Y Guivarc'h

É Le Page

Spectral gap properties for linear random walks and Pareto's asymptotics for affine stochastic recursions

Keywords: spectral gap, renewal theorem, Pareto asymptotics, random matrices, affine random recursions

Let V = R d be the Euclidean d -dimensional space, µ (resp λ) a probability measure on the linear (resp affine) group G = GL(V ) (resp H = Aff(V )) and assume that µ is the projection of λ on G. We study asymptotic properties of the iterated convolutions µ n * δ v (resp λ n * δ v ) if v ∈ V , i.e asymptotics of the random walk on V defined by µ (resp λ), if the subsemigroup T ⊂ G (resp.

Σ ⊂ H ) generated by the support of µ (resp λ) is "large". We show spectral gap properties for the convolution operator defined by µ on spaces of homogeneous functions of degree s ≥ 0 on V , which satisfy Hölder type conditions. As a consequence of our analysis we get precise asymptotics for the potential kernel Σ ∞ 0 µ k * δ v , which imply its asymptotic homogeneity. Under natural conditions the H -space V is a λ-boundary; then we use the above results and radial Fourier Analysis on V \ {0} to show that the unique λ-stationary measure ρ on V is "homogeneous at infinity" with respect to dilations v → t v (for t > 0), with a tail measure depending essentially of µ and Σ. Our proofs are based on the simplicity of the dominant Lyapunov exponent for certain products of Markov-dependent random matrices, on the use of renewal theorems for "tame" Markov walks, and on the dynamical properties of a conditional λ-boundary dual to V .

Résumé

Soit V l'espace Euclidien de dimension d , µ (resp. λ) une probabilité sur le groupe linéaire (resp.affine) G = GL(V ) (resp. H = Aff(V )) et supposons que µ soit la projection de λ sur G. Nous étudions certaines propriétés asymptotiques des convolutions itérées de µ (resp. λ) appliquées à un vecteur non nul v ∈ V , c'est à dire de la marche aléatoire sur V définie par µ (resp. λ), si le semigroupe T ⊂ G (resp. Σ ⊂ H ) engendré par le support de µ (resp. λ) est « grand ». Nous montrons des propriétés d'isolation spectrale pour l'opérateur de convolution défini par µ sur des espaces de fonctions homogènes de degré s ≥ 0 sur V , qui satisfont des conditions du type de Hölder. Comme conséquence de notre analyse nous obtenons des asymptotiques précises pour le noyau potentiel Σ ∞ 0 µ k * δ v , qui impliquent son homogénéité à l'infini.Sous des conditions naturelles, le H -espace V est une λ-frontière ; nous utilisons alors les résultats précédents et l'analyse de Fourier radiale sur V \ {0} afin de montrer que l'unique mesure λstationnaire est homogène à l'infini, par rapport aux dilatations v → t v ( pour t > 0),avec une mesure de queue qui dépend essentiellement de µ et Σ. Nos preuves sont basÈes sur la simplicité de l'exposant de Lyapunov dominant de certains produits de matrices en dépendance markovienne, sur l'utilisation de théorèmes de renouvellement pour certaines marches markoviennes et sur les propriétés dynamiques d'une λ-frontière duale de V .

Introduction, statement of results

We consider the d -dimensional Euclidean space V = R d , endowed with the natural scalar product (x, y) → 〈x, y〉, the associated norm x → |x|, the linear group G = GL(V ), and the affine group H = Aff(V ). Let λ be a probability measure on H with projection µ on G, such that suppλ has no fixed point in V . We denote by T = [suppµ], (resp. Σ = [suppλ]) the closed subsemigroup of G (resp H) generated by suppµ (resp suppλ). Under natural conditions, including negativity of the dominant Lyapunov exponent L µ corresponding to µ, for any v ∈ V , the sequence of iterated convolutions λ n * δ v converges weakly to ρ; the probability measure ρ is the unique probability which solves the convolution equation λ * ρ = ρ, and suppρ is unbounded if [suppµ] contains an expanding element. Then, an important property of ρ is the existence of α > 0 such that |x| s d ρ(x) < ∞ for s < α and |x| s d ρ(x) = ∞ for s ≥ α, if suppλ is compact. One of our main results below (Theorem C) gives the α-homogeneity of ρ at infinity, i.e. Pareto's asymptotics of ρ (see [START_REF] Resnick | Heavy-tail phenomena[END_REF], p. 74).

In general, for the asymptotic behaviour of λ n * δ v and the "shape at infinity" of ρ there are four cases:

1. The "contractive" case where the elements of suppµ have norms less than 1, ρ exists and is compactly supported.

2. The "expansive" case where L µ > 0 and ρ does not exist.

3. The 'critical" case where L µ = 0 and ρ does not exist.

4. The "weakly contractive" case where L µ < 0 and ρ exists with unbounded support.

Heuristically, cases 3, 4, mentioned above, can be considered as transitions between the cases 1, 2, which appear to be extreme cases. In this paper we are mainly interested in case 4 and in the shape at infinity of ρ; in the corresponding analysis we use the approach of [START_REF] Kesten | Random difference equations and renewal theory for products of random matrices[END_REF], based on the associated linear random walk, we develop methods and prove results which are of independent interest for products of random matrices. An important tool here is the "Radon transform" of ρ, i.e. the function on V defined by

ρ(v ) = ρ{x ∈ V ; 〈x, v 〉 > 1},
which allow us to transfer the shape problem for ρ into an asymptotic problem for ρ.

In [START_REF] Kesten | Random difference equations and renewal theory for products of random matrices[END_REF] the shape problem was connected to the study of a Poisson equation on the Gspace V \{0} satisfied by ρ and by a convolution operator associated with µ; the measure λ was assumed to be supported on the positive matrices or to have a density on H. In the first case an important result was the validity of Pareto's asymptotics for the projection of ρ on the positive directions.We observe that special cases of the above problem and various consequences of Pareto's asymptotics have been considered in the litterature (see for example [START_REF] Davis | Point processes and partial sum convergence for weakly dependent random variables with infinite variance[END_REF], [START_REF] Ch | Implicit renewal theory and tails of solutions of random equations[END_REF], [START_REF] Kluppelberg | Extremal behaviour of models with multivariate random recurrence representation[END_REF]), especially if µ has a density on G, a condition which implies spectral gap properties for the convolution operators associated to µ or λ in suitable Hilbert spaces. In contrast,our basic hypothesis which involves only T and L µ , implies that the above operators satisfy Doeblin-Fortet inequalities (see [START_REF] Tulcea | Théorie ergodique pour des classes d'opérations non complètement continues[END_REF]), hence also spectral gap properties in spaces of Hölder functions. Here our main result, partly contained in Theorem C below, describes the general case and the homogeneity at infinity stated in the theorem gives new results even for d = 1 (see [START_REF] Ch | Implicit renewal theory and tails of solutions of random equations[END_REF]) or for the multidimensional situations considered in [START_REF] Kesten | Random difference equations and renewal theory for products of random matrices[END_REF], [START_REF] Kluppelberg | Extremal behaviour of models with multivariate random recurrence representation[END_REF].

We observe that, more generally, the homogeneous behaviour at infinity of certain invariant measures is of interest for various questions in Probability Theory and Mathematical Physics (see [START_REF] Davis | Point processes and partial sum convergence for weakly dependent random variables with infinite variance[END_REF], [START_REF] De Calan | On the distribution of a random variable occuring in i.i.d. disordered systems[END_REF], [START_REF] Dolgopyat | Quenched limit theorems for nearest neighbour random walks in 1D random environment[END_REF], [START_REF] Durrett | Fixed points of the smoothing transformation[END_REF], [START_REF] Kesten | Random difference equations and renewal theory for products of random matrices[END_REF], [START_REF] Resnick | Heavy-tail phenomena[END_REF]) but also in some geometrical questions such as dynamical excursions of geodesic flow and winding around cusps in hyperbolic manifolds (see [START_REF] Babillot | Asymptotic laws for geodesic homology on hyperbolic manifolds with cusps[END_REF], [START_REF] Pollicott | Limiting distributions for geodesic excursions on the modular surface[END_REF], [START_REF] Sullivan | Disjoint spheres, approximation by imaginary quadratic numbers and the logarithm law for geodesics[END_REF]), or analysis of the H-space (V, ρ) as a λ-boundary and its dynamical consequences (see [START_REF] Benoist | Mesures stationnaires et fermés invariants des espaces homogènes[END_REF], [START_REF] Furman | Random walks on groups and random transformations[END_REF], [START_REF] Furstenberg | Boundary theory and stochastic processes on homogeneous spaces[END_REF], [START_REF] Guivarc | Orbits of linear group actions, random walks on homogeneous spaces and toral automorphisms[END_REF]).

