Yves Guivarc

Spectral gap properties and limit theorems for some random walks and dynamical systems

We give a description of some limit theorems and the corresponding proofs for various transfer operators. Our examples are closely related with random walks on homogeneous spaces. The results are obtained using spectral gap methods in Hölder spaces or Hilbert spaces. We describe also their geometrical setting and the basic corresponding properties. In particular we focus on precise large deviations for products of random matrices, Fréchet's law for affine random walks and local limit theorems for Euclidean motion groups or nilmanifolds.

Introduction

In the proofs of limit theorems for dynamical systems with hyperbolic properties, the use of transfer operators plays an important role. For transformations with expanding properties this amount to the use of spectral gap properties of Perron-Frobenius operators (see for example [START_REF] Broise | Etudes spectrales d'opérateurs de transfert et applications[END_REF], [START_REF] Gouézel | Berry-Essen theorem and local limit theorem for non uniformly expanding maps[END_REF], [START_REF] Keller | Un théorème de la limite centrale pour une classe de transformations monotones par morceaux[END_REF], [START_REF] Rousseau-Égèle | Un théorème de la limite centrale pour une classe de transformations dilatantes[END_REF]). For Anosov systems and their extensions, the relevant transfer operators are obtained through the use of a Markov partition which allows to go over to a symbolic system (see [START_REF] Bowen | Equilibrium states and the ergodic theory of Anosov diffeomorphisms[END_REF], [START_REF] Sinai Ya | Gibbs measures in ergodic theory[END_REF]) and to prove limit theorems (see [START_REF] Guivarc | Théorèmes limites pour une classe de chaines de Markov et applications aux difféomorphismes d'Anosov[END_REF]). In the context of random walks on homogeneous spaces, spectral gap properties on Hölder spaces ( [START_REF] Gao | Spectral gap properties and convergence to stable laws for affine random walks[END_REF], [START_REF] Guivarc | Théorèmes limites pour une classe de chaines de Markov et applications aux difféomorphismes d'Anosov[END_REF], [START_REF] Guivarc | On spectral properties of a family of transfer operators and convergence to stable laws for affine random walks[END_REF]), L 2 -spaces (see [START_REF] Bekka | On the spectral theory of groups of affine transformations on compact nilmanifolds[END_REF], [START_REF] Bourgain | Spectral gaps in SU(d)[END_REF], [START_REF] Conze | Ergodicity of group actions and spectral gap. Application to random walks and Markov shifts[END_REF], [START_REF] Furman | Sharp ergodic theorems for group actions and strong ergodicity[END_REF]), or Sobolev spaces (see [START_REF] Cantat | Théorème limite central pour les endomorphismes holomorphes et les correspondances modulaires[END_REF], [START_REF] Dinh | Dynamics in several complex variables : Endomorphisms of projective spaces and polynomial-like mappings[END_REF]) are valid also. Hence we are led to study a general class of Markov chains on a metric space (X, d) associated with semigroup actions on X. Let m be a probability on the Borel subsets of the complete and separable metric space (X, d), S a Polish semigroup acting measurably on X by x → g.x (x ∈ X, g ∈ S). Let p(x, g) be a non negative measurable function on (X, S) with p(x, g)dµ(g) = 1 for any x ∈ X, where µ ∈ M 1 (S) is a probability on S. We consider the Markov operator Q defined on bounded measurable functions by Qϕ(x) = ϕ(g.x)p(x, g)dµ(g). The special case where p(x, g) = 1 is often considered in the framework of the so-called "Iterated Random Functions" (see [START_REF] Guivarc | Homogeneity at infinity of stationary solutions of Multivariate Affine Stochastic Recursions[END_REF] for example). In various situations like the study of expanding transformations (see [START_REF] Broise | Etudes spectrales d'opérateurs de transfert et applications[END_REF]) or large deviations for products of random matrices (see below) it is natural to allow p(x, g) to depend on (x, g) ∈ X × S. We will assume that m is Q-invariant i.e Qm = m where Qm(ϕ) = m(Qϕ). We denote by Ω the product space Ω = S N and for ω = (g 1 , g 2 , • • •) ∈ Ω, n ∈ N, we write

s n = s n (ω) = g n • • • g 1 ∈ S, p n (x, ω) = n Π k=1 p(s k-1 .
x, g k ). We define the probability P x on Ω as the projective limit of the system of probabilities p n (x, .)µ ⊗n on the spaces Ω n = S × • • • × S (n times) and we write P = P x dm(x). The corresponding expectation with respect to P x (resp. P) will be denoted by E x (resp. E). In particular, we have

Q n ϕ(x) = E x (ϕ • s n ).
Then the Q-invariance of m implies that P is invariant under the shift θ on Ω. If m is Q-ergodic, then the dynamical system (Ω, θ, P) is also ergodic. We will be interested in limit theorems for the Birkhoff sum S n (x, ω) = n Σ 1 f (s k-1 .x, g k ) where f (x, g) is a given real valued Borel function on X × S. It is natural to consider the associated "Markov walk" on X × R given by (x, t) → (g 1 .x, t + f (x, g 1 )), its kernel denoted by . Q, leaves invariant the measure m ⊗ ℓ where ℓ is Lebesgue measure on R and .

Q commutes with the translations on X × R given by (x, t) → (x, t + t ′ ). The iterated transitions (x, t) → (s n .x, t + S n (x, .)) are given by the iterated kernel .

Q n . Here we restrict mainly to the problem of the local limit theorem i.e to the asymptotics of E[1 I (S n (x, .)] where I is an interval of the line. The Fourier transform of the random variable S n , i.e E x (e itSn(x,. ) where t ∈ R, can be calculated with the "Fourier operators" Q it given on bounded functions on X by Q it ϕ(x) = ϕ(g.x) p(x, g)e itf (x,g) dµ(g) and, with abuse of notation, we write Q it ϕ(x) = Q(e itf ϕ). In particular we have : E x (e itSn(x,.) ϕ(s n .x)) = (Q it ) n ϕ(x). The spectral methods developed below in specific examples are also useful for proving precise forms of other limit theorems like central limit thorem with remainder (see [START_REF] Gouézel | Berry-Essen theorem and local limit theorem for non uniformly expanding maps[END_REF], [START_REF] Page | Théorèmes limites pour les produits de matrices aléatoires[END_REF], [START_REF] Rousseau-Égèle | Un théorème de la limite centrale pour une classe de transformations dilatantes[END_REF]), large deviations, renewal theorem (see [START_REF] Guivarc | Théorèmes limites pour une classe de chaines de Markov et applications aux difféomorphismes d'Anosov[END_REF] for example), but also for convergence to stable laws ( [START_REF] Guivarc | On spectral properties of a family of transfer operators and convergence to stable laws for affine random walks[END_REF]) and extreme value theory ( [START_REF] Guivarc | Asymptotique des valeurs extrêmes pour les marches aléatoires affines[END_REF]). For recent results on random walks in random medium, closely connected with extreme value theory and stable laws on the affine group see [START_REF] Dolgopyat | Quenched limit theorems for nearest neighbour random walks in 1D random environment[END_REF].

In section 1 we recall basic spectral results and we show their connexion with the local limit theorem. In section 2 we describe results for products of random matrices which are used in sections 3, 4. In section 3 we apply the results of sections 1, 2 to some specific situations ; we get local limit theorems on motion groups and on nilmanifolds, as well as large deviations for products of random matrices. In section 4 we consider affine random walks on R d and we apply the above spectral results to get an analogue of Fréchet's law, in particular. Sections 3,4 and the end of section 2 corresponds to recent results (see [START_REF] Guivarc | Spectral gap properties and asymptotics of stationary measures for affine random walks[END_REF], [START_REF] Guivarc | Asymptotique des valeurs extrêmes pour les marches aléatoires affines[END_REF]). For the beginning of section 2, see the surveys [START_REF] Guivarc | Limit theorems for random walks and products of random matrices[END_REF], [START_REF] Guivarc | On contraction properties for products of Markov driven random matrices[END_REF] and the book [START_REF] Bougerol | Products of random matrices with applications to Schrödinger operators[END_REF]. We thank B. Bekka, J.-P Conze, G.-A Margulis for important remarks and the referee for very useful suggestions.

1 Spectral gap properties of Markov operators and the local limit theorem.

We will show the local limit theorem in various geometric situations and we begin by recalling a useful analytic setting first considered in [START_REF] Doeblin | Sur les chaînes à liaisons complètes[END_REF] and fully developed in [START_REF] Tulcea | Théorie ergodique pour des classes d'opérations non complètement continues[END_REF]. We assume that the metric space (X, d) is compact and we write C(X) for the space of continuous functions ; if ϕ ∈ C(X) we denote for ε > 0 |ϕ| ∞ = sup

x |ϕ(x)|, [ϕ] ε = sup
x,y |ϕ(x) -ϕ(y)| d ε (x, y) and we write H ε (X) ⊂ C(X) for the space of ε-Hölder functions i.e ϕ ∈ H ε (X) if and only if [ϕ] ε < ∞. The space H ε (X) is normed by ϕ = |ϕ| ∞ + [ϕ] ε and then it becomes a Banach space. We will say that a bounded operator Q on C(X) satisfies condition (D-F) if a) The powers Q n of Q acting on C(X) are uniformly bounded in norm | | ∞ . b) There exists numbers ρ ∈ [0, 1[, r ∈ N, C ≥ 0 such that for any ϕ

∈ H ε (X) [Q r ϕ] ε ≤ ρ[ϕ] ε + C|ϕ| ∞ .
The functional inequality in b) will we called (D-F) inequality. Then we have the

Theorem 1.1 Assume that (X, d) is compact and Q satisfies condition (D-F) on H ε (X).
Then Q has only finitely many unimodular spectral values and they are isolated in the spectrum of Q. The corresponding characteristic subspaces have finite dimension and are equal to the corresponding eigenspaces.

For more general and more precise statements see [START_REF] Doeblin | Sur les chaînes à liaisons complètes[END_REF], [START_REF] Hennion | Sur un théorème spectral et son application aux noyaux Lipchitziens[END_REF], [START_REF] Tulcea | Théorie ergodique pour des classes d'opérations non complètement continues[END_REF]. If Q is a Markov operator on X, hence in particular Q1 = 1, we say that Q is irreducible if the equation Qϕ = e iθ ϕ(θ ∈ R) implies e iθ = 1, ϕ =const. If the Markov operator Q on X is irreducible and satisfies condition (D-F) then m is the unique Q-invariant measure and the projection m ⊗ 1 on C1 satisfies lim

n→∞ Q n -m ⊗ 1 = 0. More precisely, we have the spectral decomposition Q = m ⊗ 1 + R where R(m ⊗ 1) = (m ⊗ 1) R = 0 and the spectral radius of R is less than 1. We write H ε 0 (X) = {ϕ ∈ H ε (X) ; m(ϕ) = 0} and we observe that H ε (X) = C1 ⊕ H ε 0 (X).
Under condition D-F, Q-ergodicity of m is equivalent to simplicity of the eigenvalue 1. Also, the support supp(m) of m is the smallest closed Q-invariant subset of X. In general it is smaller than X, but we can always restrict Q to supp(m). For the operator

Q it we have [Q it ϕ] ε ≤ ρ[ϕ] ε + C t |ϕ| ∞ where ρ = sup x,y p(x, g) d ε (g.x, g.y) d ε (x, y) dµ(g) C t = sup
x,y |e itf (x,g) p(x, g) -e itf (y,g) p(y, g)| d ε (x, y) dµ(g).

The corresponding quantities for (Q it ) n are conveniently expressed with E x , in particular ρ is replaced by :

ρ(n) = sup x,y E x d ε (s n .x, s n .y) d ε (x, y) .
In particular if the functions x → p(x, g), x → f (x, g) are uniformly Hölder then C t < ∞. Also, if lim n→∞ ρ(n) < 1, then condition D-F is satisfied ; hence in this case we are able to develop a detailed study of S n (x, ω).

We assume now that Q is an irreducible Markov operator with satisfies condition D-F and f (., g) is uniformly Hölder. We follow closely [START_REF] Guivarc | Théorèmes limites pour une classe de chaines de Markov et applications aux difféomorphismes d'Anosov[END_REF].

