x, t). And then, we have, by definition of C γ ω , that for any (x, t), (x ′ , t ′ ) ∈ X × R,

|f (x, t) -p -(f )(x ′ )| |f (x, t) -f (x ′ , t ′ )| + |f (x ′ , t ′ ) -p -(f )(x ′ )| f γ,ω ω((x, t), (x ′ , t ′ )) γ + |f (x ′ , t ′ ) -p -(f )(x ′ )|
So, letting t ′ go to -∞, we find that for any t ∈ R and any x, x ′ ∈ X with and this ends the proof of the lemma.

π A • π H (x) = π A • π H (x ′ ), |f (x, t) -p -(f )(x ′
In the sequel, we will use a regular function ψ on R such that lim t→-∞ ψ(t) = 1 and lim t→+∞ ψ(t) = 0

The space of functions in C γ ω (X × R) such that p -(f ) = 0 = p + (f ) has a finite codimension and this function ψ will allow us to make the projections onto this space and it's supplementary explicit.

So, from now on, we note (4.8)

ψ(t) = 1 √ 2π +∞ t e -u 2 /2 du
The choice of this particular function is arbitrary but will simplify the computations to come.

In next lemma, we prove that the projection of a function in C γ ω (X × R) to the space of functions such that p + (f ) = 0 = p -(f ) has image in the space E γ,0 (X × R) that we defined in equation (4.6).

Lemma 4.19. For any function f ∈ C 0,γ ω (X × R), let ϕ(x, t) = f (x, t) -p -(f )(x)ψ(t) -p + (f )(x)(1 -ψ(t))

where ψ is the function defined in equation (4.8).

Then, for any γ ∈ ]0, 1] there is a constant C depending only on γ such that for all f ∈ C 0,γ ω (X × R), we have that ϕ ∈ E γ,0 (X × R) and ϕ γ,0 C f γ,ω

Proof. It is clear that ϕ is continuous on X × R. And this finishes the proof of the lemma.

Lemma 4.20. Under the assumptions of theorem 4.1, for any f ∈ C 0,γ ω (X × R), any x ∈ X and any t, t ′ ∈ R, ω(gx, t + σ(g, x), gx, t ′ + σ(g, x)) γ dρ * n (g) f ω,γ e (t-t ′ )/2 -e (t ′ -t)/2 γ +∞ n=0 G e -γ|t+σ(g,x)|/2-γ|t ′ +σ(g,x)|/2 dρ * n (g) Moreover, |e (t-t ′ )/2 -e (t ′ -t)/2 | = |e |t-t ′ |/2 -e -|t-t ′ |/2 | = e -|t-t ′ |/2 e |t-t ′ | -1 e |t-t ′ | -1 So,

I(t, t ′ , x) f ω,γ e |t-t ′ | -1 γ +∞ n=0 G
e -γ|t+σ(g,x)|/2 dρ * n (g)

f ω,γ e |t-t ′ | -1 γ +∞ n=0 P n ϕ(x, t)
with ϕ(x, t) = e -γ|t|/2 . To prove the lemma, we only need to check that the series is bounded uniformly in (x, t). We would like to apply the renewal theorem (that would prove that the considered sum has finite limits at ±∞ and so is bounded) to the function ϕ but we can't do it so early because this function is not regular. This is why we use two regular functions that dominate ϕ : +∞ n=0 G e -γ|t+σ(g,x)| dρ * n (g) P n ϕ 1 (x, t) = N 0 1 R ϕ 1 (t)dt So, +∞ n=0 P n ϕ 1 is bounded on X × R. We treat +∞ n=0 P n ϕ 2 in the same way and this finises the proof of the lemma. For x, x ′ ∈ X such that π A • π H (x) = π A • π H (x ′ ) and t, t ′ ∈ R, we note ω 0 ((x, t), (x

′ , t ′ )) = |t -t ′ | 2 + d(x, x ′ ) 2 (1 + |t|)(1 + |t ′ |)
Corollary 4.21. Under the assumptions of theorem 4.1, there are constants C, K, α ∈ R * + such that for any f ∈ C 0,γ ω (X × R) with p + (f ) = 0 = p -(f ) any x, x ′ ∈ X and any t, t ′ ∈ R,