Hence, following [START_REF] Kesten | Random difference equations and renewal theory for products of random matrices[END_REF], we start with the linear situation i.e. the G-action on V \ {0} and we consider a probability measure µ on G. As in [START_REF] Furstenberg | Boundary theory and stochastic processes on homogeneous spaces[END_REF] we assume that T satisfies the socalled i-p condition (i-p for irreducibility and proximality), i.e. T is strongly irreductible and contains at least one element with a unique simple dominant eigenvalue; if d = 1, we assume furthermore that T is non arithmetic, i.e. T is not contained in a subgroup of R * of the form {±a n ; n ∈ Z} for some a > 0. We observe that for d > 1 condition i-p is satisfied by T if and only if it is satisfied by the algebraic subgroup Z c(T ),which is the Zariski closure of T , hence condition i-p is satisfied if T is "large" (see [START_REF] Goldsheid | Lyapunov exponents of a product of random matrices[END_REF], [START_REF] Prasad | Regular elements in Zariski dense subgroups[END_REF]). On the other hand, the set of probability measures µ on G such that the associated semigroup T satisfies condition i-p is open and dense in the weak topology hence µ is '"generic" if d > 1 and T satisfies i-p. Also, if d > 1, an essential aperiodicity consequence of condition i-p is the density in R * + of the multiplicative subgroup generated by moduli of dominant eigenvalues of the elements of T (see [START_REF] Benoist | Propriétés asymptotiques des groupes linéaires[END_REF], [START_REF] Guivarc | On the Spectrum of a Large Subgroup of a Semisimple Group[END_REF], [START_REF] Guivarc | Products of random matrices and convergence theorems[END_REF]). We denote by |g | the norm of g ∈ G and we write

γ(g ) = sup(|g |, |g -1 |), I µ = {s ≥ 0 : |g | s d µ(g ) < ∞},
we denote ]0, s ∞ [ the interior of the interval I µ . For simplicity of exposition, and since linear maps commute with the symmetry v → -v , it is convenient to deal with the Gfactor space V of V \ {0} by symmetry, instead of V \ {0} itself. We use the polar decomposition V = P d-1 × R * + , and the corresponding functional decompositions, where P d-1 is the projective space of V .

We consider the convolution action of µ on continuous functions on V \ {0} which are homogeneous of degree s ≥ 0, i.e. functions f which satisfy f (t v ) = |t | s f (v ) (t ∈ R). This action reduces to the action of a certain positive operator P s on C (P d-1 ), the space of continuous functions on the projective space P d-1 . More precisely, if f (v ) = |v | s ϕ( v) with ϕ ∈ C (P d-1 ), v ∈ P d-1 , then P s ϕ is given by

P s ϕ(x) = |g x| s ϕ(g • x)d µ(g ),
where x ∈ P d-1 , x → g • x denotes the projective action of g on x, and |g x| is the norm of any vector g v with |v | = 1 and v = x. Also for z = s + i t ∈ C, with s ∈ I µ and t ∈ R, we write P z ϕ(x) = |g x| z ϕ(g • x)d µ(g ). By duality P z acts also on measures on P d-1 and for a measure ν we denote by P z ν the new measure obtained from ν. The space of endomorphisms of a Banach space B will be denoted by EndB. For ε > 0 let H ε (P d-1 ) be the space of ε-Hölder functions on P d-1 , with respect to a certain natural distance. We denote by ℓ s (resp ℓ) the s-homogeneous (resp. Haar) measure on R * + and we write

ℓ s (d t ) = d t t s+1 , ℓ(d t ) = d t t , h s (v ) = |v | s .
An s-homogeneous Radon measure η on V = P d-1 × R * + is written as η = π ⊗ ℓ s where π is a bounded measure on P d-1 . For s ∈ I µ we define the function

k(s) = lim n→∞ |g | s d µ n (g ) 1/n
, where µ n is the n -th convolution power of µ on the group G and we observe that logk(s) is a convex function on I µ . A key tool in our analysis for d > 1 is the Theorem. A. Assume d > 1 and the subsemigroup T ⊂ GL(V ) generated by suppµ satisfies condition i-p. Then, for any s ∈ I µ there exists a unique probability measure ν s on P d-1 , a unique positive continuous function e s ∈ C (P d-1 ) with ν s (e s ) = 1 such that P s ν s = k(s)ν s , P s e s = k(s)e s .

For s ∈ I µ , if |g | s γ τ (g )d µ(g ) < ∞ for some τ > 0 and if ε > 0 is sufficiently small, the action of P s on H ε (P d-1 ) has a spectral gap:

P s = k(s)(ν s ⊗ e s +U s ),
where the operator ν s ⊗ e s is the projection on Ce s defined by ν s , 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 ν s ⊗e s from ]0, s ∞ [ to End H ε (P d-1 ) is analytic. The spectral radius of P z is less than k(s) if s = Rez ∈ [0, s ∞ [ and t = Imz = 0.

We observe that, since condition i-p is open, the last property is robust under perturbation of µ in the weak topology. If d = 1, k(s) is equal to |x| s d µ(x), hence k(s) is the Mellin transform of µ (see [START_REF] Widder | The Laplace transform[END_REF]) and the above statements are also valid if T is non arithmetic. However, the last property is not robust for d = 1.

If s = 0, P s reduces to the convolution operator by µ on P d-1 and convergence to the unique µ-stationary measure ν 0 = ν was studied in [START_REF] Furstenberg | Boundary theory and stochastic processes on homogeneous spaces[END_REF] using proximality of the Taction on P d-1 . In this case, spectral gap properties for P z , if Rez= s is small, were first proved in [START_REF] Page | Théorèmes limites pour les produits de matrices aléatoires[END_REF] using the simplicity of the dominant µ-Lyapunov exponent (see [START_REF] Guivarc | Products of random matrices and convergence theorems[END_REF]). Limit theorems of Probability Theory for the product S n = g n • • • g 1 of the random i.i.d. matrices g k , distributed according to µ, are consequences of this result and of radial Fourier analysis on V \ {0} used in combination with boundary theory (see [START_REF] Benoist | Random walks on reductive groups[END_REF], [START_REF] Bougerol | Products of random matrices with applications to Schrödinger operators[END_REF], [START_REF] Furman | Random walks on groups and random transformations[END_REF], [START_REF] Goldsheid | Zariski closure and the dimension of the Gaussian law of the product of random matrices[END_REF], [START_REF] Guivarc | Simplicité de spectres de Lyapunov et propriété d'isolation spectrale pour une famille d'opérateurs de transfert sur l'espace projectif[END_REF], [START_REF] Page | Théorèmes limites pour les produits de matrices aléatoires[END_REF]). If µ has a density with compact support, Theorem A is valid for any s ∈ R. In general and for d > 1, it turns out that the function k(s), as defined above, looses its analyticity at some s 1 < 0. For a recent detailed study of the operators P z (s =Rez small) and their equicontinuous extensions in a geometrical setting which allows the algebraic group Z c(T ) to be reductive and defined over a local field of any characteristic, we refer to the forthcoming book [START_REF] Benoist | Random walks on reductive groups[END_REF]. We observe that condition i-p used here and s small imply that the basic assumptions of ( [START_REF] Benoist | Random walks on reductive groups[END_REF], chap 8) are satisfied. Also, for s > 0, the properties described in the theorem were considered in [START_REF] Guivarc | Simplicité de spectres de Lyapunov et propriété d'isolation spectrale pour une famille d'opérateurs de transfert sur l'espace projectif[END_REF]; they are basic ingredients for the study of precise large deviations of S n (ω)v ( [START_REF] Letchikov | Products of unimodular independant random matrices[END_REF]).

The Radon measure ν s ⊗ ℓ s on V satisfies the convolution equation µ * (ν s ⊗ ℓ s ) = k(s)ν s ⊗ ℓ s and the support of ν s is the unique T -minimal subset of P d-1 , the so-called limit set Λ(T ) of T (see [START_REF] Benoist | Propriétés asymptotiques des groupes linéaires[END_REF], [START_REF] Benoist | Random walks on reductive groups[END_REF], [START_REF] Guivarc | Produits de matrices aléatoires et applications[END_REF]). The function e s is an integral transform of the twisted µeigenmeasure * ν s . For s > 0 and σ a probability measure on P d-1 not concentrated on a proper subspace, |g | s is comparable to |g x| s d σ(x); the uniqueness properties of e s and ν s are based on this geometrical fact. The proof of the spectral gap property depends on the simplicity of the dominant Lyapunov exponent for the product of random matrices S n = g n • • • g 1 with respect to a natural shift-invariant Markov measure Q s on Ω = G N , which is locally equivalent to the product measure Q 0 = µ ⊗N . A construction of a kernel-valued martingale (based on * ν s ) plays an essential role in the proof of simplicity and in the comparison of |S n (ω)| with |S n (ω)v |. For s = 0 this study corresponds to [START_REF] Guivarc | Products of random matrices and convergence theorems[END_REF].