Definition 1.2 We will say that the function f (x, g) on X × S is non degenerate (resp is aperiodic) if f is not additively (resp multiplicatively) cohomologous to a constant on supp(m) i.e for any g ∈ supp(µ), x ∈ supp(m) and some u

∈ H ε (X), c ∈ R (resp u ∈ H ε (X), |u| = 1, t ∈ R, θ ∈ R) we have f (x, g) = u(g.x) -u(x) + c (resp. e itf (x,g) = e iθ u(g.x)u -1 (x)).
We assume that the integral f (x, g)dm(x)dµ(g) is finite and we denote this integral by γ(f ). The function v(x) = f (x, g)dµ(g) -γ(f ) satisfies m(v) = 0, hence the equation u -Qu = v has a unique solution u with u ∈ H ε 0 (X). If f 2 (x, g)dm(x)dµ(g) is finite we can define

σ 2 = σ 2 (f ) = (f (x, g) -γ(f ) + u(x) -u(g.x)) 2 dm(x)dµ(g)
and non degeneracy of f can be expressed as σ 2 (f ) > 0. Also we see that for t small, by perturbation theory on H ε (X), the operator

Q it has a simple dominant eigenvalue k(f, it) = k(it) which is isolated in the spectrum of Q it and satisfies |k(it)| = lim n→∞ (Q it ) n 1/n < 1. If f 2 (x, g)dm(x)dµ(g) < ∞ the second derivative of k(it) at zero exists and k ′ (f, 0) = iγ(f ), k ′′ (f, 0) = -(σ 2 (f ) + γ 2 (f ))
. The non-degeneracy (resp aperiodicity) of f can be caracterized as follows :

Proposition 1.3 With the above notations, the following conditions are equivalent a) f is degenerate (resp not aperiodic). b) For any t ∈ R (resp for some t = 0), there exists θ ∈ R and

ϕ t ∈ H ε (supp(m)) such that Q it ϕ t = e iθ ϕ t on supp(m).
The following simple Lemma will be useful in the proof of the local limit theorem below. Lemma 1.4 With the above notation assume f 2 (x, g)dm(x)dµ(g) < ∞, γ(f ) = 0, and f is non degenerate. Then for any ϕ ∈ H ε (X), t ∈ R, the sequence (Q it/ √ n ) n ϕ converges to e -t 2 σ 2 /2 m(ϕ). There exists A > 0, ε 0 > 0 such that for |t/ √ n| < ε 0 the sequence

(Q it/ √ n ) n is bounded by 2 e -At 2 .
Proof : Since Q is irreducible and satisfies condition (D-F), we have for t small, by perturbation in H ε (X), Q it ϕ = k(it)m t (ϕ)e t + R(it)ϕ where m t ⊗ e t is the projection operator on the line Ce t , (m t ⊗ e t )R(it) = R(it)(m t ⊗ e t ) = 0, and the spectral radius of R(it) is less than |k(it)|. Since γ(f ) = 0 and

f 2 (x, g)dm(x)dµ(g) < ∞ we have k(it) = 1 -σ 2 t 2 /2 + •(t 2 ), hence |k(it)| < 1 for t small non zero and lim n→∞ k n (it √ n) = lim n→∞ (1 -σ 2 t 2 /2n) n = e -σ 2 t 2 /2 . Also, for some C ∈ [0, |k(it)|[, C < 1 we have (Q it ) n = k n (it)m t ⊗ e t + R n (it) with (R(it)) n ≤ C n , lim t→0 m t ⊗ e t = m ⊗ 1. This gives the first conclusion. The asymptotic expansion of k(it) gives |k(it)| ≤ e -σ 2 t 2 /4 for |t| ≤ ε 0 with ε 0 > 0, hence |k n (it/ √ n)| ≤ e -σ 2 t 2 /4 . If |t/ √ n| ≤ ε 0 we have C n ≤ exp(t 2 ε -2 0 LogC). Since (Q it/ √ n ) n ≤ |k n (it/ √ n)| + R n (it/ √ n) we have (Q it/ √ n ) n ≤ 2e -At 2 with A = sup(σ 2 /4, ε -2 0 LogC).
Theorem 1.5 Assume that the Markov kernel Q on H ε (X) is irreducible, Q it satisfies condition (D-F) for any t ∈ R, the function f on X × S is aperiodic with γ(f ) = 0, and f 2 (x, g)dm(x)dµ(g) < ∞. Then for any ϕ ∈ H ε (X) and arbitrary interval I ⊂ R :

lim n→∞ σ √ 2πn E x (ϕ(s n .x)1 I (S n (x, .))] = m(ϕ)ℓ(I),
where ℓ is Lebesgue measure on R.

Proof : As in [START_REF] Breiman | Probability[END_REF] we consider the following sequence of Radon measures ℓ n x on X ×R defined by

ℓ n x (ϕ ⊗ u) = σ √ 2πn E x (ϕ(s n .x)u(S n (x, .
))) and we need to show the weak convergence of ℓ n

x to m ⊗ ℓ. Hence it suffices to show that lim n→∞ ℓ n x (ϕ ⊗ u) = m(ϕ)ℓ(u) for any continuous u ∈ L 1 (R) with compactly supported Fourier transform u (see [START_REF] Breiman | Probability[END_REF]). The Fourier inversion formula gives with ε > 0 :

E x (ϕ(s n .x)u(S n (x, .)) = (2π) -1 ∞ -∞ (Q it ) n ϕ(x) u(t)dt ℓ n x (ϕ ⊗ u) = σ √ n(2π) 1/2 |t|≥ε ′ (Q it ) n ϕ(x) u(t)dt + ε ′ √ n -ε ′ √ n (Q it/ √ n ) n ϕ(x) u(t/ √ n)dt
Since f is aperiodic, Proposition 1.3 gives that Q it has no unimodular eigenvalue if t = 0. Since Q it satisfies condition (D-F) it follows that the spectral radius of

Q it satisfies r(Q it ) < 1.
Hence the first term in the right hand side has limit zero. On the other hand, the above lemma shows that for

ε ′ small (Q it/ √ n ) n ϕ(x)1 [-ε ′ √ n,ε ′ √ n] (t) satisfies dominated convergence
with limit e -t 2 σ 2 /2 . The convergence of the second term to m(ϕ)

∞ -∞ e -σ 2 t 2 /2 u(0)dt = (σ -1 √ 2π) -1 m(ϕ)ℓ(u) follows.
Remark 1.6 Condition (D-F) for Q it (t = 0) and aperiodicity of f are satisfied if Q it has spectral radius less than one and f is non degenerate.

Clearly the same proof in the L 2 context with L 2 0 (X) = {ϕ ∈ L 2 (X) ; m(ϕ) = 0} gives Theorem 1.7 Assume that the Markov kernel Q is irreducible and its spectral radius r(Q 0 ) on L 2 0 (X) satisfies r(Q 0 ) < 1. Assume the Borel function f (x, g) on X × S satisfies γ(f ) = 0, σ 2 (f ) > 0, and the operator Q it on L 2 (X) has spectral radius less than 1 for any t = 0. Then for ϕ ∈ L 2 (X) and arbitrary interval I ⊂ R :

lim n→∞ σ √ 2πnE(ϕ(s n .x)1 I (S n (x, .)) = m(ϕ)ℓ(I).
For various examples where the conditions of the theorem are satisfied, see [START_REF] Conze | Ergodicity of group actions and spectral gap. Application to random walks and Markov shifts[END_REF]. Clearly the hypothesis and the result are stable under perturbation of Q in the operator norm and this property gives new examples. We observe that if r(Q 0 ) < 1, Proposition 3.4 below allows one to verify that the spectral radius of Q it is less than 1 in various situations. Then the spectral arguments developed in [START_REF] Guivarc | Théorèmes limites pour une classe de chaines de Markov et applications aux difféomorphismes d'Anosov[END_REF] can also be used for the proofs of various limit theorems, under a Hilbert space spectral hypothesis on Q. The condition r(Q 0 ) < 1 of the theorem is called "strong mixing" of Q in [START_REF] Rosenblatt | Markov processes, Structure and Asymptotic behaviour[END_REF]. For a study of the central limit theorem in a similar general framework, we refer to [START_REF] Rosenblatt | Markov processes, Structure and Asymptotic behaviour[END_REF]. The classical "Doeblin's condition" is stronger than "strong mixing". More precisely "Doeblin's condition" is equivalent to the fact that the spectral radius of Q 0 in L 1 0 (m) is less than 1.

2 Law of large numbers and spectral gap properties for products of random matrices.

In this section we describe some properties of products of random matrices which are essential for the proofs of limit theorems in sections 3, 4 below. In section 3 we will show the local limit theorem and large deviations asymptotics in a certain range for products of random matrices. These results will play also an important role in section 4, in the study of extreme values for affine random walks. We need to give a brief exposition of known results on products of random matrices. We begin by describing some notation. Let G = GL(V ) be the full linear group of the Euclidean vector space V = R d (d ≥ 2), G = SL(V ) the special linear group of V , P(V ) (resp S(V )) the projective space (resp unit sphere) of V . The scalar product on V will be written x, y if x, y ∈ V . The corresponding norm of x ∈ V (resp g ∈ G) will be written |x| (resp |g|) . For g ∈ G we write γ(g) = sup(|g|, |g -1 |). If x ∈ V we will denote x(resp x) its projection on P(V ) (resp S(V )). Also for x ∈ P(V ) we will write x ∈ S(V ) for one of its corresponding representatives in S(V ). The action of g on x ∈ P(V ) (resp y ∈ S(V )) will be written g.x (resp g.y). We will use on S(V ) (resp P(V ) the distance δ(x, ỹ) = |x -ỹ| (resp δ(x, y) = inf ( δ(x, ỹ), δ(-x, ỹ)). We will also consider the distance δ on P(V ) given by δ(x, y) = |x ∧ y| = sin(x, y) which is equivalent to δ. We note that the natural norm |x ∧ y| on the wedge product Λ 2 V satisfies

|x| 2 |y| 2 = |x ∧ y| 2 + | x, y | 2 .
For s ∈ R we write s = inf(s, 1), ŝ = sup(s, 1). For a subset E of a group we write E (resp [E]) for the closed subgroup (resp semigroup) generated by E. If g ∈ G has a simple dominant eigenvalue λ g we will write V = Rv g ⊕ V < g where v g is a dominant eigenvector and V < g the corresponding supplementary hyperplane. In this case we will say that g is proximal and we denote by g + ∈ P(V ) the attracting fixed point defined by v g . For a semigroup Γ ⊂ G we will write Γ prox for the subset of proximal elements in Γ. The closure of {g + ; g ∈ Γ prox } will play an important role below in the discussion of aperiodicity for the function f (x, g) = Log|g x| with g ∈ G, x ∈ P(V ) (see Corollary 2.12). It will called limit set of Γ. The dual space of V will be denoted V * and the dual map of g by g * . We will say that a semigroup Γ ⊂ G satisfies condition (I-P) if a) No finite union of proper subspaces is Γ-invariant (strong irreducibility of Γ). b) Γ contains a proximal element. It follows from [START_REF] Goldsheid | Lyapunov exponents of a product of random matrices[END_REF] that condition (I-P) for Γ, Γ and the Zariski closure Zc(Γ) of Γ in G are all equivalent. Since Zc(Γ) is a Lie subgroup of G with a finite number of connected components, condition (I-P) is easily verified. If the field R is replaced by the local field Q p , the corresponding (I-P) condition is useful (see [START_REF] Guivarc | On the Spectrum of a Large Subgroup of a Semisimple Group[END_REF]). If Γ acts on a locally compact metric space X and if λ(resp ν) is a positive measure on Γ (resp X) we define the convolution λ * ν by (λ * ν)(ϕ) = ϕ(γ.x)dλ(γ)dν(x) where ϕ ∈ C(X) is compactly supported. We will denote by M 1 (E) the space of probabilities on a Polish space E. In particular if µ ∈ M 1 (Γ) we will write µ n for its convolution power of order n. Also we will write P = µ ⊗N for the corresponding probability on Γ N . The product of the random matrices

g k ∈ G(1 ≤ k ≤ n) will be written S n = g n • • • g 1 .
The spectral radius of a bounded linear operator U of a Banach space into itself will be denoted r(U ) = lim n→∞ |U n | 1/n . We begin by considering products of unimodular random matrices ; we follow ( [START_REF] Bougerol | Products of random matrices with applications to Schrödinger operators[END_REF] [16], [START_REF] Guivarc | Quelques propriétés asymptotiques des produits de matrices aléatoires[END_REF], [START_REF] Guivarc | On contraction properties for products of Markov driven random matrices[END_REF]) with a few changes. We start with the following (see [START_REF] Guivarc | Quelques propriétés asymptotiques des produits de matrices aléatoires[END_REF], [START_REF] Vircer | Matrix and operator random products Teor[END_REF]).

Theorem 2.1 Let V = R d (d ≥ 2), G = SL(V ), µ ∈ M 1 (G)
be such that the subgroup < supp(µ) > is non-compact and strongly irreducible, and denote by P µ the convolution operator defined by µ on the G-space V \ {0}. Let L 2 (V ) be the Hilbert space on V with respect to Lebesgue measure and r V (µ) the spectral radius of P µ acting on L 2 (V ). Then r V (µ) < 1.

We have the following corollaries. Log|g|dµ n (g) converges to a positive number :

lim n→∞ 1 n Log|g|dµ n (g) ≥ - 1 d Log r V (µ) > 0.
Corollary 2.3 Assume supp(µ) satisfies condition (I-P), denote by Pµ the operator on P(V ) defined by convolution with µ. Then Pµ has a unique invariant probability measure ν. If Log|g|dµ(g) is finite the sequence of functions on P(V ),

1 n Log|gx|dµ n (g) converges uniformly to L µ = Log|gx|dµ(g)dν(x) > 0.
This is a weak form of the well known Furstenberg's result giving exponential growth for products of unimodular random matrices : lim [START_REF] Furstenberg | Boundary theory and stochastic processes on homogeneous spaces[END_REF]). This fact follows also from the above and the subadditive ergodic theorem.

n→∞ 1 n Log|g n • • • g 1 | = L µ > 0 P -a.e (see

Corollary 2.4

We consider the d-dimensional torus

T d = R d /Z d (d > 1)
, the action of a semigroup Γ ⊂ SL(d, Z) on T d by automorphisms, a probability measure µ ∈ M 1 (Γ) such that the action of the group < supp(µ) > on R d is strongly irreducible. Then the spectral radius r 0 (µ) of the corresponding convolution action on L 2 0 (T d , m) where m is Lebesgue measure, satisfies r 0 (µ) < 1.

We observe that this corollary (see [START_REF] Furman | Sharp ergodic theorems for group actions and strong ergodicity[END_REF]) is a very special case of a general result of automorphismactions on a compact nilmanifold N/∆ used below : the action of Γ ⊂ Aut(N/∆) on L 2 0 (N/∆) does not contain weakly the identity representation if and only if there is no Γ-equivariant torus factor T of N/∆ such that the corresponding quotient group acting on T is virtually abelian (see [START_REF] Bekka | On the spectral theory of groups of affine transformations on compact nilmanifolds[END_REF]).