(G - 1 σ ρ Π 0 )f (x, t) -(G - 1 σ ρ Π 0 )f (x ′ , t ′ ) C f γ,ω ω 0 ((x, t), (x ′ , t ′ )) α
Moreover, for any x ∈ X,

lim t→±∞ (G - 1 σ ρ Π 0 )f (x, t) = 0
Proof. Let, for k ∈ N and t ∈ R, ϕ k (t) = t k e -t 1 R + (t). We already saw in lemma 4.13 that there is a constant C k such that for any f ∈ E γ,0 (X × R), f * ϕ k+1 ∈ E γ,k (X × R) and f * ϕ k+1 γ,k C k f γ,0

In particular, for any f ∈ C 0,γ ω , according to lemma 4.19,

f * ϕ k+1 γ,k C k f γ,ω
Moreover,

ϕ k+1 * (G - 1 σ ρ Π 0 )f (x, t) = (G - 1 σ ρ Π 0 )(f * ϕ k+1 )(x, t)
and so, taking k large enough, corollary 4.15 proves that I(k, x, t, x ′ , t ′ , f ) : = ϕ k+1 * (G -

1 σ ρ Π 0 )f (x, t) -ϕ k+1 * (G - 1 σ ρ Π 0 )f (x ′ , t ′ )
CC k ω 0 ((x, t), (x ′ , t ′ )) γ f γ,ω

And, as we also have that

|(G - 1 σ ρ Π 0 )f (x, t) -(G - 1 σ ρ Π 0 )f (x, t ′ )| C f γ,ω e |t-t ′ | -1 γ
we can conclude with the result by Frennemo stated in the appendix A.

We can now treat functions in C γ ω (X×R) that vanish at infinity. However, we would like to study functions on R d ×A that doesn't vanish on {0}×A but such that a∈A f (a, 0) = 0. This is why we prove the Lemma 4.22. Under the assumptions of 4.1, there is a constant C such that for any p ∈ L ∞ (A) with a∈A p(a) = 0, for any x, x ′ ∈ X and any t, t ′ ∈ R,

(G - 1 σ ρ Π 0 )f (x, t) -(G - 1 σ ρ Π 0 )f (x ′ , t ′ ) C p ∞ ω 0 ((x, t), (x ′ , t ′ ))
where we noted f (x, t) = p(π A • π H (x))ψ(t) and ψ the function we defined in equation (4.8).

Moreover, if a∈A p(a) = 0, then for any x ∈ X,

lim t→-∞ G - 1 σ ρ Π 0 f (x, t) = +∞ n=0 P n p(π A (x))
Proof. As P commutes to the derivation, e iξu e -ξ 2 /2 dξdu = -N 1 p(x)ϕ(t)

and of 1 2π

+∞ t R
e iξu (-iξ)V (-iξ)p(x)e -ξ 2 /2 dξdu = 1 2π R e iξt V (-iξ)p(x)e -ξ 2 /2 dξ (to get this equality, we derive two times). And so, finally, we get that

(G - 1 σ ρ Π 0 )f (x, t) = lim t→-∞ N 1 f (x, t) = N 1 f (x, -∞)
Now, we need to prove that

N 1 f (x, -∞) = +∞ n=0 P n f (π A • π H (x))
To do so, note that these two functions are solutions of the equation g -P g = f and, as the random walk on A is irreducible and aperiodic, there is a constant C ∈ R such that for any a ∈ A, where ω f is a modulus of uniform continuity for f .