In order to develop probabilistic consequences of Theorem A we endow Ω = G N with the shift-invariant measure P = µ ⊗N (resp Q s ). We know that if log γ(g )d µ(g ) (resp |g | s log γ(g )d µ(g )) is finite the dominant Lyapunov exponent L µ (resp L µ (s)) of S n = g n • • • g 1 with respect to P (resp Q s ) exists and

L µ = lim n→∞ 1 n log |S n (ω)|d P(ω), L µ (s) = lim n→∞ 1 n log |S n (ω)|d Q s (ω).
If s ∈]0, s ∞ [, k(s) has a continuous derivative k ′ (s) and L µ (s) = k ′ (s) k(s) . By strict convexity of log k(s), if lim s→s ∞ k(s) ≥ 1 and s ∞ > 0, we can define α > 0 by k(α) = 1.

We consider the potential kernel U on V defined by U (v, •) = ∞ Σ 0 µ k * δ v . Then we have the following multidimensional extensions of the classical renewal theorems (see [START_REF] Feller | An Introduction to Probability Theory and its Applications[END_REF]), which describes the asymptotic homogeneity of U (v, •). We recall that for v ∈ V and A ⊂ V , U (v, A) is the mean number of visits of S n (ω)v to A for n ≥ 0.

Theorem. B.

Assume T satisfies condition i-p, log γ(g )d µ(g ) < ∞ and L µ > 0. If d = 1 assume furthermore that µ is non-arithmetic. Then, for any v ∈ V , U (v, •) is a Radon measure on V and we have the vague convergence

lim t→0 + U (t v, •) = 1 L µ ν ⊗ ℓ,
where ν is the unique µ-stationary measure on P d-1 .

Theorem. B α . Assume T satisfies condition i-p, logγ(g )d µ(g ) < ∞, L µ < 0, s ∞ > 0 and there exists α > 0 with k(α) = 1, |g | α log γ(g )d µ(g ) < ∞. If d = 1 assume furthermore that µ is non-arithmetic. Then for any v ∈ P d- 1 we have the vague convergence on V lim

t→0 + t -α U (t v, •) = e α (v ) L µ (α) ν α ⊗ ℓ α .
Up to normalization the Radon measure ν α ⊗ ℓ α is the unique α-homogeneous measure which satisfies the harmonicity equation µ * (ν α ⊗ ℓ α ) = ν α ⊗ ℓ α .

For v ∈ P d-1 , we consider the random variable M(v ) = sup{|S n v | ; n ∈ N}. Then we have the following matricial version of Cramér's estimate in collective risk theory (see [START_REF] Feller | An Introduction to Probability Theory and its Applications[END_REF]).

Corollary. With the notations of Theorem B α , for any u ∈ P d-1 , we have the convergence

lim t→∞ t α P{M(u) > t } = Ae α (u) > 0.
Theorems B, B α are consequences of the arguments used in the proof of Theorem A and of a renewal theorem for a class of Markov walks on R (see [START_REF] Kesten | Renewal theory for functionals of a Markov chain with general state space[END_REF]). An essential role is played by the law of large numbers for log |S n v | under Q s (s = 0, α); the comparison of |S n | and |S n v | follows from the finiteness of the limit of (log |S n | -log|S n v |). This is the essential property used in [START_REF] Kesten | Renewal theory for functionals of a Markov chain with general state space[END_REF], in a more general framework. For the sake of brevity, we have formulated these theorems in the context of V instead of V . Corresponding statements where P d-1 is replaced by the unit sphere S d-1 are given in section 4. Also the above weak convergence can be extended to a larger class of functions.

In [START_REF] Kesten | Random difference equations and renewal theory for products of random matrices[END_REF], renewal theorems as above were obtained for non negative matrices, the extension of these results to the general case was an open problem and a partial solution was given in [START_REF] Page | Théorème de renouvellement pour les produits de matrices aléatoires[END_REF]. Theorems B and B α extend these results to a wider setting. In view of the interpretation of U (v, •) as a mean number of visits, Theorem B is a strong reinforcement of the law of large numbers for S n (ω)v , hence it can be used in some problems of dynamics for group actions on T -spaces. In this respect we observe that a specific version of the asymptotic homogeneity of U stated in Theorem B has been of essential use in [START_REF] Guivarc | Orbits of linear group actions, random walks on homogeneous spaces and toral automorphisms[END_REF] for the description of the T -minimal subsets of the action of a large subsemigroup T ⊂ SL(d , Z) of automorphisms of the torus T d .

On the other hand Theorem B α gives a description of the fluctuations of a linear random walk on V with P-a.e. exponential convergence to zero, under condition i-p and the existence in T of a matrix with spectral radius greater than one. These fluctuation properties are responsible for the homogeneity at infinity of stationary measures for affine random walks on V , that we discuss now.

Let λ be a probability measure on the affine group H of V , µ its projection on G and T , Σ as above. We assume that T satisfies condition i-p, and suppλ has no fixed point in V .

If d = 1 we assume that T is non-arithmetic. We consider the affine stochastic recursion on V X n+1 = A n+1 X n + B n+1 , (R)

where (A n , B n ) are λ-distributed i.i.d random variables. From a heuristic point of view, the corresponding affine random walk can be considered as a superposition of an additive random walk on V governed by B n and a multiplicative random walk on V \ {0} governed by A n . Here, as it appears below in Theorem C, the non trivial multiplicative part A n plays a dominant role, while the additive part B n has a stabilizing effect since λ has a unique stationary probability ρ on V and ρ is not supported on a point. If

E(log |A n |) + E(log |B n |) < ∞ and the dominant Lyapunov exponent L µ for the product A 1 • • • A n is negative, then R n = Σ n-1 0 A 1 • • • A k B k+1 converges λ ⊗N -a.e.
to R, the law ρ of R is the unique λ-stationary measure on V, (V, ρ) is a λ-boundary (see [START_REF] Furstenberg | Boundary theory and stochastic processes on homogeneous spaces[END_REF])and suppρ = Λ a (Σ) is the unique Σ-minimal subset of V . If T contains at least one matrix with an expanding direction, then Λ a (Σ) is unbounded and if I µ = [0, ∞[ there exists α > 0 with k(α) = 1, hence we can inquire about the "shape at infinity" of ρ. According to a conjecture of F. Spitzer the measure ρ should belong to the domain of attraction of a stable law with index

α if α ∈ [0, 2[ or a Gaussian law if α ≥ 2.
Here we prove a multidimensional precise form of this conjecture, and more generally the α-homogeneity at infinity of ρ, where we assume that µ satisfies the conditions of Theorem B α , λ satisfies moment conditions and suppλ has no fixed point in V . We denote by Λ(T ) the inverse image of the limit set Λ(T ) in S d-1 and by ν α the symmetric lifting of ν α to S d-1 . Then our main result implies the following

Theorem. C. With the above notation we assume that T satisfies condition

i-p, suppλ has no fixed point in V , s ∞ > 0, L µ < 0 and α ∈]0, s ∞ [ satisfies k(α) = 1. If d = 1 we assume also µ is non-arithmetic. Then, if E(|B| α+τ ) < ∞ and E(|A| α γ τ (A)) < ∞ for some τ > 0, the unique λ-stationary measure ρ on V satisfies the following vague convergence on V \ {0} lim t→0 + t -α (t • ρ) = C σ α ⊗ ℓ α ,
where C > 0, σ α is a probability measure on Λ(T ) and σ α ⊗ ℓ α is a µ-harmonic Radon measure supported on R * Λ(T ). If T has no proper convex invariant cone in V , we have σ α = ν α . The above convergence is also valid on any Borel function f such that the set of discontinuities of f is σ α ⊗ ℓ α -negligible and such that for some ε > 0 the function

|v | -α | log |v || 1+ε | f (v )| is bounded.
Briefly, we say that ρ satisfies Pareto's asymptotics of index α (see [START_REF] Resnick | Heavy-tail phenomena[END_REF], page 74). The convergence in Theorem C can be considered as a Cramér type estimate for the random variable R and was stated in [START_REF] Guivarc | Heavy tail properties of multidimensional stochastic recursions[END_REF].This statement gives the homogeneity at infinity of ρ, hence the measure C σ α ⊗ℓ α defined by the theorem can be interpreted as the "tail measure" of ρ. In the context of extreme value theory for the process X n , the convergence stated in the theorem implies that ρ has "multivariate regular variation" and this property plays an essential role in the theory (see [START_REF] Gnedenko | Sur la distribution limite du terme maximum d'une série aléatoire[END_REF], [START_REF] Resnick | Heavy-tail phenomena[END_REF]). If T has a proper convex invariant cone, then C σ α can be decomposed as

C σ α = C + ν α + +C -ν α -,
where C + , C -≥ 0 and ν α + ⊗ ℓ α , ν α -⊗ ℓ α are µ-harmonic extremal measures on V \ {0}. In section 5 the discussion of positivity for C , C + , C -in terms of Λ(T ) and Λ a (Σ) lead us, via Radon transforms, to consider an associated linear random walk on the vector space V × R, which plays a dual role to the original λ-random walk X n on V . The proof of positivity for C depends on the use of Kac's recurrence theorem for this dual random walk. On the other hand, the proof of α-homogeneity of ρ at infinity follows from a Choquet-Deny type property for the linear µ-random walk on V \ {0}. The spectral gap property, stated in theorem A, plays an essential role in this study.