Corollary 2.5 With the hypothesis of Corollary 2.3 we assume that for some τ > 0, |g| τ dµ(g) is finite. Then there exists ε 0 > 0 such that for any ε ∈]0, ε 0 ] :

lim n→∞ sup |x|=1 |gx| -ε dµ n (g) 1/n < 1.
Then in particular if dim V = 2, and 2ε ∈]0, ε 0 ] one has :

ρ(ε) = lim n→∞ sup x,ȳ∈P(V ) δ ε (g.x, g.ȳ) δ ε (x, ȳ) dµ n (g)) < 1,
and the spectral radius of Pµ on

H ε 0 (P(V )) = {ϕ ∈ H ε (P(V )); ν(ϕ) = 0} is at most ρ(ε) < 1.
Remark 2.6 a) As the proof below shows, we can replace in the statement of the theorem V by Z d , i.e SL(V ) by SL(d, Z) ; then r V (µ) is replaced by the spectral radius of the convolution by µ on Z d \ {0}, less than 1 under the same condition. b) Strong irreducibility is a necessary hypothesis for the validity of the theorem :

if µ = 1 2 (δ a + δ b ) with a = λ 0 0 1/λ , b = 0 1 -1 0 , λ > 1 we get with standard notations on P 1 (R 2 ) : ν = 1 2 (δ ē1 + δ ē2 ), L µ = lim n→∞ 1 n Log|g|dµ n (g) = 0. c) If d = 2, condition (I-P) for [supp(µ)
] is equivalent to strong irreducibility and noncompactness of < supp(µ) >.

For the proof of the theorem we need the lemma (see [START_REF] Furstenberg | Boundary theory and stochastic processes on homogeneous spaces[END_REF]) :

Lemma 2.7 Assume that a semigroup Γ ⊂ SL(V ) is strongly irreducible and non-compact.

Then there is no Γ-invariant probability measure on P(V ).

Proof of Lemma 2.7 Assume ν ∈ M 1 (P(V )) satisfies g.ν = ν for every g ∈ Γ. Since Γ is unbounded there exists g n ∈ Γ with lim

n→∞ |g n | = ∞. We denote u n = |g n | -1 g n and we observe that |det u n | = |g n | -d converges to zero. Since |u n | = 1,
we can extract a convergent subsequence in EndV , again denoted u n and we have lim

n→∞ u n = u, with |u| = 1, det u = 0.
We write W = Ker u, W ′ = Im u and we denote by W and W ′ the corresponding projective subspaces with W c = P(V ) \ W . We decompose ν as ν = ν 1 + ν 2 where

ν 1 (resp ν 2 ) is the restriction of ν 1 (respν 2 ) to W c (resp W ) and we observe that ν = lim n→∞ g n .ν = lim n→∞ g n .ν 1 + u.ν 2 where u.ν 2 is supported by W ′ . Since P(V ) is compact we can assume, taking a subsequence, that g n .ν 1 converges to ν ′ 1 which is supported on the subspace W 1 = lim n→∞ g n .W . Since g.ν = ν it follows that ν(W 1 U W ′ ) = 1.
We consider the set Φ of subsets F of P(V ) which are finite unions of proper subspaces with ν(F ) = 1.

Since any decreasing sequence in Φ is finite, Φ has a least element F 0 . Since g.ν = ν, we have g.F 0 = F 0 for any g ∈ Γ, hence Γ is not strongly irreducible. This gives the required contradiction.

Proof of Theorem 2.1 Assume r V (µ) = 1 and let z = e iθ , |z| = 1 be a spectral value of P µ = P . Then either lim

n→∞ P ϕ n -zϕ n 2 = 0 for some sequence ϕ n ∈ L 2 (V )
with ϕ n 2 = 1 or Im(P -zI) is not dense in L 2 (V ). In the second case, duality gives zP * ϕ = ϕ for some ϕ ∈ L 2 (V ), ϕ = 0. Since SL(V ) preserves Lebesgue measure on V , P is a contraction of L 2 (V ), hence the fixed points of z P and zP * in L 2 (V ) are the same i.e P ϕ = zϕ. Hence it suffices to consider the first case. Then the condition lim

n→∞ P ϕ n -zϕ n 2 = 0 is equivalent to lim n→∞ < P ϕ n , zϕ n >= 1. Since |P ϕ n | ≤ P |ϕ n | and P |ϕ n |, |ϕ n | ≤ 1 we get lim n→∞ P |ϕ n |, |ϕ n | = 1, i.e lim n→∞ |ϕ n | • g, |ϕ n | dµ(g) = 1. Since |ϕ n | • g, |ϕ n | ≤ 1 it follows that, taking a subsequence, lim n→∞ |ϕ n | • g, |ϕ n | = 1 µ -a.e, i.e lim n→∞ |ϕ n | • g -|ϕ n | 2 = 0 µ -a.e.
On the other hand we have

|ϕ n | 2 • g -|ϕ n | 2 1 ≤ |ϕ n | • g -|ϕ n | 2 |ϕ n | • g + |ϕ n | 2 ≤ 2 |ϕ n | • g -|ϕ n | 2 , hence lim n→∞ |ϕ n | 2 • g -|ϕ n | 2 1 = 0 µ -a.e.
We consider the probability ν n on V \ {0} defined by ν n = |ϕ n | 2 ℓ, where ℓ is Lebesgue measure on V , and its projection ν n on P(V ). Then we have in variation norm lim n→∞ gν n -ν n = 0, hence lim n→∞ g.ν n -ν n = 0 µ -a.e, and also lim n→∞ g.ν n -ν n = 0 weakly for any g ∈ supp(µ). Taking a weak limit ν of a subsequence ν n k we get g.ν = ν for any g ∈ supp(µ), hence ν is < supp(µ) >-invariant. From the lemma this is impossible because Γ =< supp(µ) > is non-compact and strongly irreducible.

Proof of Corollary 2.2

We have 1 = det g ≤ |g| d , hence |g| ≥ 1 for g ∈ G, and ∞ > Log|g|dµ n (g) = u n ≥ 0. Also by subadditivity of Log|g| :

u m+n ≤ u m + u n . It follows n -1 u n ≤ u 1 < ∞, lim n→∞ n -1 u n = inf n≥1 n -1 u n = c ≥ 0.
For ε > 0 we consider the L 2 functions f ε and 1 C on V given by :

f ε (v) = inf(1, |v| -d-ε ), C = {1 ≤ |v| ≤ 2}.
The theorem gives : lim sup

n→∞ | < P n µ f ε , 1 C > | 1/n ≤ r V (µ) < 1. But : P n µ f ε , 1 C ≥ C |gv| -d-ε dµ n (g)dv ≥ 2 -d-ε ℓ(C) |g| -d-ε dµ n (g),. It follows Log P n µ f ε , 1 C ≥ -(d + ε)u n + Log(2 -d-ε ℓ(C)), (d + ε) lim n→∞ n -1 u n ≥ -Log r V (µ). Hence c ≥ -d -1 Log r V (µ) > 0 Proof of Corollary 2.3
The uniqueness of the P µ -invariant measure ν ∈ M 1 (P(V )) will be shown later (Theorem 2.11). This uniqueness implies the uniform convergence of the sequence

1 n n-1 Σ 0 P k µ ϕ(x) to ν(ϕ), for any ϕ ∈ C(P(V )). We can write : Log|gx|dµ n (g) = Log|g n g n-1 • • • g 1 x|dP(ω) = n Σ 1 Log|g k (g k-1 • • • g 1 .x)|dP(ω) = n Σ 1 Log|g(h.x)|dµ(g)dµ k-1 (h) = n Σ 1 P k-1 µ ϕ(x) with ϕ(x) = Log|gx|dµ(g).
The condition Log|g|dµ(g) < ∞ implies the continuity of ϕ on P(V ). From above we get the uniform convergence of 1 n Log|gx|dµ n (g) to ϕ(x)dν(x). The positivity of the limit follows from 2.2.

Proof of Corollary 2.4

We observe that m is preserved by any γ ∈ SL(d, Z) and we can write

L 2 (T d ) = L 2 (T d , m) as L 2 (T d ) = C1 ⊕ L 2 0 (T d ). For any ϕ ∈ L 2 (T d ) we have T µ ϕ(x) = µ * ϕ(x) = Σ γ ϕ(γ -1 x)µ(γ). We observe that the Fourier transform of ϕ, i.e ϕ(k) =< ϕ, e k > where e k (x) = e 2iπ<k,x> and k ∈ Z d , gives an isometry ϕ → ϕ from L 2 (T d ) to ℓ 2 (Z d ), in which L 2 0 (T d ) corresponds to ℓ 2 (Z d \ {0}). Also for g ∈ SL(d, Z), we have ϕ • g = ϕ • g * .
Clearly the stated hypothesis on < supp(µ) > carries over to < supp(µ) > * . If < supp(µ) > is strongly irreducible on R d , then supp(µ) ⊂ SL(d, Z) is infinite, hence non-compact. Hence we can use Theorem 2.1 and the Remark 2.6 to deduce that the spectral radius r 0 (µ) of T µ on L 2 0 (T d ) satisfies r 0 (µ) < 1.

Proof of Corollary 2.5 We denote u n (ε) = sup

|x|=1 |gx| -ε dµ n (g). Since |gx| ≥ |g -1 | -1 ≥ |g| -(d-1) the integral is finite for ε(d -1) ≤ τ . We have u m+n (ε) ≤ u m (ε)u n (ε) because : |gx| -ε dµ m+n (g) = |g(h.x)| -ε |hx| -ε dµ m (g)dµ n (h) ≤ sup |y|=1 |gy| -ε dµ m (g) |hx| -ε dµ n (h) ≤ u m (ε)u n (ε). Hence lim n→∞ |u n (ε)| 1/n = inf n (u n (ε)) 1/n . It follows that it suffices to show u n (ε) < 1 for some n ≥ 1.
We write |gx| -ε = exp(-εLog|gx|) and use the inequality e u ≤ 1 + u + u 2 e |u| which gives

|gx| -ε ≤ 1 -εLog|gx| + (ε 2 Log 2 |g|)|g| (d-1)ε , |gx| -ε dµ n (g) ≤ 1 -ε Log|gx|dµ n (g) + ε 2 |g| (d-1)ε Log 2 |g|dµ n (g).
The last integral is bounded by ( (Log|g|) 4 dµ n (g)) 1/2 ( |g| 2ε(d-1) dµ n (g))

(g) ≥ c > 0, hence lim inf ε→0 + 1 -u n (ε) ε ≥ c > 0, i.e u n (ε) < 1 for ε ≤ ε 0 small.
We observe that δ(x, ȳ) = |x ∧ y| where |x| = |y| = 1 is also a distance on P(V ) which is equivalent to δ since δ(x, ȳ) = |sin(x, y)|. Also δ(g.x, g.ȳ) = |gx| |gy|δ(x, ȳ) since g ∈ SL(2, V ). If follows :

δ ε (g.x)(g.ȳ) δ ε (x,ȳ dµ n (g) = 1 |gx| ε |gy| ε dµ n (g) ≤ ( |gx| -2ε dµ n (g)) 1/2 ( |gy| -2ε dµ n (g)) 1/2 ≤ sup |x|=1 |gx| -2ε dµ n (g). The inequality ρ(ε) < 1 follows if 2ε ≤ ε 0 . For any ϕ ∈ H ε (P(V )) we have : ϕ(x) -ν(ϕ) = (ϕ(x) -ϕ(y))dν(y). It follows that |ϕ -ν(ϕ)| ∞ ≤ [ϕ] ε , hence |P n µ ϕ| ∞ ≤ [P n µ ϕ] ε if ϕ ∈ H ε 0 (P(V ), and the norms ϕ → ϕ , ϕ → [ϕ] ε on H ε 0 (P(V )) are equivalent. Also [P n µ ϕ] ε ≤ [ϕ] ε sup x,ȳ∈P(V ) δ ε (g.x, g.ȳ) δ ε (x, ȳ) dµ n (g). Since |P n µ ϕ| ∞ ≤ [P n µ ϕ] ε on H ε 0 (P(V )) we get P n µ ≤ 2 sup x,ȳ∈P(V ) δ ε (g.x, g.ȳ) δ ε (x, ȳ) dµ n (g) on H ε 0 (P(V )).
The last conclusion follows from the above inequality ρ(ε) < 1.

Corollary 2.5 has a natural extension to any dimension d ≥ 2. The proof is based on a basic inequality between the two first Lyapunov exponents associated with µ ∈ M 1 (GL(V )) (see [START_REF] Goldsheid | Lyapunov exponents of a product of random matrices[END_REF], [START_REF] Guivarc | Products of random matrices and convergence theorems[END_REF]). We consider the ergodic dynamical system (Ω, θ, π) where Ω = G N , θ is the shift, π is a θ-invariant probability, f a GL(d, R)-valued function on Ω which satisfies

(Log + |f (ω)| + Log + |f -1 (ω)|)dπ(ω) < ∞, and we write S n (ω) = f (θ n-1 ω) • • • f (ω) for n ≥ 0. Then the top Lyapunov exponent L 1
f is defined by

L 1 f = lim n→∞ 1 n Log|S n (ω)|dπ(ω).
Using the wedge products Λ k V we define the other Lyapunov exponents L k f by :

L 1 f + • • • + L k f = lim n→∞ 1 n Log|Λ k S n (ω)|dπ(ω).
Here we consider the two first Lyapunov exponents L 1 µ , L 2 µ for products of independant random matrices with law µ i.e π = µ ⊗N = P. We have the following result (see [START_REF] Guivarc | Products of random matrices and convergence theorems[END_REF]).