N
Proof. This lemma is a corollary of lemma 1 in [START_REF] Frennemo | On general Tauberian remainder theorems[END_REF] when we remark that if f is uniformly continuous then

- inf x t x+1/V f (t) -f (x) sup x t x+1/V ω f (|x -t|) ω f (1/V )
Proof of theorem A.2. For any s ∈ R, let u s be the function defined for t ∈ R by u s (t) = f (t)e -1 2 (t-s) 2 . Then, for any t, t ′ ∈ R,

|u s (t) -u s (t ′ )| = e -1 2 (t-s) 2 (f (t) -f (t ′ )) + f (t ′ )(e -1 2 (t-s) 2 -e -1 2 (t ′ -s) 2 ) ω f (|t -t ′ |) + f ∞ |t -t ′ | sup u∈R |u|e -u 2 /2
So, the function u s is uniformly continuous. As it is integrable, we have, according to the lemma, that for any V ∈ R * + and any t, s ∈ R,

|f (t)| = |u s (t)| C ω f 1 V + f ∞ V sup u∈R |u|e -u 2 /2 + sup τ V -V e iξτ 1 - |ξ| V u s (ξ)dξ
Moreover, Frennemo proves that

sup τ V -V e iξτ 1 - |ξ| V u s (ξ)dξ C(1 + V ) k sup t ′ ∈R e -|t ′ | |ϕ k * f (t -t ′ )|
This is what we intended to prove.

Let (X, d) be a compact metric space and γ ∈ ]0, 1]. For any (x, t), (x ′ , t ′ ) ∈ X × R, we note ω 0 ((x, t), (x

′ , t ′ )) = |t -t ′ | 2 + d(x, x ′ ) 2 (1 + |t ′ |)(1 + |t|)
Corollaire A.4. Let (X, d) be a compact metric space and γ ∈ ]0, 1].

For any k ∈ N, there is a constant C k ∈ R + and α ∈ R * + such that for any continuous bounded function f on X × R such that

|f (x, t) -f (x, t ′ )| e |t-t ′ | -1 γ C(f ), f ∞ C(f ) and |ϕ k * f (x, t) -ϕ k * f (x ′ , t ′ )| C(f )ω((x, t), (x ′ , t ′ ))
We have that |f (x, t) -f (x ′ , t ′ )| C k C(f )ω((x, t), (x ′ , t ′ )) α

Proof. For any x, x ′ ∈ X and s ∈ R, we note f x,x ′ ,s the function defined for any t ∈ R by f x,x ′ ,s (t) = f (x, t) -f (x ′ , t + s) Thus, for an other constant C that doesn't depend on f ,

|f x,x ′ ,s (t)| CC(f ) inf V ∈[1,+∞[ 1 V γ + (1 + V ) k ω((x, t), (x ′ , t + s))
Noting now, for some δ ∈ R * + that we will choose later, V = ω((x, t), (x ′ , t + s)) -δ we find that for some constant C that doesn't depend on f , |f (x, x ′ , s)(t)| CC(f ) ω γδ + (1 + ω -γδ ) k ω CC(f )ω α for δ small enough and some α (where we used that ω is bounded on X × R). This ends the proof of the corollary.

  that, for some L ∈ R and any t0 ∈ R * + , supt∈R |t| t 0 1 |t| L (I d -P (it)) -1 is finitewhere the norm is taken in some space of hölder-continuous functions on the sphere.Contents1. Introduction 1.1. Main results 1.2. Proofs 1.3. Notations and conventions 2. Unitary perturbations of Markov operators 2.1. Preliminaries 2.2. Control of the resolvant of the perturbated operator 3. Diophantine properties in linear groups 1. Introduction 1.1. Main results. Let ρ be a borelian probability measure on SL d (R) (2 d ∈ N) and let x ∈ R d \ {0}. We define a random walk on R d \ {0} starting at x by

  Then, for any γ ∈ R * + and any continuous function f on R d such that sup v∈R d \{0} |f (v)| v γ and sup v∈R d v γ |f (v)| are finite we have that

  f γ := sup x,y∈R d \{0} x =y a∈A (1 + x ) γ (1 + y ) γ |f (x,a) -f (y, a)| x -y γ < +∞, and such that for any a ∈ A, lim x→∞ f (x, a) = 0 and a∈A f (0, a) = 0

  g) M and sup g∈G e σsup(g) N (g) M are finite and, for σ ∈ Z M N (X), we note (2.4) [σ] M = sup g∈G σ Lip (g) N (g) M and [σ] ∞ = sup g∈G e σsup(g) N (g) M