For d = 1, positivity of C = C + + C -was proved in [START_REF] Ch | Implicit renewal theory and tails of solutions of random equations[END_REF] using Levy's symmetrisation argument, positivity of C + and C -was tackled in [START_REF] Guivarc | Heavy tail properties of multidimensional stochastic recursions[END_REF] by a complex analytic method introduced in [START_REF] De Calan | On the distribution of a random variable occuring in i.i.d. disordered systems[END_REF]. For d = 1 we have ν α + = δ 1 , ν α -= δ -1 and the precise form of Theorem C gives that the condition C + = 0 is equivalent to suppρ ⊂] -∞, c] with c ∈ R. In the Appendix we give an approach to part of Theorem C using tools familiar in Analytic Number Theory like Wiener-Ikehara's theorem and a lemma of E. Landau but also results for Radon transforms of positive measures which are only valid for α ∉ N (see [START_REF] Boman | Support theorems for the Radon transform and Cramer-Wold theorems[END_REF], [START_REF] Weiss | Measures that vanish on half-spaces[END_REF]). However the discussion of positivity for C + , C -seems to be not possible using only these analytical tools.

A natural question is the speed of convergence in Theorem C. For d = 1 see [START_REF] Ch | Implicit renewal theory and tails of solutions of random equations[END_REF], if λ has a density. For d > 1 and under condition i-p, this question is connected with the possible uniform spectral gap for the operator P z of Theorem A,if z = α + i t .

To go further we observe that Theorem C gives a natural construction for a large class of probability measures in the domain of attraction of a stable law. Using also spectral gaps and weak dependence properties of the process X n , Theorem C allow us to prove convergence to stable laws for normalized Birkhoff sums along the affine λ-walk on V (see [START_REF] Gao | Spectral gap properties and convergence to stable laws for affine random walks[END_REF]) Furthermore if d = 1, and conditionally on regularity assumptions usual in extreme value theory, such convergences were shown in [START_REF] Davis | Point processes and partial sum convergence for weakly dependent random variables with infinite variance[END_REF]; hence the results of [START_REF] Gao | Spectral gap properties and convergence to stable laws for affine random walks[END_REF] improve and extends the results of [START_REF] Davis | Point processes and partial sum convergence for weakly dependent random variables with infinite variance[END_REF] in the case of GARCH processes (d ≥ 1). If d > 1 the above convergence is robust under perturbation of λ in the weak topology. These convergences to stable laws are connected with the study of random walk in a random medium on the line or the strip (see [START_REF] Dolgopyat | Quenched limit theorems for nearest neighbour random walks in 1D random environment[END_REF]) if α < 2. On the other hand the study of the extremal value behaviour of the process X n can be fully developed on the basis of Theorem C and on the above weak dependence properties of X n ; in particular the asymptotics of the extremes of |X n | are given by Fréchet type laws with index α ( [START_REF] Guivarc | Asymptotique des valeurs extrêmes pour les marches aléatoires affines[END_REF]), a result which extends the main result of [START_REF] Kluppelberg | Extremal behaviour of models with multivariate random recurrence representation[END_REF] to the generic case. In a geometrical context, as observed in [START_REF] Pollicott | Limiting distributions for geodesic excursions on the modular surface[END_REF] for excursions of geodesic flow around the cusps of the modular surface, the famous Sullivan's logarithm law is a simple consequence of Fréchet's law for the continuous fraction expansion of a real number uniformly distributed in [0, 1].

Here also a logarithm law is valid for the random walk X n (see [START_REF] Guivarc | Asymptotique des valeurs extrêmes pour les marches aléatoires affines[END_REF]). The arguments developed in the proof of homogeneity at infinity for ρ can also be used in the study of certain quasi-linear equations which occur in various domains related to branching random walks in particular (see [START_REF] Ch | Implicit renewal theory and tails of solutions of random equations[END_REF]), [START_REF] Liu | On generalized multiplicative cascades[END_REF]). For example the description of the shape at infinity of the fixed points of the multidimensional version of the "smoothing transformation" considered in [START_REF] Durrett | Fixed points of the smoothing transformation[END_REF] in the context of infinite particle systems in interaction depends on such arguments (see [START_REF] Buraczewski | On multidimensional Mandelbrot's Cascades[END_REF]). In an econometrical context, the stochastic recursion (R) can be interpreted as a mechanism which, in the long run, produces debt or wealth accumulation with a specific homogeneous structure at large values; hence,in the natural setting of affine stochastic recursions, this mechanism "explains" the remarkable power law asymptotic shape of wealth distribution empirically discovered by the economist V. Pareto ([43]).

For information on the role of spectral gap properties in limit theorems for Probability theory and Ergodic theory we refer to [START_REF] Babillot | Asymptotic laws for geodesic homology on hyperbolic manifolds with cusps[END_REF], [START_REF] Benoist | Random walks on reductive groups[END_REF], [START_REF] Bougerol | Products of random matrices with applications to Schrödinger operators[END_REF], [START_REF] Gao | Spectral gap properties and convergence to stable laws for affine random walks[END_REF], [START_REF] Goldsheid | Zariski closure and the dimension of the Gaussian law of the product of random matrices[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 | Products of random matrices and convergence theorems[END_REF], [START_REF] Page | Théorèmes limites pour les produits de matrices aléatoires[END_REF]. For information on products of random matrices we refer to [START_REF] Benoist | Random walks on reductive groups[END_REF], [START_REF] Bougerol | Products of random matrices with applications to Schrödinger operators[END_REF], [START_REF] Furman | Random walks on groups and random transformations[END_REF], [START_REF] Guivarc | On contraction properties for products of Markov driven random matrices[END_REF]. Theorem A (resp B, B α and C) is proved in sections 2,3 (resp 4 and 5).

We thank Ch.M. Goldie, I. Melbourne and D. Petritis for useful informations on stochastic recursions and extreme value theory.We thanks also the referees for careful reading and very useful sugestions.

Ergodic properties of transfer operators on projective spaces

In this section we study the qualitative properties of transfer operators on P d-1 or S d-1 . As mentioned in the introduction one can find in [START_REF] Benoist | Random walks on reductive groups[END_REF] a detailed study of a general class of transfer operators on flag manifolds with equicontinuity properties. Here, our transfer operators depend on a complex parameter z which is typically large with Rez> 0. Hence, for self-containment reasons in particular, we develop from scratch our study on P d-1 or S d-1 for Rez= s ≥ 0. A first step is to reduce these transfer operators to Markov operators (Theorems 2.6, 2.16) with equicontinuity properties.

Notation and preliminary results

Let V = R d be the Euclidean space endowed with the scalar product 〈x, y〉 = d 1 x i y i and the norm |x| = d 1 |x i | 2 1/2 , V the factor space of V \ {0} by the finite group {±I d }. We denote by P d-1 (resp S d-1 ), the projective space (resp unit sphere) of V and by v (resp ṽ) the projection of v ∈ V on P d-1 (resp S d-1 ). The linear group

G = GL(V ) acts on V , V by (g , v ) → g v . If v ∈ V \ {0}, we write g • v = g v
|g v | and we observe that G acts on S d-1 by (g , x) → g • x. We will also write the action of g ∈ G on x ∈ P d-1 by g • x; we define |g x| as |g x| if x ∈ P d-1 and x ∈ S d-1 has projection x ∈ P d-1 . Also, if x, y ∈ P d-1 , |〈x, y〉| is defined as |〈 x, y〉| where x, y ∈ S d-1 have projections x, y. Corresponding notations will be taken when convenient. For a subset A ⊂ S d-1 the convex envelope Co(A) of A is defined as the intersection with S d-1 of the closed convex cone generated by A in V . We denote by O(V ) the orthogonal group of V and by m the O(V )-invariant measure on P d-1 . A positive measure η on P d-1 will be said to be proper if η(U ) = 0 for every proper projective subspace U = P d-1 .We denote by EndV the space of endomorphisms of the vector space V .