Theorem 2.8 Let µ ∈ M 1 (GL(d, R)) be such that Logγ(g) is µ-integrable and the semi- group [supp(µ)] of GL(d, R) satisfies condition (I-P). Then L 1 µ > L 2 µ .
Condition (I-P) is satisfied for exemple, if the projection of the group < supp(µ) > on SL(d, R) is Zariski-dense in SL(d, R) (see [START_REF] Goldsheid | Lyapunov exponents of a product of random matrices[END_REF]).

The geometrical meaning of the theorem is that for any x, y ∈ V \ {0}, the angle between the random vectors S n (ω)x, S n (ω)y decreases P-a.e to zero at exponential speed exp(L 2 µ -L 1 µ ) < 1. The proof of this fact is based on a martingale argument due to H. Furstenberg and given below (see Theorem 2.11) : if ν is a P µ -invariant probability on P(V ) then the sequence of random measures g 1 • • • g n .ν on P(V ) converges weakly to a Dirac measure. The uniqueness of ν as a P µ -invariant measure follows (see below). A useful algebraic tool for passing from ergodic properties of P µ to Lyapunov exponents is the following cocycle σ : if ξ ∈ T 1 is a unit tangent vector to P(V ) defined by x ∈ V and x ∧ y ∈ ∧ 2 (V ), σ(g, ξ) = |g(x∧y)||x| 2 |gx| 2 |x∧y| . Then, using the above convergence, one shows that, under condition (I-P) (see [START_REF] Guivarc | Products of random matrices and convergence theorems[END_REF]) :

lim n→∞ 1 n sup ξ∈T 1 Logσ(g, ξ)dµ n (g) < 0.
The extension of Corollary 2.4 is as follows :

Theorem 2.9 Assume that µ ∈ M 1 ( G) satisfies γ τ (g)dµ(g) < ∞ for some τ > 0 and the semigroup [supp(µ)] satisfies (I-P). Then for ε sufficiently small one has :

ρ(ε) = lim n→∞ sup x,ȳ δ ε (g.x, g.ȳ) δ ε (x, ȳ) dµ n (g) 1/n < 1.
In particular the spectral radius of P µ acting on

H ε 0 (P(V )) = {ϕ ∈ H ε (P(V )); ν(ϕ) = 0} is at most ρ(ε) < 1.
In order to deduce limit theorems for products of random matrices (see [START_REF] Page | Théorèmes limites pour les produits de matrices aléatoires[END_REF]), we are led to consider the "Fourier operators" P it µ on H ε (P(V )) defined by P it µ ϕ(x) = |gx| it ϕ(g.x)dµ(g).

Corollary 2.10 We denote for any k ≥ 0, ε > 0 :

C k (ε) = |g| ε dµ k (g), ρ k (ε) = sup x,ȳ∈P(V ) δ ε (g.x, g.ȳ) δ ε (x, ȳ) dµ k (g)
Then, for any t ∈ R, ϕ ∈ H ε (P(V )), k ≥ 0 :

[(P it µ ) k ϕ] ε ≤ ρ k (ε)[ϕ] ε + |ε -1 t|C k (ε)|ϕ| ∞ .
In particular for ε small and k large the operator P it µ satisfies the (D-F) inequality,

[(P it µ ) k ϕ] ε ≤ ρ ′ (ε)[ϕ] ε + |ε -1 t|C ′ (ε)|ϕ| ∞ with ρ ′ (ε) < 1, C ′ (ε) < ∞ Proof of Theorem 2.9 We have if |x| = |y| = 1, δ(g.x, g.ȳ) δ(x, ȳ) = |gx ∧ gy| |gx||gy||x ∧ y| .
Hence, by Schwarz inequality δ ε (g.x,g.ȳ)

δ ε (x,ȳ) dµ n (g) 2 ≤ |gx∧gy| |gx| 2 |x∧y| ε dµ n (g) |gx∧gy| |gy| 2 |x∧y| ε dµ n (g).
Hence it suffices to show

lim n→∞ sup ξ∈T 1 σ ε (g, ξ)dµ n 1/n < 1.
The proof is the same as for Corollary 2.5 where |gx| -ε replaces σ ε (g, ξ), once we have

lim n→∞ 1 n sup ξ∈T 1
Logσ(g, ξ)dµ n (g) < 0 which, as explained above, follows from

L 1 µ > L 2 µ .
The spectral gap assertion on H ε 0 (P(V )) is proved as in Corollary 2.5.

Proof of Corollary 2.10 We use the following mean value inequality for

|x| = |y| = 1, t ∈ R, ε ∈]0, 1] : ||gx| it -|gy| it | ≤ |ε -1 t| |g| ε |x -y| ε . Hence, with x, ȳ ∈ P(V ), |x| = |y| = 1 : ||gx| it -|gy| it | ≤ |ε -1 t| |g| ε δε (x, ȳ), ||gx| it ϕ(g.x) -|gy| it ϕ(g.ȳ)| ≤ |ϕ| ∞ ||gx| it -|gy| it | + [ϕ] ε δε (g.x, g.ȳ).
The first inequality follows by integration will respect of µ k . Let ε > 0 with 0 < 2ε < ε 0 as in Corollary 2.5. Since lim

k→∞ sup x,ȳ∈P(V ) δ ε (g.x, g.ȳ) δ ε (x, ȳ) dµ k (g) 1/k < 1 we can fix k > 0 with ρ ′ (ε) = ρ k (ε) < 1. Taking ε with ε < τ , we have C k (ε < ∞). Hence (D-F) inequality follows. Theorem 2.11 Assume µ ∈ M 1 ( G), ν ∈ M 1 (P(V )
) is P µ -invariant and [supp(µ)] satisfies condition (I-P). Then the sequence σ n .ν = g 1 g 2 • • • g n .ν converges P -a.e to a Dirac measure δ z (ω). In particulier ν is unique as a P µ -invariant probability and gives zero mass to any proper subspace of P(V ).

The following corollaries will play an essential role in the proofs of the local limit theorem for Log|S n (ω)x|.

Corollary 2.12 Let ∆ be a closed subsemigroup of G which satisfies condition (I-P). Then the action of ∆ on P(V ) has a unique minimal subset Λ(∆) and Λ(∆) is not contained in a countable union of proper subspaces. Furthermore Λ(∆) is the closure of the set of attractive points of ∆.

The set Λ(∆) is called the limit set of ∆. This corollary (see Proposition 3 in [START_REF] Guivarc | Products of random matrices and convergence theorems[END_REF]) is useful in the verification of the aperiodicity conditions in the proofs of the local limit theorems. In view of its essential role in limit theorems we give a detailed proof below.

Corollary 2.14 Assume Γ ⊂ GL(V ) is a subsemigroup which satisfies (I-P), d ≥ 2 and define S Γ = {Log|λ g | ; g ∈ ∆ prox }. Then S Γ generates a dense subgroup of the additive group R.

This property plays an essential role in the study of renewal theorems for products of random matrices (see [START_REF] Guivarc | Spectral gap properties and asymptotics of stationary measures for affine random walks[END_REF]), and also in some geometric situations (see [START_REF] Broise | Etudes spectrales d'opérateurs de transfert et applications[END_REF]).

For the proof of Theorem 2.11 we use the following lemmas.

Lemma 2.15 Let ϕ ∈ C(P(V )) and denote

f n (ω) = σ n .ν, f r n (ω, η) = σ n η 1 • • • η r .ν for (ω, η) ∈ Ω × Ω.
Then for r fixed the sequence f r n (ω, η) -f n (ω) converges P × P -a.e to zero.

Lemma 2.16 For any proper subspace W of P(V ) one has ν(W ) = 0.

Proof of Lemma 2.15 Since g → g.ν(ϕ) is µ-harmonic, f n (ω) is a martingale and we have for any r ≥ 0 :

E(f n+r -f n ) 2 = E(f 2 n+r ) -E(f 2 n ).
Hence for any N > 0,

N Σ 1 E(f 2 n+r -f 2 n ) ≤ 2r|ϕ| 2 ∞ . It follows that the series ∞ Σ 1 E(f 2 n+r -f 2 n ) is convergent. Also : E(f 2 n+r -f 2 n ) = |f r n (ω, η) -f n (ω)| 2 d(P ⊗ P)(ω, η).
Hence the series

∞ Σ 1 |f r n (ω, η) -f n (ω)| 2 is P ⊗ P -a.e convergent.
In particular for r fixed and P -a.e η = (η

1 , • • • , η r ) : lim n→∞ |f r n (ω, η) -f n (ω)| = 0.
Proof of Lemma 2. [START_REF] Furstenberg | Boundary theory and stochastic processes on homogeneous spaces[END_REF] We consider the set Φ of proper projective subspaces

H of P(V ) of minimal dimension r such that ν(H) > 0. Clearly if H, H ′ ∈ Φ with H = H ′ , then ν(H ∩ H ′ ) = 0. It follows that for any ε > 0 the set Φ ε = {H ∈ Φ ; ν(H) ≥ ε} is finite. In particular, there exists H 0 ∈ Φ with ν(H 0 ) = sup{ν(H) ; h ∈ Φ} and the set Φ ′ = {H ∈ Φ ; ν(H) = ν(H 0 )} is finite. Since g -1 H 0 and H 0 have the same dimension r the equation ν(H 0 ) = (g.ν)(H 0 )dµ(g) implies ν(g -1 H 0 ) = ν(H 0 ) for µ-almost every g. Hence g -1 H 0 ∈ Φ ′ , g -1 Φ ′ ⊂ Φ ′ for any g ∈ supp(µ). Since Φ ′ is finite this contradicts strong irreducibility of [suppµ]. Hence Φ = φ.
Proof of Theorem 2.11 Using Lemma 2.15 we know that the sequence f n (ω) is a bounded martingale. The martingale convergence theorem implies the P -a.e convergence of f n (ω). Since C(P(V )) is separable, we get the weak convergence of σ n .ν to the random probability ν ω . We denote by Ω ′ ⊂ Ω the set of ω such that σ n .ν converges weakly and such that the convergence in Lemma 2.15 for any ϕ ∈ C(P(V )) takes place with P⊗P-probability 1. Then, P(Ω ′ ) = 1 and for any ω ∈ Ω ′ , and for P -a.e η :

lim n→∞ σ n η 1 • • • η r .ν = lim n→∞ σ n .ν.
Let n k (ω) be a subsequence such that |σ n k | -1 σ n k converges to a linear map τ ω with kernel H ω , hence the quasi projective map τ ω is defined and continuous on P(V ) \ H ω . Then by Lemma 2.16, ν and η 1 • • • η r .ν gives zero mass to any proper subspace, hence

τ ω (η 1 • • • η r .ν) = τ ω .ν = ν ω .
This equality is valid for a dense subset of (supp(µ)) ⊗r , hence for any γ ∈ (supp(µ)) r : τ ω (γ.ν) = τ ω .ν. Then for any γ in [supp(µ)] we have τ ω .(γ.ν) = ν ω = τ ω .ν. Let γ 1 be a proximal element of [supp(µ)] with γ + 1 = x, and γ 0 ∈ [supp(µ)] with γ 0 .x / ∈ H ω (γ 0 exists by strong irreducibility of [supp(µ)], see [START_REF] Goldsheid | Zariski closure and the dimension of the Gaussian law of the product of random matrices[END_REF]). Then taking γ = γ 0 γ n 1 ∈ [supp(µ)] we get lim n→∞ γ 0 γ n 1 .ν = γ 0 .δ x . The continuity of x → τ ω .x outside H ω gives finally :

τ ω .ν = ν ω = τ ω (γ 0 .δ x ) = δ τωγ 0 .x . It follows that for ω ∈ Ω ′ , ν ω is a Dirac measure δ z(ω) and τ ω (P(V ) \ H ω ) = z(ω).
The martingale convergence theorem implies that the P-law of z(ω) is ν. Since τ ω .ν = δ z(ω) and τ ω is independant of ν, this gives the uniqueness of ν.

Proof of Corollary 2.12 Let µ ∈ M 1 ( G), be such that [supp(µ)] = ∆, and ν ∈ M 1 (P(V )) be the unique P µ -invariant probability. If F is a closed ∆-invariant subset of P(V ), then Kakutani's fixed point theorem implies that there exists a P µ -invariant probability η ∈ M 1 (F ) ; the theorem gives η = ν, hence F ⊃ supp(η) = supp(ν). The first assertion follows. If supp(ν) is contained in a countable union of subspaces, then at least one of them has positive measure, which contradicts Theorem 2.11. The last assertion follows from the minimality of Λ(∆).