  Then, as n β ln |t|, we also have that e κn |t| e κβ ln |t| |t| |t| 1+βκ

σ

  sup (g) = sup x∈X |σ(g, x)| and σ Lip (g) = sup x,y∈X π A (x)=π A (y) x =y |σ(g, x) -σ(g, y)| d(x, y) And, for M ∈ R + and N : G → [1, +∞[ a submultiplicative function on G, Z M N (X) = σ is a cocycle on X sup g∈G σ Lip (g) N (g) M and sup g∈G e σsup(g) N (g) M are finite Finally, for σ ∈ Z M N (X), we set [σ] M = sup g∈G σ Lip (g) N (g) M and [σ] ∞ = sup g∈G e σsup(g) N (g) M

Figure 1 .

 1 Figure 1. Shape of the zone D η,C,L Lemma 4.5. Under the assumptions of proposition 4.3, if σ ρ > 0 then there are η, t, C ∈ R * + such that for any s ∈ [0, η] and any n ∈ N, sup x∈X G

  )| e γ(t-|t|)/2 f γ,ωAnd this proves that p-(f )(x) = p -(f )(x ′ ) and that, for any t ∈ R -, |f (x, t) -p -(f )(x)| e γt f γ,ω For t ∈ R + , we have that |f (x, t) -p -(f )(x)| 2 f ∞ 2e γt f γ,ω

+∞ n=0 P

 n=0 n f (x, t) -+∞ n=0 P n f (x, t ′ ) C f ω,γ e |t ′ -t| -1 γ Proof. Let f ∈ C 0,γ ω (X × R), x ∈ X and t, t ′ ∈ R. Then, I(t, t ′ , x) : = +∞ n=0 P n f (x, t) -+∞ n=0 P n f (x, t ′ ) +∞ n=0 G |f (gx, t + σ(g, x)) -f (gx, t ′ + σ(g, x))|dρ * n (g) f ω,γ +∞ n=0 G

P

  n f ′ (x, u)du But, according to proposition 4.14,+∞ n=0 P n f ′ (x, u) = 1 σ ρ +∞ u N 0 f ′ (x, s)ds + 1 2π R e iξu U (-iξ)p(x) f ′ (ξ)dξ But f ′ (ξ) = -e -ξ 2 /2 and +∞ u N 0 f ′ (x, s)ds = -N 0 f (x, u) So, +∞ n=0 P n f ′ (x, u) = -1 σ ρ N 0 f (x, u) + 1 2π R e iξu U (-iξ)p(x)e -ξ 2 /2 dξThis proves that+∞ n=0 P n f (x, t) = 1 σ ρ +∞ t N 0 f (x, u)du -1 2π +∞ t Re iξu U (-iξ)p(x)e -ξ 2 /2 dξdu Moreover, noting N 1 = U (0) and V (z) such that U (z) = N 1 + zV (z), we have that the second term (that only depend on π A • π H (x)) is the sum of1 2π N 1 p(x) +∞ t R

N

  0 f (x, u)du + N 1 f (x, t) -1 2π R e iξt V (-iξ)p(x)e -ξ 2 /2 dξSo, according to Riemann-Lebesgue's lemma and using the definition of Π 0 , lim t→-∞

P

  n p(a) + |A|C = |A|C

  0,γ (X), . (t) ) are isomorphic as Banach spaces. Moreover, P (it) is better controlled with . (t) as shown by next

	Lemma 2.25. Under the assumptions of proposition 2.22, for any t ∈ R with |t| and any n ∈ N, P (it) n (t) 2C 2	2

  -u 2 /2 du = 1, we find that for any t ∈ R and any x ∈ X,|ϕ(x, t)| |f (x, t) -p -(f )(x)|ψ(t) + |f (x, t) -p + (f )(x)|(1 -ψ(t)) f γ,ω e γt ψ(t) + e -γt (1 -ψ(t)) So, e γ|t| |ϕ(x, t)| f γ,ω e γ(|t|+t) ψ(t) + e γ(|t|-t) (1 -ψ(t))This proves that there exists some C depending only on γ such that sup