Let P be a positive kernel on a Polish space E and let e be a positive function on E which satisfies Pe = ke for some k > 0. Then we can define a Markov kernel Q e on E by Doob's relativisation procedure: Q e ϕ = 1 ke P (ϕe). This procedure will be used frequently here. For a Polish G-space E we denote by M 1 (E ) the space of probability measures on E . If ν ∈ M 1 (E ), and P is as above, ν will be said to be P -stationary if P ν = ν, i.e for any Borel function ϕ, ν(P ϕ) = ν(ϕ). We will write C (E ) (resp C b (E ) for the space of continuous (resp bounded continuous) functions on E . If E is a locally compact Gspace, µ ∈ M 1 (G), and ρ is a Radon measure on E , we recall that the convolution µ * ρ is defined as a Radon measure by µ * ρ = δ g x d µ(g )d ρ(x), where δ y is the Dirac measure at y ∈ E . A µ-stationary measure on E will be a probability measure ρ ∈ M 1 (E ) such that µ * ρ = ρ. In particular, if E = V or V and µ ∈ M 1 (G) we will consider the Markov kernel P on V (resp P on V ) defined by

P (v, •) = µ * δ v , (resp P (v, •) = µ * δ v ). On P d-1 (resp S d-1 ) we will write P (x, •) = µ * δ x (resp P (x, •) = µ * δ x ).
If u is an endomorphism of V , we denote u * its adjoint map, i.e. 〈u * x, y〉 = 〈x, u y〉 if x, y ∈ V . If µ ∈ M 1 (G) we will write µ * for its push forward by the map g → g * and we define the kernel * P on V by * P (v, .) = µ * * δ v . For s ≥ 0 we denote ℓ s (resp h s ) the s-homogeneous measure (resp function) on

R * + = {t ∈ R ; t > 0} given by ℓ s (d t ) = d t t s+1 (resp h s (t ) = t s ).
For s = 0 we write ℓ(d t ) = d t t .

Using the polar decomposition V \ {0} = S d-1 × R * + and the corresponding functional decompositions on V \{0}, every s-homogeneous measure η (resp function ψ) on V \{0} can be written as

η = π ⊗ ℓ s (resp. ψ = ϕ ⊗ h s ),
where π (resp. ϕ) is a measure (resp. function) on S d-1 . Similar decompositions are valid on

V = P d-1 × R * + . If g ∈ G, and η = π ⊗ ℓ s (resp ψ = ϕ ⊗ h s ) the directional com- ponent of g η (resp. ψ • g ) is given by ρ s (g )(η) = |g x| s δ g •x d η(x), ρ s (g )(ψ)(x) = |g x| s ψ(g • x).
The representations ρ s and ρ s extend to measures on G by the formulae

ρ s (µ)(η) = |g x| s δ g •x d µ(g )d π(x), ρ s (µ)(ψ)(x) = |g x| s ψ(g • x)d µ(g ).
We will write, for ϕ ∈ C (P d-1 ) (resp. ψ ∈ C (S d-1 ))

P s ϕ = ρ s (µ)(ϕ), (resp. P s ψ = ρ s (µ)(ψ)), * P s ϕ = ρ s (µ * )(ϕ), (resp. * P s ψ = ρ s (µ * )(ψ)).
We endow S d-1 (resp. P d-1 ) with the distance δ (resp. δ) defined by

δ(x, y) = |x -y| (resp. δ( x, ȳ) = inf{|x -y|; |x| = |y| = 1}). For ε > 0, ϕ ∈ C (P d-1 ) (resp. ψ ∈ C (S d-1
)), we denote

[ϕ] ε = sup x =y |ϕ(x) -ϕ(y)| δ ε (x, y) (resp. [ψ] ε = sup x =y |ψ(x) -ψ(y)| δε (x, y) ), |ϕ| = sup{|ϕ(x)|; x ∈ P d-1 } (resp. |ψ| = sup{|ψ(x)|; x ∈ S d-1 }),
and we write

H ε (P d-1 ) = {ϕ ∈ C (P d-1 ); [ϕ] ε < ∞}, (resp. H ε (S d-1 ) = {ψ ∈ C (S d-1 ); [ψ] ε < ∞}.
The set of positive integers will be denoted by N. We denote by µ n the n th convolution power of µ, i.e for

ψ ∈ C b (G), µ n (ψ) = ψ(g n • • • g 1 )d µ ⊗n (g 1 , • • • , g n ). Definition 2.1. If s ∈ [0, ∞[ and µ ∈ M 1 (G), we denote k(s) = k µ (s) = lim n→∞ |g | s d µ n (g ) 1/n , I µ = {s ≥ 0; k µ (s) < +∞}.
We observe that the above limit exists, since by subadditivity of g → log |g |, the quant-

ity u n (s) = |g | s d µ n (g ) satisfies u m+n (s) ≤ u m (s)u n (s). Also k µ (s) = inf n∈N (u n (s))
µ ′′ = µ + (1-c) µ ′ then k µ ′′ (s) = ck µ (s) + (1-c) k µ ′ (s), if s ∈ I µ ∩ I µ ′ .
Definition 2.2.

1. An element g ∈ EndV is said to be proximal if g has a unique eigenvalue λ g ∈ R of maximum modulus and λ g is simple.

2. A semigroup T ⊂ G is said to be strongly irreducible if no finite union of proper subspaces is T -invariant.

Proximality of g means that we can write

V = Rv g ⊕V < g with g v g = λ g v g , g V < g ⊂ V <
g and the restriction of g to V < g has spectral radius less than |λ g |. In this case lim

n→+∞ g n • x = vg if
x ∉ V < g and we say that λ g is the dominant eigenvalue of g . If E ⊂ G we denote by E pr ox the set of proximal elements of E .The closed subsemigroup (resp. group) generated by E will be denoted [E ] (resp. 〈E 〉). In particular we will consider below the case E = suppµ where suppµ is the support of µ ∈ M 1 (G).

Definition 2.3. A semigroup T ⊂ G is said to satisfy condition i-p if T is strongly irredu- cible and T prox = .
As shown in ( [START_REF] Goldsheid | Lyapunov exponents of a product of random matrices[END_REF], [START_REF] Prasad | Regular elements in Zariski dense subgroups[END_REF]) this property of T is satisfied if it is satisfied by Z c(T ), the Zariski closure of T . It can be proved that condition i-p is valid if and only if the connected component of the closed subgroup Z c(T ) is locally the product of a similarity group and a semi-simple real Lie group without compact factor which acts proximally and irreducibly on P d-1 . In this sense T is "large". For example, if T is a countable subgroup of G which satisfies condition i-p then T contains a free subgroup with two generators.

We recall that in C n , the Zariski closure of E ⊂ C n is the set of zeros of the set of polynomials which vanish on E . The group G = GL(V ) can be considered as a Zariski-closed subset of R d 2 +1 . If T is a semigroup, then Z c(T ) is a closed subgroup of G with a finite number of connected components. If d = 1, condition i-p is always satisfied. Hence, when using condition i-p, d > 1 will be understood.

Remark.

The above definitions will be used below in the analysis of laws of large numbers and renewal theorems. A corresponding analysis has been developed in [START_REF] Kesten | Random difference equations and renewal theory for products of random matrices[END_REF] for the case of non-negative matrices. We observe that proximality of an element in G is closely related to the Perron-Frobenius property for a positive matrix. If T is not irreducible, we can consider the subspace V + (T ) generated by the dominant eigenvectors of the elements of T prox ; then V + (T ) is T -invariant (see [START_REF] Goldsheid | Zariski closure and the dimension of the Gaussian law of the product of random matrices[END_REF] p 120) and, if T + is the restriction of T to V + (T ), then T + satisfies condition i-p. In that way our results below could be used in reducible situations (see for example [START_REF] Buraczewski | On multidimensional Mandelbrot's Cascades[END_REF]). Definition 2.4. Assume T is a subsemigroup of G which satisfies condition i-p. Then the closure of the set { vg ; g ∈ T prox } will be called the limit set of T and will be denoted Λ(T ).

We recall that for a semigroup T acting on a topological space E , a subset X ⊂ E is said to be T -minimal if any orbit T y(y ∈ X ) is contained in X and dense in X . For the minimality of Λ(T ) ⊂ P d-1 , see [START_REF] Benoist | Propriétés asymptotiques des groupes linéaires[END_REF], [START_REF] Guivarc | Produits de matrices aléatoires et applications[END_REF].