Proof of Corollary 2.13 Let Λ(∆) be the inverse image of Λ(∆) in V \ {0} and let φ be the function on Λ(∆) defined by φ(v) = ϕ(v)|v| it where we can assume t ≥ 0. Then the relation satisfied by ϕ can be written as φ(γv) = e iθ φ(v), hence φ(δv) = e irθ φ(v) if δ ∈ A r , r ∈ N. Assume t > 0, hence ρ = e 2πt > 1. We have also φ(λv) = |λ| it φ(v) and in particular φ(±ρ k v) = φ(v), k ∈ Z. We observe that, since ∆ acts minimally on Λ(∆) and ϕ is continuous, the invariance condition on ϕ implies |ϕ| =const, hence we can assume |ϕ| = 1. It follows that the condition φ(λv) = φ(v) for some v ∈ Λ(∆) is equivalent to λ = ±ρ k (k ∈ Z). We define Λ c ⊂ Λ(∆) by Λ c = φ-1 (c) where c is a value of φ hence Λ c = ±ρ Z Λ c = φ. It follows that Λ c is closed as well as its projection in Λ(∆). We note that for β ∈ R, one has

Λ ce iβ = exp(βt -1 )Λ c , γ(Λ c ) ⊂ Λ ce iθ for γ ∈ A, hence δ(Λ c ) ⊂ exp(rθt -1 )Λ c if δ ∈ A r .
Then the above projection of Λ c in Λ(∆) is ∆invariant, hence equal to Λ(∆) by minimality. Let γ k be sequence in ∆ and n k ∈ Z such that u = lim k→∞ ρ -n k γ k = 0. Then, since γ k (Λ c ) ⊂ Λ ce iθ k and ±ρ Z Λ c = Λ c , we have also u(Λ c ) ⊂ Λ ce iα U {0} with e iα = lim k→∞ e iθ k , passing if necessary to a subsequence. Let δ ∈ A r be a proximal element and γ k = δ k , hence Keru = W = {0} has codimension 1 and Imu = Rv δ for some v δ = 0. From the above we get u(Λ c ) ⊂ ±ρ Z v δ U {0}. Since u -1 ({v δ }) = a + W with a = 0, we get Λ c \ W ⊂ ±ρ Z (a + W ), hence Λ c has a countable projection in V /W . Let F c be the family of subspaces W ′ in V such that Λ c has a countable projection in V /W ′ . We observe that the condition Λ ce iβ = exp(βt -1 )Λ c gives

F ce iβ = F c for β ∈ R. Also if W 1 , W 2 ∈ F c , then W 1 ∩ W 2 ∈ F ; this follows since V /W 1 ∩ W 2 is a subspace of V /W 1 × V /W 2 . Then F c has a minimum element W c = ∩ W ′ ∈Fc W ′ . Since F c = F ce iθ , we have W c = W ce iθ , hence the condition γΛ c ⊂ Λ ce iθ for γ ∈ A implies γ(W c ) ⊂ W c .
Since the projection of Λ c in P(V ) is Λ(∆) and, since dimV ≥ 2, we know from the theorem that Λ(∆) is uncountable, it follows that W c = {0}. This contradicts irreducibility of ∆. Hence we have t = 0, ϕ(x) = e iθ ϕ(γ.x) for any x ∈ Λ(∆), γ ∈ A. Taking γ proximal and x = γ + we get e iθ = 1 since |ϕ(x)| = 1, hence ϕ =const by minimality of the ∆-action on Λ(∆).

We need the following lemmas, the first of them is well known (see [START_REF] Bowen | Equilibrium states and the ergodic theory of Anosov diffeomorphisms[END_REF] p 90-94, Lemma 2.16). The second one is basic.

Lemma 2.17 Let A be a finite set, Ω the compact metric space A N and θ the shift on Ω. Lemma 2.18 Assume g, h ∈ GL(V ) are such that h is proximal and g.h + / ∈ V < h . Then for n = 2p large, g h n is proximal and :

lim n→∞ (g h n ) + = g.h + , lim n→∞ V < g h n = V < h .
Proof : We consider the sequence u n = |h n | -1 h n and we observe that u n converges to a map π h proportional to the projection on R v h along the hyperplane V < h . Hence lim

n→∞ g u n = g π h and Im(g π h ) = R(g.h + ), Ker(g π h ) = V < h . Since g.h + / ∈ V < h
, g π h is collinear to a projection onto a one-dimensional subspace. Since g π h has a simple dominant eigenvalue, the same is true for g u n for n large. Hence for n large, g h n is proximal and we have the required convergences.

Lemma 2.19 Assume Γ ⊂ GL(V ) is a subsemigroup which satisfies condition (I-P). Then there exists a, b ∈ Γ prox with a

+ = b + , V < a = V < b and a + / ∈ V < b , b + / ∈ V < a .
Proof :We consider the dual semigroup Γ * acting on the dual space V * and the projective space P(V * ). Then condition (I-P) is also valid for Γ * and we can consider the corresponding limit set Λ(Γ * ). We fix a ∈ Γ prox and observe that we can find b ∈ Γ prox with

V < b = V < a , a + / ∈ V < b .
Otherwise there would be a dense subset of Λ(Γ * ) contained in the union of the two projective subspaces in P(V * ) defined by V < a and a + . Hence Λ(Γ * ) would be contained in their union, but from Corollary 2.12, this is impossible. If b + / ∈ {a + } ∪ V < a we have found the required pair (a, b). If not, we consider g ∈ Γ and the sequence g b n (n ∈ 2N). By strong irreducibility we can choose g ∈ Γ such that g.b + / ∈ V < b ∪ V < a ∪ {a + }. Then we apply the above lemma and replace b by g b n = b ′ for n large. Then V < b ′ is close to V < b and the above relations are still satisfied. Since (b ′ ) + is close to g.b + and g.b

+ / ∈ {a + } ∪ V < a , the condition (b ′ ) + / ∈ {a + } ∪ V <
a is also satisfied. Hence we can take (a, b ′ ) as the required pair.

Proof of Corollary 2.14 Since Γ satisfies (I-P) we can choose a 1 , a 2 in Γ according to Lemma 2. [START_REF] Goldsheid | Zariski closure and the dimension of the Gaussian law of the product of random matrices[END_REF]

. Let C 1 , C 2 be closed disjoint neighbourhoods of a + 1 , a + 2 in P(V ) such that (C 1 ∪ C 2 ) ∩ (V < a 1 ∪ V < a 2 ) = φ and let 0 ∈ P(V ) be a point outside V < a 1 ∪ V < a 2 ∪ C 1 ∪ C 2 . Since a, b are injective then for i = 1, 2, lim n→∞ a n i .(C 1 ∪ C 2 ) = a + i , lim n→∞ a n i .0 = a + i . Taking n large and a = a n 1 , b = a n 2 we get a.0 ∈ C 1 , b.0 ∈ C 2 , a.(C 1 ∪ C 2 ) ⊂ Int C 1 , b.(C 1 ∪ C 2 ) ⊂ Int C 2 .
(ω) = Log|a 1 (ω)z(θω)|, S n ϕ(ω) = n-1 Σ 0 ϕ • θ k (ω), we have, with γ = a 1 • • • a n-1 ∈ Γ and x = z(θ n ω) ∈ Λ(Γ), S n ϕ(ω) = Log|γx|.
Then if ψ is Hölder on Ω, we define a Hölder function ψ on Λ(Γ) by ψ(z(ω)) = ψ(ω) and we have ψ(ω) -ψ(θ n ω) = ψ(γ.x) -ψ(x). In particular, if ω is periodic with period p, and ϕ is as above then z(ω) is a dominant eigenvector of

γ = a 1 • • • a p-1 with Log|λ γ | = S p ϕ(ω).
If S Γ = {0}, then Lemma 2.17 implies the existence of u continuous on Λ(Γ) such that |γx|u(γ.x) = u(x) for γ ∈ Γ, x ∈ Λ(Γ), which contradicts Corollary 2.13. If S Γ does not generate a dense subgroup of R, then for some c > 0, S Γ ⊂ cZ hence S p ϕ(ω) ∈ cZ for any periodic point ω and we can apply Lemma 2.17 to the function c -1 ϕ. In particular the function e 2iπc -1 ϕ can be written in the form e 2iπ(ψ-ψ•θ) where ψ is a Hölder function on Ω. We can define ψ as above and write u = e 2iπψ . Then u is continuous and we have with

γ = a 1 • • • a n , x = z(θ n ω), |γx| 2iπc -1 = u(γ.x)u -1 (x). Then we have for Lemma 2.24 If f ∈ C(P(V )) is real valued and satisfies Q s f ≤ f , then f is constant on Λ([suppµ]
) and equal to its infimum on P(V ).

Lemma 2.25 For any g ∈ SL(V ) x, y ∈ V with |x| = |y| = 1 : ||gx| s -|gy| s | ≤ (s + 1)|g| s δs (x, y), δ(g.x, g.y) ≤ 2|g||gx| -1 δ(x, y), |q s (x, g) -q s (y, g)| ≤ b s |g| s k -1 (s) δs (x, y) with b s > 0.
To go further, in analogy with the case s = 0 in Theorem 2.9, we need to consider a dynamical system (Ω, θ, Q s ) where Ω = G N , θ is the shift and Q s is a Markov measure defined as follows. We denote by π s a Q s -invariant extremal probability on P(V ), we observe that with ω = (g

1 , • • • , g n ) ∈ Ω n = G ⊗n , S n (ω) = g n • • • g 1 ∈ G we have q s n (x, S n (ω)) = n Π k=1 q s (s k-1 .x, g k ), q s n (x, S n (ω))dµ ⊗n (ω) = 1.
Definition 2.26 We denote by Q s x ∈ M 1 (Ω) the limit of the projective system of probabilities q s n (x, .)µ ⊗n on Ω n , and we write

Q s = Q s x dπ s (x).
We observe that Q s is invariant under the shift θ on Ω, we write E s for the corresponding expectation : E s (ϕ) = E s x (ϕ)dπ s (x) where E s x is the expectation with respect to Q s x . Since π s is extremal, it follows that Q s is θ-ergodic. Using the above Lemmas it is easy to show the following. Proposition 2.27 There exists a constant b > 0 such that for any x, y ∈ P(V ) the total variation measure of Q s x -Q s y is bounded by bδ s(x, y)Q s . For any

x ∈ P(V ), Q s x is equivalent to Q s .
Using the Markov operator * Q s on P(V * ), given by * Q s ϕ(x) = * q s (x, g)ϕ(g * .x)dµ(g) where * q s is the dual analogue of q s , the corresponding * Q s -invariant measure * π s and function * e s 0 on P(V * ), one obtains a * P s µ -eigenmeasure by the formula * ν s = ( * e s 0 ) -1 * π s . Then the positive kernel x → ν s x from P(V ) to P(V * ) given by ν s x = | x,. | s e s 0 (x) * ν s is continuous in variation and satisfies g * .ν s g.x q s (x, g)dµ(g) = ν s x , hence S * n .ν s Sn.x is a bounded martingale with respect to Q s

x and the natural filtration. Then, in analogy with Theorem 2.11 we have the following.

Proposition 2.28 The martingale S * n .ν s Sn.x converges Q s x -a.e to a Dirac measure

δ z * s (ω) , the law of z * s (ω) is * π s , * π s is proper, g * 1 .z * s (θω) = z * (ω) Q s -a.e and if < x, z * (ω) > = 0, then lim n→∞ σ(S n , ξ) = 0 Q s -a.e.
In particular, * π s is the unique * Q s -invariant probability. a property which is stronger than the density property stated in corollary 2.14. This allows one to get a multidimensional local limit theorem which extends Theorem 3.1, under the condition that [supp(µ)] has a Zariski dense projection in SL(V ).

Using the operators Q s+it (s ≥ 0, t ∈ R) and the probability Q s

x of section 2 instead of P µ , Q it µ and P we get in the same way, using Theorem 2.30 and the "anomalous" centering by L 1 µ (s) for s > 0: Theorem 3.3 Assume d > 1, [supp(µ)] satisfies condition (I-P) and for some δ > 0, s ∈ I µ , |g| s γ δ (g)dµ(g) is finite. Then there exists σ s > 0 such that for any ϕ ∈ C(P(V )), x ∈ P(V ) and any compactly supported function ψ on R which is Riemann integrable we have

lim n→∞ σ s √ 2πn k n (s) |gx| s ψ[Log|gx| -n L 1 µ (s)] ϕ(g.x)dµ n (g) = ℓ(ψ)e s (x)ν s ((e s ) -1 ϕ).
This gives precise large deviations for the "anomalous mean values" γ s = L 1 µ (s) of Log|S n (ω)x|. This corresponds to estimating the probability that Log|S n (ω)x| belongs to an interval of fixed length centered at nγ s , and gives exponential speed k -n (s) of convergence. One can show, using Theorem 2.9 and perturbation theory that there exists s 1 < 0 such that Theorem 3.2 remains walid for s ∈]s 1 , s ∞ [. We recall that L 1 µ (s) = k ′ (s)k -1 (s), hence the strict convexity of Logk(s) implies that for s ≥ s 1 , L 1 µ (s) is increasing and takes only once any value between L 1 µ (s 1 ) and lim s→s∞ L 1 µ (s). However, in general for s < 0, the function, k(s) as defined above, loses its analyticity at a negative value of s, hence the analogue of Theorem 3.2 is no longer valid for any s ∈ R, and a general µ. For an analogous statement see [START_REF] Letchikov | Products of unimodular independant random matrices[END_REF].

We consider now the following map θ of the circle T = {|z| = 1} ⊂ C defined by θ(z) = z 2 and the C-valued function f (z) = z. We are interested in the asymptotic properties of the

Birkhoff sums S n (z) = n Σ 1 z 2 k = n-1 Σ 0 f (θ k z).
Following [START_REF] Guivarc | Applications d'un théorème limite local à la transience et à la récurrence de marches de Markov[END_REF], we sketch the proof of the following local limit in C : Theorem 3.4 There exists a positive number σ > 0 such that, for any bounded Borel set B ⊂ C with negligible boundary :

lim n→∞ 2π n σ m{z ∈ T ; S n (z) ∈ B} = ℓ(B)
where ℓ(B) is the Lebesgue measure of B. In particular for m -a.e z ∈ T the sums S n (z) are dense in C.