	Moreover, using that	1 √ 2π R	e

t e γ|t| |ϕ(x, t)| C f γ,ω Moreover, for any x, x ′ such that π A • π H (x) = π A • π H (x ′ ), |ϕ(x, t) -ϕ(x ′ , t)| = |f (x, t) -f (x ′ , t)| e -γ|t| d(x, x ′ ) γ f γ,ω

  (t) 1 R -e γt et ϕ 2 (t) e -γt 1 R + (t)

								+∞			+∞
									P n ϕ 1 (x, t) +	P n ϕ 2 (x, t)
								n=0		n=0
	where we noted
	ϕ 1 (t) =	2 √ 2π	e γt	t	+∞	e -u 2 /2 du et ϕ 2 (t) =	2 √ 2π	e -γt	-∞ t	e -u 2 /2 du
	to get that
	ϕ 1 Then, the renewal theorem that we proved for regular functions (corollary 4.15) and
	Riemann Lebesgue's lemma, prove that
		+∞							+∞
	lim t→+∞	n=0	P n ϕ 1 (x, t) = 0 and lim t→-∞	n=0

  Then, for any t, t ′ ∈ R,|f x,x ′ ,s (t) -f x,x ′ ,s (t ′ )| = |f (x, t) -f (x ′ , t + s) -f (x, t ′ ) + f (x ′ , t ′ + s)| 2C 0 e |t-t ′ | -1 * f x,x ′ ,s (t)| = |ϕ k * f (x, t) -ϕ k * f (x ′ , t + s)| C(f )ω((x, t), (x ′ , t + s))And so, according to Frennemo's theorem,|f x,x ′ ,s (t)| CC(f ) inf -|t ′ | ω((x, t -t ′ ), (x ′ , t -t ′ + s)) -|t ′ | ω((x, t -t ′ ), (x ′ , t -t ′ + s)) ω((x, t), (x ′ , t + s)) sup t ′ ∈R e -|t ′ | (1 + |t ′ |)

		V ∈R * +	2(e 1/V -1) γ +	2 V
	But, for any t, t ′ ∈ R, This proves that	+(1 + V ) k sup t ′ ∈R 1 1 + |t -t ′ |	1 + |t ′ | 1 + |t|
	sup t ′ ∈R

γ and |ϕ k e e

There is ε ∈ R * + such that G g ε dρ(g) is finite.

For any non negative continuous function f on X, P f is non negative.

And so, C = 0 since a∈A N 1 p(a) = 0 = a∈A +∞ n=0 P n p(a). Finally, according to proposition 4.3, R e iξt V (-iξ)p(x)e -ξ 2 /2 dξ p ∞ R C(1 + |ξ|) L e -ξ 2 /2 dξ and

This finishes the proof of the lemma (because p(x) only depend on the projection of x onto A).

Appendix A. Remainder terms in Wiener's tauberian theorem In this appendix, we state Frennemo's result about remainder terms in Wiener's tauberian theorem. The aim of this section is the study of the following problem : given two functions f, ϕ on R for whoch we now the rate of convergence to 0 of |f * ϕ| at infinity, can we get the one of |f | ?

The first answer to this question is a corollary of a tauberian theorem of Wiener that says that if f is bounded, ϕ is integrable, the Fourier-Laplace transform of ϕ doesn't vanish on iR and if f * ϕ converges to 0 at ±∞, then, for any integrable function g on R, f * g also converge to 0 at ±∞. This kind of result is interesting for us in the study of the renewal theorem because to make our method work, we have to assume that our test functions are regular so we will have to regularize them by convoling them with regular functions and these remainder terms estimates will allow us to keep the rate of convergence to 0. Définition A.1. Let f be a uniformly continuous function on R.

We say that a non decreasing function ω : R + → R + , that is continuous at 0 and such that ω(0) = 0 is a modulus of uniform continuity for f if for any x, y ∈ R,

The following theorem is an adaptation of the second theorem of Frennemo in [START_REF] Frennemo | On general Tauberian remainder theorems[END_REF].

Then, there is a constant C depending only on k such that for any uniformly continuous and bounded function f on R and any x ∈ R,

where ω f is a modulus of uniform continuity for f . Lemme A.3. There is a constant C such that for any integrable any uniformly continuous function f on R,