With these definitions we have the Proposition 2.5. Assume T ⊂ G is a subsemigroup which satisfies condition i-p and S ⊂ T generates T . Then T Λ(T ) = Λ(T ) and Λ(T ) is the unique T -minimal subset of

P d-1 . If µ ∈ M 1 (G) is such that T = [suppµ] satisfies i-p, there exists a unique µ-stationary measure ν on P d-1 . Also suppν = Λ(T ) and ν is proper. Furthermore, if d > 1, the subgroup of R * + generated by the set {|λ g |; g ∈ T prox } is dense in R * + . In particular, if ϕ ∈ C (Λ(T )) satisfies for some t ∈ R, |e i θ | = 1 : ϕ(g • x) |g x| i t = e i θ ϕ(x) for any g ∈ S, x ∈ Λ(T ) then t = 0, e i θ = 1, ϕ = constant.
Remark. The first part of the above statement is essentially due to H. Furstenberg ([17], Propositions 4.11, 7.4) . The second part is proved in ( [START_REF] Guivarc | Products of random matrices and convergence theorems[END_REF]. Proposition 3). For another proof and extensions of this property see [START_REF] Benoist | Propriétés asymptotiques des groupes linéaires[END_REF], [START_REF] Guivarc | On the Spectrum of a Large Subgroup of a Semisimple Group[END_REF]. This property plays an essential role in the renewal theorems of section 4 as well as in section 5 for d > 1. In the context of non negative matrices a modified form is also valid (see [START_REF] Buraczewski | On multidimensional Mandelbrot's Cascades[END_REF]); in ( [START_REF] Kesten | Random difference equations and renewal theory for products of random matrices[END_REF]), Theorem A) under weaker conditions on T , its conclusion is assumed as an hypothesis. If d = 1, we will need to assume it, i.e we will assume that T is non-arithmetic ; if T = [suppµ] satisfies this condition, we say that µ is non-arithmetic.

Uniqueness of eigenfunctions and eigenmeasures on P d-1

Here we consider s ∈ I µ and the operator P s (resp * P s ) on C (P d-1 ) defined by

P s ϕ(x) = |g x| s ϕ(g • x)d µ(g ), (resp. * P s ϕ(x) = |g x| s ϕ(g • x)d µ * (g )).
For a measure ν on P d-1 , P s ν is defined by duality against C (P d-1 ). For z = s +i t ∈ C we will also write The Markov operator Q s on P d-1 defined by Q s ϕ = 1 k(s)e s P s (ϕe s ) has a unique stationary measure π s given by π s = e s ν s and we have for any ϕ ∈ C (P d-1 ) the uniform convergence of

P z ϕ(x) = |g x| z ϕ(g • x)d µ(g
(Q s ) n ϕ towards π s (ϕ). If z = s +i t , t ∈ R and Q z is defined by Q z ϕ = 1 k(s)e s P z (e s ϕ),then the equation Q z ϕ = e i θ ϕ with ϕ ∈ C (P d-1 ), ϕ = 0 implies e i θ = 1, t = 0, ϕ = constant.

Remark.

1. If s = 0, then e s = 1, and ν s = ν is the unique µ-stationary measure [START_REF] Furstenberg | Boundary theory and stochastic processes on homogeneous spaces[END_REF]. The fact that ν is proper is of essential use in [START_REF] Page | Théorèmes limites pour les produits de matrices aléatoires[END_REF], [START_REF] Bougerol | Products of random matrices with applications to Schrödinger operators[END_REF] and [START_REF] Guivarc | Products of random matrices and convergence theorems[END_REF], for the study of limit theorems.

2. In section 3 we will also construct a suitable kernel-valued martingale which allows to prove that ν s is proper (see Theorem 3.2), if s ∈ I µ . We note that analyticity of k(s) is proved in Corollary 3.20 below. Continuity of the derivative of k will be essential in sections 3,4 and is proved in Theorem 3.10. If s = 0 the corresponding martingale construction was done in [START_REF] Furstenberg | Boundary theory and stochastic processes on homogeneous spaces[END_REF].

3. By definition of P s and P , the function (resp. measure) e s ⊗ h s (resp. ν s ⊗ ℓ s ) satisfies the equation

P (e s ⊗ h s ) = k(s)(e s ⊗ h s ), (resp. P (ν s ⊗ ℓ s ) = k(s)ν s ⊗ ℓ s )
The proof of the theorem depends of a proposition and the following lemmas improving corresponding results for positive matrices in [START_REF] Kesten | Random difference equations and renewal theory for products of random matrices[END_REF].

Lemma 2.7. Assume σ ∈ M 1 (P d-1 ) is not supported by a hyperplane. Then, there exists a constant c s (σ) > 0 such that, for any u in EndV

|ux| s d σ(x) ≥ c s (σ)|u| s .
Proof. Clearly it suffices to show the above inequality if |u| = 1. The fonction u → |ux| s d σ(x) is continuous on EndV , hence its attains its infimum c s (σ) on the compact subset of EndV defined by |u| = 1. If c s (σ) = 0, then for some u ∈ EndV with |u| = 1, we have |ux| s d σ(x) = 0 hence, ux = 0, σa.e. In other words, suppσ ⊂ Ker(u), which contradicts the hypothesis on σ. Hence c s (σ) > 0.

Lemma 2.8. If s ∈ I µ there exists σ ∈ M 1 (P d-1 ) such that P s σ = kσ for some k > 0. For any such σ, we have k = k(s) and σ is not supported on a hyperplane. Furthermore for every n ∈ N

|g | s d µ n (g ) ≥ k n (s) ≥ c s (σ) |g | s d µ n (g ).
Proof. We consider the non-linear operator P s on M 1 (P d-1 ) defined by P s σ = P s σ (P s σ) [START_REF] Babillot | Asymptotic laws for geodesic homology on hyperbolic manifolds with cusps[END_REF] . Since |g | s d µ(g ) < +∞, this operator is continuous in the weak topology. Since M 1 (P d-1 ) is compact and convex, Schauder-Tychonov theorem implies the existence of k > 0 and σ ∈ M 1 (P d-1 ) with P s σ = kσ, hence k = (P s σ) [START_REF] Babillot | Asymptotic laws for geodesic homology on hyperbolic manifolds with cusps[END_REF]. For such a σ, the equation

kσ(ϕ) = ϕ(g • x)|g x| s d µ(g )d σ(x) implies that if x ∈ suppσ, then g • x ∈ suppσ, µ-a.e.
Then for any g ∈ suppµ we have g • suppσ ⊂ suppσ. In particular the projective subspace H generated by suppσ satisfies [suppµ] • H = H. Since [suppµ] satisfies i-p, we have H = P d-1 . Then Lemma 2.7 gives, for any g ,

|g x| s d σ(x) ≥ c s (σ)|g | s . The relation (P s ) n σ = k n σ implies k n = |g x| s d µ n (g )d σ(x); hence, using Lemma 2.7 we get c s (σ) |g | s d µ n (g ) ≤ k n ≤ |g | s d µ n (g ). It follows that k = lim n→+∞ |g | s d µ n (g ) 1/n = k(s).
Assume e ∈ C (P d-1 ) is positive and satisfies P s e = k(s)e. Then we can define the Markov kernel Q s e and the cocycle θ s e by

Q s e ϕ(x) = 1 k(s) ϕ(g • x) e(g • x) e(x) |g x| s d µ(g ), θ s e (x, g ) = |g x| s e(g • x) e(x)
.

In view of the cocycle property of θ s e (x, g ) we can calculate the iterate (Q s e ) n by the formula (Q s e ) n ϕ(x) = ϕ(g • x)q s e,n (x, g )d µ n (g ) with q s e,n (x, g ) = 1

k n (s) e(g •x)
e(x) |g x| s , and q s e,n (x)(g ) = 1. For s = 0 we have e = 1, Q s e = P . Also we write q s e,1 = q s e .

Lemma 2.9. Assume e is as above, f ∈ C (P d-1 ) is real valued and satisfies Q s e f ≤ f . Then, on Λ([suppµ]), f is constant and equal to its infimum on P d-1 .

Proof. Let M -= {x ∈ P d-1 ; f (x) = inf { f (y); y ∈ P d-1 }. The relation f (x) ≥ q s e (x, g ) f (g • x)d µ(g ) implies that if x ∈ M -then g • x ∈ M -, µ-a.e. Hence [suppµ] • M -⊂ M -. Since Λ([suppµ]) is the unique [suppµ] -minimal subset of P d-1 , we get Λ([suppµ]) ⊂ M -, i.e. f (x) = inf{ f (y); y ∈ P d-1 }, if x ∈ Λ([suppµ]).
Using Lemma 2.8, the existence of e ∈ C (P d-1 ) with P s e = k(s)e is obtained by the following Lemma 2.10. Assume σ ∈ M 1 (P d-1 ) and k > 0 satisfy * P s σ = kσ. Then the function σ s on P d-1 defined by, σ s (x) = |〈x, y〉| s d σ(y) satisfies P s σ s = k σ s . Furthermore σ s is positive and Hölder of order s = inf(1, s).