Proof : We transfer the situation to the interval I = [0, 1], the map θ(x) = {2x} and the function f (x) = e 2iπx = (cos2πx, sin2πx), replacing m by Lebesgue measure m on I. Then the adjoint Q of θ with respect to m is given by Qϕ(x) = 1 2 (ϕ(ax) + ϕ(bx)) where ax = 1 2 x, bx = 1 2 (x + 1). We write As in [START_REF] Guivarc | Applications d'un théorème limite local à la transience et à la récurrence de marches de Markov[END_REF] the recurrence and density properties of S n (z) follow. The function f is R 2 -valued and we admit the multidimensional extension of Theorem 1.5. Here aperiodicity of f means that for λ ∈ R 2 \ {0}, α ∈ R the equation :

x k = a k • • • a 1 x (a i = a,
(1) e 2iπ λ,f = e 2iπα (u • a)u -1 = e 2iπα (u • b)u -1 has no Hölder solution u. We write here Q iλ ϕ = Q(e 2iπ λ,f ϕ). The Lipchitz coefficients of ϕ, Qϕ, Q iλ ϕ satisfy.

[

Qϕ] 1 ≤ 1 2 [ϕ] 1 , |Q iλ ϕ] 1 ≤ 1 2 [ϕ] 1 + 2π|λ||ϕ| ∞ , hence condition (D-F) is satisfied by Q, Q iλ on H 1 (I), C(I).
Clearly Q is ergodic as is θ. Then, the local limit theorem will follow from Theorem 1.7 if f is aperiodic. Since f (0) = f (1), from equation (1) we get u(0) = u(1), hence we can write u = e 2iπv where v is a Hölder 1-periodic function on R. Then, using (1), the aperiodicity of f is not valid if there exists α ∈ R, such a v and a Z-valued function k(x) with λ, f (2x) = α + v(x) -v(2x)) + k(x). By continuity we get that k(x) is independant of x. Integrating with respect to m, we get λ, f (2x) = v(x) -v(2x). Now with λ = (a, b) = (0, 0) we write λ, f (x) = a cos2πx + b sin 2πx, and we use the Fourier series expansion of v,

v(x) = ∞ Σ -∞ c k e 4iπkx . Then the above equation gives a = c 1 , b = c -1 , c 0 = 0, c 2k = c k , c 2k+1 = 0. Hence c 2 k or c -2 k
is equal to a or b, which contradicts the fact that v is square integrable on I.

For the use of the Perron-Frobenius operator and spectral gaps on Sobolev spaces in the more general situation of holomorphic endomorphisms of the complex projective space P d (C) see [START_REF] Cantat | Théorème limite central pour les endomorphismes holomorphes et les correspondances modulaires[END_REF] and ( [START_REF] Dinh | Dynamics in several complex variables : Endomorphisms of projective spaces and polynomial-like mappings[END_REF], Theorem 1.83). The following is useful in dealing with L 2 -spectral gaps in the two situations which follow, i.e motion groups and nilmanifolds, but also in the general context of "strong mixing". For other examples see [START_REF] Conze | Ergodicity of group actions and spectral gap. Application to random walks and Markov shifts[END_REF] where the following was proved in a slightly different form. Proposition 3.5 Let (X, m) be a probability space, Γ a countable group acting on X by measure preserving transformations, A a generating set for Γ, µ ∈ M 1 (Γ) a probability measure on Γ with supp(µ) = A. Let c(a, x) = c a (x) (a ∈ A, x ∈ X) be a measurable function with |c(a,

x)| = 1, Q(resp Q c ) the operator on L 2 (X, m) = L 2 (X) defined by Qϕ = Σ a∈A µ(a)ϕ • a (resp Q c ϕ = Σ a∈A µ(a)c a ϕ • a). Assume that the restriction Q 0 of Q to L 2 0 (X) satisfies r(Q 0 ) < 1.
Then one has r(Q c ) = 1 if and only if there exists a measurable function α(x) with value in [0, 2π[ and θ ∈ [0, 2π[such that for any a ∈ A c(a, x) = e i(α(ax)-α(x)+θ) m -a.e. In particular any unimodular spectral value of Q c is an eigenvalue of Q c . Proof : The proof begins like the proof of Theorem 2.1. Assume r(Q c ) = 1 and let e iθ be a spectral value of Q c . Then, either the subspace Im(e iθ -Q c ) is not dense in L 2 (X) or there exists ϕ n ∈ L 2 (X) with ϕ n 2 = 1 such that lim n→∞ Q c ϕ n -e iθ ϕ n 2 = 1. In the first case there exists ϕ ∈ L 2 (X) with e iθ (Q c ) * ϕ = ϕ. Since e -iθ Q c is a contraction of L 2 (X), ϕ is also a fixed point of e -iθ Q c , hence Q c ϕ = e iθ ϕ and it suffices to consider the second case . We have : ). From above we get lim n→∞ c a e i(αn•a-αn-θ) -1 2 = 0, therefore for a subsequence n k : c a (x)e -iθ = lim k→∞ e -i(αn k (ax)-αn k (x) , m -a.e.

0 ≤ Q c ϕ n -e iθ ϕ n 2 = Q c ϕ n 2 2 + ϕ n 2 2 -2Re Q c ϕ n , e iθ ϕ n , 0 ≤ 2 -2Re Q c ϕ n , e iθ
Hence for any γ ∈ A = Γ, σ(γ, x) = lim k→∞ e -i(αn k (γx)-αn k (x)) exists m -a.e and is a T-valued cocycle. On the other hand, the spectral gap property r(Q 0 ) < 1 implies the strong ergodicity (see [START_REF] Schmidt | Asymptotically invariant sequences and action of SL(2, Z) on the 2sphere[END_REF] Proposition 2.10) of the action of Γ on X, hence proposition 2.3 of [START_REF] Schmidt | Asymptotically invariant sequences and action of SL(2, Z) on the 2sphere[END_REF] gives the existence of a measurable function α on X such that : σ(γ, x) = c γ (x)e -i|γ|θ = e -i(α(γx)-α(x)) , m -a.e with |γ| equal to the length of γ in A. The result follows.

Remark 3.6 The proof of [START_REF] Schmidt | Asymptotically invariant sequences and action of SL(2, Z) on the 2sphere[END_REF] shows that, up to unimodular coefficients, the sequence ϕ n converges in measure to an eigenfunction ϕ of Q c with eigenvalue e iθ . We write the group of complex motions of C d in semi-direct product form M d = SU (d)αC d , we denote by µ a finitely supported probability on M d , and by µ its projection on SU (d) ; let B = supp(µ), A = supp(µ). If A generates a dense subgroup of SU (d), i.e A = SU (d), and if the elements of A have algebraic coefficients in some base of C d , then it is known (see [START_REF] Bourgain | Spectral gaps in SU(d)[END_REF]) that for d > 1 the convolution operator ρ 0 (µ) by µ on L 2 0 (S 2d-1 ) satisfies r(ρ 0 (µ)) < 1. For d = 1, the problem of equidistribution of µ n * δ v (v ∈ C) was considered by D.A Kazdhan ( [START_REF] Kazhdan | Uniform distribution on the plane, (Russian)[END_REF]). In this case M 2 is solvable and μ defines a random walk on T, which has no spectral gap in L 2 (T), in general ; however M 2 is non-abelian and unitary representations are still useful (see [START_REF] Guivarc | Equirépartition dans les espaces homogènes[END_REF], [START_REF] Vorobets | Action of Finitely Generated Groups and Semigroups on the Plane by Means of Isometries[END_REF]) and allow one to obtain equidistribution. For d > 1 the analogous problem remained unsolved for a long time, and the corresponding question was asked by G.-A Margulis. Using the result of [START_REF] Bourgain | Spectral gaps in SU(d)[END_REF] we give an affirmative answer (see [START_REF] Conze | Ergodicity of group actions and spectral gap. Application to random walks and Markov shifts[END_REF]). We note that the same question has been recently solved in full generality in [START_REF] Varjú | Random walks in Euclidean spaces[END_REF] i.e without using the spectral gap property in SU (d). Our method here is of general interest in the larger context of Markov operators with a Hilbert space spectral condition, i.e Q strongly mixing in the sense of [START_REF] Rosenblatt | Markov processes, Structure and Asymptotic behaviour[END_REF].

d , any v ∈ C d , lim n→∞ σ(µ)(2πn) d (µ n * δ v )(ϕ) = ℓ(ϕ)
where ℓ is Lebesgue measure on C d ≃ R 2d .

Proof : We recall some notations from unitary representations of M d , where C d is indentified with V = R 2d . The Fourier transform of ϕ ∈ L 1 (V ) is given by ϕ(x) = ϕ(v)e 2iπ x,v dℓ(v). For x = tu with t > 0, u ∈ S 2d-1 we write ϕ t (u) = ϕ(tu) so that the Fourier inversion formula for ϕ gives ϕ

(v) = ϕ t (u)e -2iπt u,v dℓ(x). If h ∈ M d is given by hv = av + b where a ∈ SU (d), b ∈ V , we get ϕ(h -1 v) = ϕ t (a -1 u)e 2iπt u,b e -2iπt u,v dℓ(x) so that the function ϕ • h -1 has t-Fourier component given by (ρ t (h)ϕ t )(u) = ϕ t (a -1 u)e 2iπt u,b . For t > 0 the map h → ρ t (h)
is a unitary representation of M d in the Hilbert space L 2 (S 2d-1 ) which is irreducible. We denote by Q(resp Q it ) the operator on L 2 (S 2d-1 ) given by Qϕ

(u) = Σ a∈A ϕ(a -1 u)µ(a) (resp Q it ϕ(u) = Σ h∈B e 2iπt u,b ϕ(a -1 u)µ(h)) so that Q it = ρ t (µ), μn * δ v (ϕ) = ((Q it ) n ϕ t )(u)e -2iπt u,v dℓ(x). With the notation of Proposition 3.4 we have X = S d-1 , Q it = Q c where c(u, h) = e 2iπt u,b .
In view of Theorem 1.7, in order to prove the statement, it suffices to show that the restriction Q 0 of Q to L 2 0 (S 2d-1 ) satisfies r(Q 0 ) < 1, and r(Q it ) < 1 if t = 0. Since < A >= SU (d) and A has algebraic coefficients the first assertion follows from [START_REF] Bourgain | Spectral gaps in SU(d)[END_REF]. Using strict convexity in L 2 (S 2d-1 ), the second assertion follows from Proposition 3.4 if there is no function ϕ t ∈ L 2 (S 2d-1 ) such that for some θ ∈ R, t > 0, any h ∈ B : ρ t (h -1 )ϕ t = e iθ ϕ t . But this relation implies that the group ρ t ( B ) = ρ t (M d ) has a one dimensional invariant subspace ; since the representation ρ t of M d is irreducible, this is impossible.

Let X = N/∆ be a compact nilmanifold, H a subgroup of the automorphism group Aut(X) of X, m the Haar measure on X. It is known (see [START_REF] Bekka | On the spectral theory of groups of affine transformations on compact nilmanifolds[END_REF]) that, if there is no N -equivariant torus factor T of X such that the projection of H in Aut(X) is virtually abelian, then the natural representation ρ 0 of H in L 2 0 (X) does not weakly contain the identity representation. It follows that if A ⊂ H generates H and µ ∈ M 1 (A) is aperiodic, i.e supp(µ) is not contained in a coset of a proper subgroup of H, then the corresponding Markov operator Q 0 = ρ 0 (µ) satisfies r(Q 0 ) < 1. We recall that a finitely generated group Γ is said to be virtually abelian if it contains a normal group Z r such that the quotient group Γ/Z r is finite. We consider the random walk on X defined by µ and we denote by x n = a n • • • a 1 x (a i ∈ suppµ, x ∈ X) its trajectories, where x is distributed according to m. Let f be an R-valued function on

X with m(f ) = 0, m(f 2 ) < ∞ and let us consider the Birkhoff sums S n (x, ω) = n-1 Σ 0 f (x k )
with respect to the product measure P m = m ⊗ P on X × H N . We denote by E m the corresponding expectation symbol. Then Theorem 1.7 gives : Theorem 3.8 With the above notations, assume that the group H = supp(µ) ⊂ Aut X is ergodic on X, that µ is aperiodic on H and there is no N -equivariant torus factor of X such that the corresponding projection of H is virtually abelian. Let f ∈ L 2 0 (X) be such that the equation e itf (x) = e iθ is not satisfied m -a.e for any (t, θ) ∈ R 2 \ {0}.

Then there exists σ(µ, f ) > 0 such that for any ϕ ∈ L 2 (X) any ψ continuous with compact support on R :

lim n→∞ σ(µ, f ) √ 2πnE m (ϕ(x n )ψ(S n )) = m(ϕ)ℓ(ψ).
We observe that there are natural subgroups of automorphims of X without any hyperbolic element which satisfy the above hypothesis (see [START_REF] Conze | Ergodicity of group actions and spectral gap. Application to random walks and Markov shifts[END_REF]).

Proof of Theorem 3.6 As in the above example we use Theorem 1.7. The condition r(Q 0 ) < 1 follows from [START_REF] Bekka | On the spectral theory of groups of affine transformations on compact nilmanifolds[END_REF]. Here we have Q it ϕ(x) = Σ a∈A ϕ(ax)e itf (ax) µ(a). Hence the condition r(Q it ) < 1 for t = 0 from Proposition 3.4 will be satisfied if there is no t = 0, ϕ t ∈ L 2 (X), θ ∈ R such that, for any a ∈ supp(µ) and m -a.e : e itf (ax) = e iθ ϕ t (ax) ϕ t (x)

This equation implies e itf (ax) ϕ -1 t (ax) = e iθ ϕ -1 t (x), hence e itf (ax) ϕ -1 t (ax) is independant of a ∈ suppµ. Since (supp(µ)) (supp(µ)) -1 generates H and H is ergodic we get e itf (x) ϕ -1 t (x) = c, m -a.e with |c| = 1. It follows e iθ ϕ -1 t (x) = c, hence ϕ t (x) = c -1 e iθ , e itf (x) = e iθ , m -a.e, which is excluded.