Proof. We have |g x| s σ s (g • x) = |〈x, g * • y〉| s |g * y| s d σ(y) and * P s σ = kσ; hence

P s σ s (x) = k |〈x, z〉| s d σ(z) = k σ s (x).
If σ s (x) = 0 for some x, then |〈x, y〉| = 0, σ-a.e.; hence suppσ ⊂ {y ∈ P d-1 ; 〈x, y〉 = 0}. This contradicts Lemma 2.8, since [suppµ] satisfies i-p. Hence σ s is positive.

In order to show the Hölder property of σ s , we use the inequality |a sb s | ≤ s|a -b| s where a, b ∈ [0, 1] s = sup(s, 1), s = inf(s, 1). Then

||〈x, y〉| s -|〈x ′ , y〉| s | ≤ s|x -x ′ | s |y| s , | σ s (x) -σ s (x ′ )| ≤ sδ s (x, x ′ ).
Lemma 2.11. Let e be a positive and s-Hölder function on P d-1 with s = inf(s, 1), s > 0. There exists a constant b s > 0 such that for any (x, y)

∈ S d-1 × S d-1 , g ∈ G ||g x| s -|g y| s | ≤ (s + 1)|g | s δ s (x, y), δ(g • x, g • y) ≤ 2 |g | |g x| δ(x, y), |θ s e (x, g ) -θ s e (y, g ) ≤ b s |g | s δ s (x, y).
Proof. We use the inequality In view of the first inequality the first term satisfies the required bound. The last term also satisfies it, since 1 e(x) is s-Hölder and |g y| s ≤ |g | s . For the second term we write :

|a s -b s | ≤ |a -b| s if a, b ≥ 0, s ≤ 1 to get ||g x| s -|g y| s | ≤ ||g x| -|g y|| s ≤ |g (x -y)| s ≤ |g | s |x -y| s . Hence ||g x| s -|g y| s | ≤ |g | s δ s (x, y). If s > 1, we use 1 s |a s -b s | ≤ sup(a, b) s-1 |a -b| if a, b ≥ 0. We get ||g x| s -|g y| s | ≤ s|g | s-1 ||g x| -|g y|| ≤ s|g | s-1 |g (x -y)| ≤ s|g | s |x -y|.
|e(g • x) -e(g • y)| ≤ [e] s δ s (g • x, g • y) ≤ 2 s [e] s |g | |g y| s δ s (x, y).
Hence this term is bounded by :

|e -1 |2 s [e] s δ s (x, y)|g y| s-s |g | s ≤ 2 s |e -1 |[e] s |g | s δ s (x, y).
The above inequalities imply the lemma.

The following is well known (see for example [START_REF] Rosenblatt | Equicontinuous Markov operators[END_REF] Theorem 6) Lemma 2.12. Let X be a compact metric space, Q a Markov operator on X , which preserves C (X ). Assume that all the Q-invariant continuous functions are constant and for any ϕ ∈ C (X ), the sequence Q n ϕ is equicontinuous. Then Q has a unique stationary measure π, hence for any ϕ ∈ C (X ) the sequence

1 n n-1 Σ 0 Q k ϕ converges uniformly to π(ϕ).
Furthermore if the equation Qψ = e i θ ψ, ψ ∈ C (X ) implies e i θ = 1, then, for any ϕ ∈ C (X ), Q n ϕ converges uniformly to π(ϕ).

Proposition 2.13. Let µ ∈ M 1 (G) and assume that the semigroup

[suppµ] satisfies i -p. Let s ∈ I µ , s > 0 ε ∈]0, s] with s = inf(1, s), e ∈ C (P d-1
) is positive, s-Hölder with P s e = k(s)e. Then there exists a s ≥ 0 such that for any n ∈ N and any ε-Hölder function ϕ on

P d-1 [(Q s e ) n ϕ] ε ≤ a s |ϕ| + ρ n,s (ε)[ϕ] ε , where ρ n,s (ε) = sup x,y q s e,n (x, g ) δ ε (g •x,g •y) δ ε (x,y) d µ n (g )
δ ε (g • x, g • y) ≤ 2 ε |g | ε |g x| ε δ ε (x, y), ρ n,s (ε) ≤ 2 ε k n (s) sup x e(g • x) e(x) |g x| s-ε |g | ε d µ n (g ).
We denote c = sup

g ,x e(g • x) e(x) < ∞, hence using s ≥ ε, |g x| s-ε ≤ |g | s-ε we get e(g • x) e(x) |g x| s-ε |g | ε ≤ c|g | s , ρ n,s (ε) ≤ c2 ε k n (s) |g | s d µ n (g ) ≤ c 2 ε c s .
Assume that ϕ ∈ C (P d-1 ) satisfies Q s e ϕ = ϕ and denote

M + = {x ∈ P d-1 ; ϕ(x) = sup y∈P d -1 ϕ(y)}, M -= {x ∈ P d-1 ; ϕ(x) = inf y∈P d -1 ϕ(y)}.
Then, as in the proof of Lemma 2.9, suppµ

• M + ⊂ M + , suppµ • M -⊂ M -, hence by minimality of Λ([suppµ]) we have Λ([suppµ]) ⊂ M + ∩ M -. It follows M + ∩ M -= φ, ϕ =constant on P d-1 . If ψ ∈ C (P d-1
) is ε-Hölder the above inequality gives for any x, y y). Since ρ n,s (ε) is bounded this shows that the sequence (Q s e ) n ψ is equicontinuous. By density this remains valid for any ψ ∈ C (P d-1 ).

∈ P d-1 |(Q s e ) n ϕ(x) -(Q s e ) n ϕ(y)| ≤ (a s |ϕ| + ρ n,s (ε))[ϕ] ε )δ ε (x,
Hence we can apply Lemma 2.12 to Q = Q s e : there is a unique Q s e -stationary measure. If s = 0, we have ε = 0, hence the above inequality does not show the equicontinuity of P n ϕ. In this case the equicontinuity follows from Theorem 3.2 in the next section;

we have for ε > 0 the convergence lim n→∞ δ ε (g

• x, g • y)d µ n (g ) = 0, which implies for ϕ ∈ H ε (P d-1 ), |P n ϕ)(x) -P n ϕ(y)| ≤ [ϕ] ε δ ε (g • x, g • y)d µ n (g ), |P n ϕ(x) -ν(ϕ)| ≤ [ϕ] ε δ ε (g • x, g • y)d ν(y)d µ n (g ), lim n→∞ |P n ϕ(x) -ν(ϕ)| = 0.

Remark.

1. If for some τ > 0 and s

∈ [0, s ∞ [, |g | s γ τ (g )d µ(g ) < ∞ with γ(g ) = sup(|g |, |g -1 |
) then it is proved in section 3, Corollary 3.18 that lim n→∞ ρ n,s (ε) = 0, hence ρ n,s (ε) < 1 for some n = n 0 . Hence (Q s e ) n 0 satisfies a Doeblin-Fortet inequality (see [START_REF] Guivarc | Théorèmes limites pour une classe de chaines de Markov et applications aux difféomorphismes d'Anosov[END_REF]). ). Then the inequality and its proof remain valid for Qs e instead of Q s e . In particular for any ψ ∈ C (S d-1 ) the sequence ( Qs e ) n ψ is equicontinuous. This fact will be used in the next paragraph.

Proof of Theorem 2.6: As in the proof of Lemma 2.8, we consider the non linear operator * P s on M 1 (P d-1 ) defined by * P s σ = we can apply Proposition 2.13 with e = σ s ; then we get existence and uniqueness of e s ν s ∈ M 1 (P d-1 ) with P s e s = k(s)e s , P s ν s = k(s)ν s , ν s (e s ) = 1 and e s satisfies p(s)e s (x) = |〈x, y〉| s d σ(y) where p(s) = ν s ( σ s ). Also Q s = Q s e s has a unique stationary measure π s . The uniqueness of ν s ∈ M 1 (P d-1 ) with P s ν s = k(s)ν s follows. Also σ = * ν s by the same proof.

Lemma 2.8 implies that if some η ∈ M 1 (P d-1 ) satisfies P s η = kη, then k = k(s). Since suppν s is [suppµ]-invariant and Λ([suppµ]) is minimal we get suppν s ⊃ Λ([suppµ]). We can again use Schauder-Tychonoff theorem in order to construct σ ′ ∈ M 1 (P d-1 ) with suppσ ′ ⊂ Λ([suppµ]), P s σ ′ = kσ ′ . Since σ ′ = ν s , we get finally suppν s = Λ([suppµ]).