4 Extreme value theory for affine random walks.

We consider the affine group H of the vector space V = R d , a probability λ on H with projection µ on G = GL(V ). We denote by P the product measure λ ⊗N on Ω = H N , we consider the dynamical system ( Ω, θ, P) where θ is the shift on Ω, and we study some limit theorems for the stochastic recursion x n+1 = g n+1 x n + b n+1 , x 0 = x where (g n , b n ) are i.i.d random variables with law λ. Under hypothesis (C) below there exists a unique λ-invariant probability ρ on V . We recall below the "homogeneity at infinity" property of ρ (see [START_REF] Guivarc | Spectral gap properties and asymptotics of stationary measures for affine random walks[END_REF], [START_REF] Guivarc | Homogeneity at infinity of stationary solutions of Multivariate Affine Stochastic Recursions[END_REF]), first observed in [START_REF] Kesten | Random difference equations and renewal theory for products of random matrices[END_REF] in a weak form, and we describe its consequences for extreme value theory of the process (x n ) n∈N . The proofs of homogeneity are given here in a restricted setting and are based on the properties of the operators P s µ already studied in section 2. Here we consider also the case V = R ; then we say that a probability µ on R * is arithmetic if the semigroup [supp(µ)] is contained in a subgroup ±a Z of R * (a > 0). If d ≥ 1, our main hypothesis will be the existence of ρ, condition (I-P) for [supp(µ)] and the existence of g, g ′ in [supp(µ)] with r(g) < 1, r(g ′ ) > 1. More precisely condition (C) is the following.

a S n x = 0, we have the equation : R(ω) = g 1 R( θω)+b 1 , hence the law ρ of R is the unique λ-invariant probability. We observe that the above formula for R implies |x| s dρ(x) < ∞ for s < α. Property b) of condition (C) implies the unboundedness of supp(ρ). We denote by η → t.η the extension of the dilation v → tv to measures on V , by ℓ s the measure dt t s+1 on R + , by Λ([supp(µ)]) the inverse image in S(V ) of the limit set Λ([supp(µ)]) ⊂ P(V ) considered in section 2. Then we have the following "homogeneity property of infinity" for ρ, which is basic for the whole development of extreme value theory (see [START_REF] Leadbetter | Extremes and Related Properties of Random Sequences and Processes[END_REF]) : where c > 0, σ α is a probability measure on Λ([supp(µ)]) and σ α ⊗ ℓ α is a µ-harmonic measure supported by R Λ([suppµ]). If [suppµ] has no proper convex invariant cone then σ α is equal to ν α , the unique symmetric measure on S(V ) with projection ν α on P(V ). If not, σ α is a convex combination of the two extremal P α -invariant probabilities on S(V ) with projection ν α on P(V ).

We observe that, in various situations, 'homogeneity at infinity" of solutions of the corresponding functional equations can be deduced from the above theorem. This is the case (see [START_REF] Buraczewski | On multidimensional Mandelbrot cascades[END_REF]) in the study of the multidimensional "smoothing process" considered by Durett-Liggett in the context of statistical mechanics. Below, we assume condition (C) and we will restrict ourselves to the case where [supp(µ)] has no proper convex invariant cone. The proof of Theorem 4.1 in the general case is still valid but with a parameter 0 < p < c, an inequality due to the clustering of extreme values, which corresponds to the mixing with speed properties of the weakly dependant sequence x n (see Lemma 4.7 below). For analogous results on different homogeneous spaces see [START_REF] Kirseböm | Extreme value theory for random walks on homogeneous spaces[END_REF]. Proposition 4.5 With the notations and hypothesis of Theorem 4.1, the law of n -1/α M n converges to the Fréchet law Φ α,p with p < c.

As observed in [START_REF] Pollicott | Limiting distributions for geodesic excursions on the modular surface[END_REF] in the case of geodesic flow on the modular surface, the famous Sullivan's logarithm law (see [START_REF] Sullivan | Disjoint spheres, Approximation by imaginary quadratic numbers and the logarithm law for geodesics[END_REF]) is a simple consequence of such a result. Here we have the The proofs of extreme value properties for the sequence x n are based on the nice spectral properties of the family of operators P f,t defined below. Let f be a Lipchitz function which compact support on [0, 1] × V . For χ > 0, and ϕ ∈ C(V ), we write |ϕ| χ = sup |ϕ(v)| (1+|v|) χ . For ε ∈]0, 1[, λ ≥ 0 with λ + ε < χ we write [ϕ] ε,λ = sup |ϕ(x)-ϕ(y)| |x-y| ε (1+|x|) λ (1+|y| λ ) , we define the space H χ,ε,λ by |ϕ| χ + [ϕ] ε,λ < ∞ and we norm H χ,ε,λ by : ϕ = |ϕ| χ + [ϕ] ε,λ . Then H χ,ε,λ endowed with this norm is a Banach space and for suitable values of χ, ε, λ the "Laplace operator" P f,t defined by P f,t ϕ(v) = e -f (t,hv) ϕ(hv)dλ(h) acts continuously on it. The use of the spectral properties of operators P f,t allows one to prove multiple mixing properties of the sequence x n , in particular the following Lemma 4.7 below. We consider the Markov probability P ρ on the product space V N∪{0} , associated with the affine random walk defined by λ and the invariant probability ρ. We denote r n = [n] s with 0 < s < α and f j,n (v, ω) = exp(-f (n -1 j, n -1/α < v, x j >) (1 ≤ j ≤ n). For the use of complex versions of the operators P f,t in the proofs of limit theorems, in particular for spectral gap properties and convergence to stable laws of Birkhoff sums n Σ j=1

x i , after normalisation, see ( [START_REF] Gao | Spectral gap properties and convergence to stable laws for affine random walks[END_REF], [START_REF] Guivarc | On spectral properties of a family of transfer operators and convergence to stable laws for affine random walks[END_REF]).

Corollary 2 . 2

 22 With µ as in the theorem and Log|g|dµ(g) < ∞, the sequence 1 n

Corollary 2 . 13

 213 With the notations and hypothesis of Corollary 2.12 and d ≥ 2, let A ⊂ ∆ with [A] = ∆, assume that ϕ ∈ C(P(V )) satisfies for some t ∈ R, θ ∈ [0, 2π[ and any γ ∈ A, x ∈ Λ(∆), |γx| it ϕ(γ.x) = e iθ ϕ(x). Then t = 0, e iθ = 1, ϕ =const.

For a function ϕ 1 Σ

 1 on Ω we denote S n (ω) = n-k=0 ϕ • θ k (ω). Suppose ϕ is Hölder continuous and for any periodic point ω of period p, S p ϕ(ω) ∈ Z. Then there exists a Hölder Z-valued function ϕ ′ on Ω and a Hölder function ψ on Ω such that ϕ = ϕ ′ + ψ -ψ • θ. If for any p-periodic ω we have S p ϕ(ω) = 0, then ϕ is of the form ψ -ψ • θ with ψ Hölder on Ω.

  These relations imply that the semigroup Γ(a, b) generated by a, b is free, discrete, and any γ ∈ Γ(a, b) is proximal. The limit set of Γ(a, b) generates a subspace W ⊂ V with dimension at least 2 and Γ(a, b) satisfies condition (I-P) in GL(W ). In order to prove the corollary we can assume Γ = Γ(a, b) and replace V by W . Then we consider the {0, 1}-valued metric δ on {a, b} and endow Ω = {a, b} N with the metric δ(ω, ω ′ ) = ∞ Σ k=1 2 -k δ(ω k , ω ′ k ). We define a homeomorphism z between Ω and Λ(Γ) as follows. If ω = (a k ) k∈N then the sequence a 1 • • • a n .0 converges to z(ω) ∈ C 1 ∪ C 2 and we see that z is a bi-Hölder homeomorphism ; hence we can transfer properties of (Ω, θ) to the action of Γ on Λ(Γ). If z(ω) ∈ V with |z(ω)| = 1 has projection z(ω), then |a 1 (ω)z(θω)| depends of ω only. It follows that, if we set ϕ

1 Σ

 1 b) for a trajectory of the Markov chain on I defined by Q and S n (x) = n-k=0 f (x k ). By stationarity the laws of S n (z) and S n (x) are the same, hence it suffices to show : lim n→∞ 2π n σ m{x ∈ I ; S n (x) ∈ B} = ℓ(B).

2 .

 2 ϕ n . Then the condition lim n→∞ Q c ϕ n -e iθ ϕ n = 0 is equivalent to lim n→∞ Re Q c ϕ n , e iθ ϕ n = 1 hence to lim n→∞ Q c ϕ n , e iθ ϕ n = 1. Since ϕ n 2 = 1 and |c(a, x)| = 1 we have :|Qϕ|(x) ≤ Q|ϕ|(x), 0 ≤ | Q c ϕ n , e iθn ϕ n | ≤< Q|ϕ n |, |ϕ n | >≤ 1. It follows lim n→∞ Q|ϕ n |, |ϕ n | = 1, i.e lim n→∞ Q|ϕ n | -|ϕ n | 2 = 0. We can write |ϕ n | = |ϕ n |, 1 + ψ n where ψ n := |ϕ n | -|ϕ n |, 1 1 ∈ L 2 0 (X), hence lim n→∞ (Q -I)ψ n 2 = 0 and |ϕ n |, 1 ≤ ϕ n 2 = 1, 1 = ϕ n , 1 2 + ψ n 2 Since r(Q 0 ) < 1, the restriction of Q -I to L 2 0 (X)is invertible, hence the condition lim n→∞ (Q -I)ψ n 2 = 0 implies lim n→∞ ψ n 2 = 0. Hence lim n→∞ |ϕ n | -1 2 = 0. On the other hand, the condition lim n→∞ Q c ϕ n , e iθ ϕ n = 1 can be written as lim n→∞ Σ a∈A c a ϕ n • a, e iθ ϕ n µ(a) = 1 with | c a ϕ n • a, e iθ ϕ n | ≤ 1 for any a ∈ A. It follows lim n→∞ c a ϕ n • ae iθ ϕ n 2 = 0. We write ϕ n = |ϕ n |e iαn with α n (x) ∈ [0, 2π[, hence : c a ϕ n • a -e iθ ϕ n = c a e iαn•a (|ϕ n • a| -1) -e iθ (|ϕ n | -1)e iαn + (c a e iαn•a -e i(αn+θ)

Theorem 3 . 7

 37 With the above notations and d > 1 assume that B = M d and A ⊂ SU (d) consists of matrices with algebraic coefficients. Then there exists σ(µ) > 0 such that for any continuous compactly supported function ϕ on C

) 1 n 1 g 1 •

 111 The semigroup [supp(λ)] has no fixed point in V . b) There exists α > 0 with k(α) = lim n→∞ |g| α dµ n (g) some δ > 0 we have :|g| α γ δ (g) + |b| α+δ dλ(h) < ∞. d) If d > 1, [supp(µ)]satisfies condition (I-P) and if d = 1, [supp(µ)] is non-arithmetic. Condition (C) implies that Logk ′ (0) = lim n→∞ Log|S n | < 0. We observe that x n -S n x has the same law as n Σ k=1 g 1 • • • g k-1 b k , i.e as the partial sums of the convergent series R = ∞ Σ • • g k-1 b k . Since lim n→∞

Theorem 4 . 1

 41 Under condition (C), the λ-invariant probability ρ satisfies :lim t→0 + t -α (t.ρ) = c σ α ⊗ ℓ α

Corollary 4 . 6

 46 For any v ∈ V , x 0 = v, we have the P -a.e convergence :

Lemma 4 . 7

 47 With r n , f j,n as above, k n = [nr -1 n ] we have :

  1/2 , hence is finite for any n if 2ε(d -1) ≤ τ . Using Corollary 2.3, we fix n such that Log|gx|dµ n

any x ∈ Λ(Γ), γ ∈ Γ : |γx| 2iπc -1 = u(γ.x)u -1 (x) hence Corollary 2.13 implies 2iπc -1 = 0, and this is impossible. Now we consider Lyapunov exponents and spectral gap properties for a more general situation than above (see [START_REF] Guivarc | Spectral gap properties and asymptotics of stationary measures for affine random walks[END_REF]). They will be useful for large deviations results and for extreme value theory of affine random walks. Let s ≥ 0 and let us define a kind a Mellin transform of µ ∈ M 1 (GL(V )), by k(s) = lim n→∞ |g| s dµ n (g)

. Since the function g → |g| s is submultiplicative and s → |g| s is Log-convex, k(s) exists and is finite for s in an interval I µ of the form [0, s ∞ ] or [0, s ∞ [ with k(s) = ∞ for s > s ∞ . For z ∈ C the operator P µ acting on C(V \ {0}) leaves invariant the space C z (V ) of z-homogeneous continuous functions f i.e f ∈ C z (V ) means f (λv) = |λ| z f (v) for λ = 0, v = 0. The action of P µ on C z (V ) reduces to an operator P z µ on C(P(V )) : (P z µ ϕ)(x) = |g x| z ϕ(g.x)dµ(g) where x ∈ S(V ) has projection x in P(V ). We denote z = s + it where s > 0, t ∈ R and our main result here is a spectral gap result for the operator P s µ (see [START_REF] Guivarc | Spectral gap properties and asymptotics of stationary measures for affine random walks[END_REF]). Theorem 2.20 Assume d > 1 and the semigroup generated by supp(µ) satisfies condition (I-P). Then for any s ∈ I µ , there exists a unique probability measure ν s on P(V ), a unique positive continuous function e s ∈ C(P(V )) with ν s (e s ) = 1, P s µ ν s = k(s)ν s , P s µ e s = k(s)e s . For s ∈ I µ , if |g| s γ δ (g)dµ(g) < ∞ for some δ > 0, then for ε sufficiently small we have on H ε (P(V )) : P s µ = k(s)(ν s ⊗ e s + U s ) where ν s ⊗ e s is the projection on Ce s defined by ϕ → ν s (ϕ)e s and U s is an operator with spectral radius less than 1 which satisfies U s (ν s ⊗ e s ) = (ν s ⊗ e s )U s = 0. Furthermore the function k(s) is analytic, strictly convex on ]0, s ∞ [ and the function s → ν s ⊗ e s from ]0, s ∞ [ to End(H ε (P(V )) is analytic. If t = Imz = 0, the spectral radius of P z µ is less than k(s).

The proof follows the same lines as in Theorem 2.9. We give below corresponding indications.

The following lemmas and Propositions are basic.

Lemma 2.21 Assume σ ∈ M 1 (P(V )) is not supported by an hyperplane and s > 0. There there exists C s (σ) > 0 such that for any u ∈ End V :

Proof : By homogeneity it suffices to show the above inequality if |u| = 1.

The function u → |ux| s dσ(x) is continuous on the unit sphere Σ of End V . Since Σ is compact, this function reaches its infimum C s (σ) at u 0 ∈ Σ. If C s (σ) = 0, then |u 0 x|dσ(x) = 0, hence supp(σ) ⊂ Ker u 0 , which contradicts the hypothesis on σ.

Lemma 2.22 If s ∈ I µ , there exists σ ∈ M 1 (P(V )), k > 0 such that P s µ σ = kσ. We have k(s) = k and σ is not supported on a hyperplane. Furthermore for any n ∈ N :

Proof : We consider the non linear operator P s on M 1 (P(V )) defined by P s σ = (P s µ σ(1)) -1 P s σ. Since s ∈ I µ this operator is continuous on the compact convex set M 1 (P(V )) endowed with the weak topology, hence Schauder-Tychonov theorem implies the existence of k > 0 and σ ∈ M 1 (P(V )) with P s σ = σ, P s µ σ = kσ, hence k = (P s µ σ) [START_REF] Bekka | On the spectral theory of groups of affine transformations on compact nilmanifolds[END_REF]. For such a σ the equation kσ(ϕ) = ϕ(g.x)|g x| s dµ(g)dσ(x) implies g.x ∈ supp(σ), µ -a.e if x ∈ supp(σ), hence for any g ∈ supp(µ) : g.supp(σ) ⊂ supp(σ). Then the projective subspace W generated by supp(σ) is [supp(µ)]-invariant and if W is proper this contradicts condition (I-P). Since k ≤ ( |g| s dµ n (g)) 1/n the two sided inequality for k follows from Lemma 2.21, hence k = k(s).

The dual version σ ′ of the measure σ considered above is useful to construct a P s µeigenfunction as follows.

Lemma 2.23 Assume σ ′ ∈ M 1 (P(V * )) and k ′ > 0 satisfy * P s µ σ ′ = k ′ σ ′ . Then k ′ = k(s) and the function e s 0 on P(V ) defined by :

where |x| = |ỹ| = 1, satisfies P s µ e s 0 = k(s)e s 0 . Furthermore e s 0 is positive and Hölder of order s = inf(s, 1)

hence using the dual version of Lemma 2.22 we get k ′ = k(s). Also :

. If e s 0 (x) = 0 for some x, then x, ỹ = 0 σ ′ -a.e, hence supp(σ ′ ) is contained in the hyperplane of P(V * ) orthogonal to x. Since [supp(µ)] satisfies (I-P), this contradicts the dual version of Lemma 2.22. Hence e s 0 is positive. The Hölder property of e s 0 follows from the inequality

) and we observe that Lemma 2.23 implies q s n (x, g)dµ n (g) = 1. Also we consider the Markov operator

Then we have the following simple lemmas which allow to control the iterates (Q s ) n .

Then we deduce the following (see [START_REF] Guivarc | Spectral gap properties and asymptotics of stationary measures for affine random walks[END_REF]).

Theorem 2.29 The Lyapunov exponents

One has, using the same arguments, the analogues of Theorem 2.9, Corollary 2.10.

(See [START_REF] Guivarc | Spectral gap properties and asymptotics of stationary measures for affine random walks[END_REF], Theorem 3.17).

Theorem 2.30 Assume that µ ∈ M 1 ( G) satisfies |g| s γ δ (g)dµ(g) < ∞ for some δ > 0, s ∈ I µ and [suppµ] satisfies (I-P). Then for ε sufficiently small one has

and the operator

Then the spectral gap property of P s µ , P z µ stated in Theorem 2.20 follows. The analyticity of the function s → k(s) follows from perturbation theory. The inequality r(Q z ) < 1, hence r(P z µ ) < k(s), is a consequence of the geometric Corollary 2.13.

Local limit theorems for some transfer operators

In this section we concentrate mainly on random walks on some homogeneous spaces. For analogous methods developed in the context of hyperbolic transformations or expanding transformations see [START_REF] Guivarc | Théorèmes limites pour une classe de chaines de Markov et applications aux difféomorphismes d'Anosov[END_REF] , [START_REF] Broise | Etudes spectrales d'opérateurs de transfert et applications[END_REF]. We apply the general local limit theorems 1.5, 1.7 to the specific situations detailed below.

For products of i.i.d matrices, with law µ ∈ M 1 ( G), and with the notations of section 1 we observe that precise large deviations can also be considered as local limit theorems for the function f (x, g) = Log|gx| on G × P(V ) and suitable centerings. We begin with the centering by L 1 µ which corresponds to s = 0. In view of Theorem 2.9 and corollaries 2.10, 2.13, the conditions of Theorem 1.5 are satisfied, hence (see [START_REF] Page | Théorèmes limites pour les produits de matrices aléatoires[END_REF]) : Theorem 3.1 Assume that d > 1, the semigroup [supp(µ)] satisfies condition (I-P) and for some δ > 0 γ δ (g)dµ(g) < ∞. Then there exists σ > 0 such that for any ϕ ∈ C(P(V )), x ∈ P(V ) and ψ compactly supported on R and Riemann integrable, we have : In ( [START_REF] Guivarc | On the Spectrum of a Large Subgroup of a Semisimple Group[END_REF], Theorem 3) this result is extended to a semi-simple group. The Gaussian law has full dimension and the proof is based on the fact that the set {λ γ ; γ ∈ [supp(µ)] prox } is large long (see [START_REF] Guivarc | Spectral gap properties and asymptotics of stationary measures for affine random walks[END_REF]). Here, with

), we show only the convergence of t -α (t.ρ)(H + v ) under a reinforced hypothesis, and we deduce the homogeneity at infinity of ρ if α / ∈ N. The proof of the positivity of c in the general case is based on the well known Kac's return lemma and the method of ladder indices expounded in [START_REF] Feller | An Introduction to Probability Theory and its Applications[END_REF] for the case of random walks on the line ; here we will give an analytic argument in case d = 1 only. If d ≥ 2 we observe that the above homogeneity of ρ is stable under perturbation of λ in the weak topology ; it follows that convergence to Fréchet's law for affine random walks is robust if d ≥ 2. This is also the case for the convergence to stable laws associated with affine random walks on R d (see [START_REF] Gao | Spectral gap properties and convergence to stable laws for affine random walks[END_REF]). In the special case considered above, the operator P z µ defined by P z µ ϕ(x) = |gx| z ϕ(g.x)dµ(g) (z = s+it , s ≥ 0, t ∈ R) on the unit sphere S(V ) has spectral properties similar to those of P z µ in section 3. In particular, there exists a unique P s µ -eigenmeasure (resp eigenfunction) on S(V ) : P s µ ν s = k(s) ν s , P s µ ẽs = k(s) ν s and ν s (resp ẽs ) is the symmetric lift of ν s (resp e s ) to S(V ). The proof of Theorem 4.1 is based on a lemma and a proposition as follows.

Lemma 4.2 There exists an open set D ⊂ C which contains the set {Rez ∈]0, α]} such that (I -P z µ ) -1 is meromorphic in D with a unique simple pole at z = α. Furthermore :

Proof : If d = 1, the lemma follows from elementary arguments and non-arithmeticity of [supp(µ)]. We assume now d > 1. From the analogue of Theorem 2.20 in case S(V ) it follows that for some ε > 0 and any z with |z -α| < ε, there exists a holomorphic function k(z) such that k(z) is a dominant simple eigenvalue of P z µ with k(z) = 1+k(α)(z -α)+•(z -α). Since k ′ (α) = 0, we have k(z) = 1 for z = α and |z -α| small. Also for |z -α| small we have in H ε (S(V )) the decomposition P z µ = k(z) ν z ⊗ e z + U (z) where ν z ⊗ e z is a projection on C e z , U (z) satisfies U (z)( ν z ⊗ e z ) = ( ν z ⊗ e z )U (z) = 0, r(U (z)) < 1 and ν z ⊗ e z , U (z) depend holomorphically on z. We write :

where

) and (I -P z µ ) -1 is meromorphic in a disk D 0 centered at α with radius ε ′ ≤ ε, with unique pole at z = α. For z = α + it with |t| ≥ ε ′ , from the last assertion in Theorem 2.20 we get that there exists a disk D t centered at α + it such that r( P z µ ) < 1 for z ∈ D t , hence (I -P z µ ) -1 is a bounded operator depending holomorphically on z for z

We denote for v ∈ V * and Rez = s ∈]0, α[ :

. Then we have the Proposition 4.3 We have the convergence : lim

where |u| = 1, c ≥ 0 and ν α satisfies µ * (

where dα is the restriction of d α to S(V * ).

Proof : We write the above functional equation in the form

By homogeneity of f z , d z and writing fz , dz for the restrictions of f z , d z to S(V ) we get :

(I - * P z µ ) fz = dz . The function dz (u) is dominated by :

Hence, using the Hölder inequality and the moment hypothesis, we get that for u fixed the function z → dz (u) is holomorphic in the domain Rez ∈]0, α + δ[ ; on the other hand the above lemma shows the meromorphicity of z → (I - * P z µ ) -1 in an open set D which contains {Rez ∈]0, α]}. The above estimation of dz (u) shows that the same property is valid for fz = (I - * P z µ ) -1 ( dz ). If we denote by ρ u the law of < R, u > + , we have fs (u) = x s dρ u (x), hence fs (u) is the Mellin transform of the positive measure ρ u . Then, since (I - * P z µ ) -1 has a simple pole at z = α, we can apply Wiener-Ikehara theorem (see [START_REF] Widder | The Laplace transform[END_REF]) to fs (u), ρ u and obtain the tail of ρ u in the form : lim t→∞ t α ρ u (t, ∞) = lim s→α - α -1 (α -s) fs (u). Hence, using Lemma 4.2 :

Using the formula for * e α (u) given in Lemma 2.23, we get :

For d = 1 and some additionnal hypothesis we show lim t→∞ t α ρ(t, ∞) > 0, using an analytic argument sketched in [START_REF] Guivarc | Heavy tail properties of multidimensional stochastic recursions[END_REF]. This argument uses a lemma of E. Landau (see [START_REF] Widder | The Laplace transform[END_REF]) which says the following : if the Mellin transform of a positive measure ρ supported on [0, ∞[ can be extended holomorphically to a neighbourbood of α ∈ R + , containing [α, α + ε[ then we have

, we get for s < α :

Here we assume supp(µ

for some positive K and B, hence the functions k(z) = E(g z 1 ), E(|b 1 | z ) are holomorphic for Rez > 0. Using the mean value theorem we get

Then as above, the Landau Lemma and equation (E) imply that the Mellin transform f (z) is holomorphic for Rez > 0, and f (s) = ∞ 0 x s dρ(x) < ∞ for s ≥ 0. Now equation (E 0 ) gives for s ≥ 1 k 1/s (s)f 1/s (s) = E((R -b 1 ) s + ) 1/s ≤ E(R s + ) 1/s + B. Hence (k 1/s (s) -1) f 1/s (s) ≤ B. Since 1 < lim s→∞ k 1/s (s) = K 1 ≤ K, it follows : lim s→∞ f 1/s (s) ≤ B(K 1 -1) -1 < ∞. Hence we have : supp(ρ) ⊂] -∞, B(K 1 -1) -1 ], which contradict the hypothesis supp(ρ) ∩ R + unbounded.

Proof of Theorem 4.1 We consider only the case where [supp(µ)] has no proper convex invariant cone and α / ∈ N. Then using Proposition 4.3, and the properties of Radon transforms of positive measures studied in [START_REF] Boman | Support theorems for the Radon transform and Cramer-Wold theorems[END_REF], we get the homogeneity at infinity result for ρ if α / ∈ N. For α ∈ N we use radial Fourier analysis (see [START_REF] Guivarc | Spectral gap properties and asymptotics of stationary measures for affine random walks[END_REF]).

Theorem 4.1 allows to describe the asymptotic behaviour of large values of x n , following the framework of extreme value theory (see [START_REF] Kluppelberg | Extremal behaviour of models with multivariate random recurrence representation[END_REF], [START_REF] Leadbetter | Extremes and Related Properties of Random Sequences and Processes[END_REF]). We denote M n = sup{|x i | ; 1 ≤ i ≤ n}. We recall that Fréchet's law with parameters p, α is the probability measure Φ α,p on R + given by Φ α,p (0, t) = e -pt -α . For i.i.d random variables x k with law ν such that lim t→∞ t α ν(t, ∞) = c > 0, it was observed by M. Fréchet (see [START_REF] Leadbetter | Extremes and Related Properties of Random Sequences and Processes[END_REF]) that the law of n -1/α M n converges to Φ α,c . For affine random walks it can be shown (see [START_REF] Guivarc | Spectral gap properties and asymptotics of stationary measures for affine random walks[END_REF]) that Fréchet's law is