In order to show the continuity of s → ν s , s → e s we observe that, from the above argument, ν s is uniquely defined by

P s ν s = k(s)ν s , ν s ∈ M 1 (P d-1
ν s n = η ∈ M 1 (P d-1
). We have

P s n ν s n (ϕ) = ν s n (P s n ϕ), lim s n →s 0 P s n ν s n (ϕ) = lim s n →s 0 k(s n )ν s n (ϕ) = k(s 0 )η(ϕ).
Then the uniform continuity in (s, x) of P s ϕ(x) implies P s 0 η = k(s 0 )η. The uniqueness of ν s 0 implies ν s 0 = η, and the arbitrariness of s n gives the continuity of s → ν s at s 0 . The same property is true for the operator * P s and the measure * ν s defined by * P s ( * ν s ) = k(s) ( * ν s ), * ν s ∈ M 1 (P d-1 ). Lemma 2.10, implies p(s)e s (x) = |〈x, y〉| s d * ν s (y), and since the set of functions x → |〈x, y〉| s (y ∈ P d-1 , s ∈ I µ ) is locally equicontinuous we have lim

s→s 0 |e s -e s 0 | = 0.
In order to show the strict convexity of log k(s) we take s, t ∈ I µ , p ∈ (0, 1) and we observe that from Hölder inequality, P ps+ (1-p)t [(e s ) p (e t ) 1-p ] ≤ k p (s)k 1-p (t )(e s ) p (e t ) 1-p . We denote f = (e s ) p (e t ) 1-p and assume k(ps + (1-p)t ) = k p (s)k 1-p (t ) for some s = t . Then Lemma 2.9 can be used with e = e ps+(1-p)t and Q pq+(1-p)t e ϕ = 1 k(ps+(1-p)t)e P ps+(1-p)t (ϕe). It gives on Λ([suppµ]) : f = ce ps+(1-p)t for some constant c > 0.

Hence, on Λ([suppµ]) we have

P ps+(1-p)t [(e s ) p (e t ) 1-p ] = k p (s)k (1-p) (t )(e s ) p (e t ) 1-p .
This means that there is equality in the above Hölder inequality. It follows that, for some positive function c(x) and any x in Λ([suppµ]), g ∈ suppµ

|g x| s e s (g • x) e s (x) = c(x)|g x| t e t (g • x) e t (x) .
Integration with respect to µ gives: c(x) = k(s) k(t) . Since s = t , we get, for some constant c > 0 and ϕ ∈ C (P d-1 ) positive, for any (x, g ) as above

|g x| = c ϕ(g •x) ϕ(x) . It follows, if g ∈ (suppµ) n and x ∈ Λ([suppµ]), |g x| = c n ϕ(g •x) ϕ(x) . If g ∈ [suppµ]
pr ox , we get |λ g | ∈ c N . This contradicts Proposition 2.5.

In order to show the convergence of (Q s ) n ϕ, since by Proposition 2.13 the family (Q s ) n ϕ is equicontinuous, it suffices to show in view of Lemma 2.12 that the relation Q s ϕ = e i θ ϕ with ϕ ∈ C (P d-1 ), |e i θ | = 1 implies e i θ = 1, ϕ =constant. Taking absolute values we get |ϕ| ≤ Q s |ϕ|. As in Lemma 2.9, we get that for any x in Λ([suppµ]), |ϕ(x)| = sup{|ϕ(y)|; y ∈ P d-1 }. Hence we can assume |ϕ(x)| = 1 on Λ([suppµ]). Now we can use the equation

e i θ ϕ(x) = q s (x, g )ϕ(g • x)d µ(g ), where q s (x, g ) = 1 k(s) e s (g •x) e s (x) |g x| s , hence q s (x, g )d µ(g ) = 1. Strict convexity yields the equality e i θ ϕ(x) = ϕ(g • x), for any x ∈ Λ([suppµ]), g ∈ suppµ.
We know, from Proposition 2.13 that P n ϕ converges uniformly to ν(ϕ) where ν is the unique P -stationary measure on P d-1 . Furthermore, on Λ([suppµ]) we have P n ϕ = e i nθ ϕ. The above convergence gives e i θ = 1, since ϕ = 0 on Λ([suppµ]). The fact that ϕ is constant follows from Proposition 2.13.

In order to show the last assertion in case d > 1 we write Q z ϕ(x) as

Q z ϕ(x) = |g x| i t q s (x, g )ϕ(g • x)d µ(g ).
We observe that the absolute value of the function Q z ϕ is bounded by the function Q s |ϕ|. Hence, from above, the equation

Q z ϕ = e i θ ϕ gives Q s |ϕ| ≥ |ϕ|, hence Q s |ϕ| =
|ϕ| and |ϕ| =cte. Then the equation Q z ϕ = e i θ ϕ gives for any x and g ∈ suppµ, since q s (x, g )d µ(g ) = 1 we have |g x| i t ϕ(g •x) ϕ(x) = e i θ , µ-a.e. This contradicts Proposition 2.5 if t = 0. If t = 0,from above we have e i θ = 1, ϕ =constant .

Eigenfunctions, limit sets, and eigenmeasures on S d-1

Here we study the operator P s on S d-1 defined by P s ϕ(x) = ϕ(g • x)|g x| s µ(d g ). We show that there are 2 cases, depending on the existence of a [suppµ]-invariant proper convex cone in V or not. We still denote by e s the function on S d-1 lifted from e s ∈ C (P d -1) . We denote Qs the operator on S d-1 defined by Qs ϕ = 1 k(s)e s P s (ϕe s ).

We already know, using the remark which follows Proposition 2.13, that for s > 0 and any given ϕ ∈ C (S d-1 ), the sequence ( Qs ) n ϕ is equicontinuous. For any subsemigroup T of G satisfying condition i-p, we denote by Λ(T ) the inverse image of Λ(T ) in S d-1 . We begin by considering the dynamics of T on S d-1 . For analogous results in more general situations see [START_REF] Guivarc | On contraction properties for products of Markov driven random matrices[END_REF]. We recall that a convex cone in V is said to be proper if it does not contain a line.

Proposition 2.14. Assume T ⊂ G is a subsemigroup which satisfies condition ip. If d = 1, we assume that T is non-arithmetic. Then the action of T on S d-1 has one or two minimal sets whose union is Λ(T ) : I There is no T -invariant proper convex cone in V and in that case, Λ(T ) is the unique

T -minimal subset of S d-1 .

II T preserves a closed proper convex cone C ⊂ V and then the action of T on S d-1 has two and only two minimal subsets

Λ + (T ), Λ -(T ) with Λ -(T ) = -Λ + (T ), Λ + (T ) ⊂ S d-1 ∩ C . The convex cone generated by Λ + (T ) is proper and T -invariant.
The proof depends of the following lemma.

Lemma 2.15.

Let V i (1 ≤ i ≤ r ) be vector subspaces of V . If condition i-p is valid, then there exists g ∈ T prox such that the hyperplane V < g does not contain any V i (1 ≤ i ≤ r ).
Proof. The dual semigroup T * of T satisfies also condition i-p hence we can also consider its limit set Λ(T * ) ⊂ P(V * ). Let v(g * ) be the point of P(V * ) corresponding to a dominant eigenvector of g * . Observe that the condition that an hyperplane contains V i defines a subspace of V * . If for any g * ∈ (T * ) pr ox the hyperplane v (g * ) contains some V i then by density any x ∈ Λ(T * ) contains some V i . Then the T * -invariance of Λ(T * ) implies that T * leaves invariant a finite union of subspaces of P(V * ), which contradicts condition i-p.

Proof of Proposition 2.14:

Let x ∈ Λ(T ) and S = T • y. We observe that if y ∈ S d-1 , then T • y contains x or -x, since the projection of T • x in P d-1 contains Λ(T ). Assume first -x ∉ T • x. If y ∈ T • x, then T • y ⊂ S, hence x ∈ T • y. This shows the T -minimality of S.
The same argument shows that -y ∉ S, hence S ∩ -S = φ. Since the projection of S in P d-1 is Λ(T ), we see that the projection of S d-1 on P d-1 gives a T -equivariant homeomorphism of S on Λ(T ). Since -x ∉ S, there are two T -minimal sets, S and -S.

Since for any y ∈ S d-1 , T • y contains S or -S, these sets are the unique minimal sets.

Assume now -x ∈ T • x, hence S = -S. Since the projection of

S in P d-1 is Λ(T ), we see that S = Λ(T ). Assume now that C is a T -invariant closed proper convex cone. Then C ∩ S d-1 is T - invariant and closed, hence C ∩S d-1 ⊃ Λ + (T ) or Λ -(T ) in the first situation, (-x ∉ T • x).
In the second situation C cannot exists, since C ∩ S d-1 would contain Λ(T ), which is symmetric.

It remains to show that, in the first situation, there exists a T -invariant closed proper convex cone. Let C be the convex cone generated by Λ + (T ) and let us show [START_REF] Feller | An Introduction to Probability Theory and its Applications[END_REF] shows that there exists

C ∩ -C = {0}. Assume C ∩ -C = {0}; then we can find y 1 , . . . , y p ∈ C , z 1 , . . . , z q ∈ -C and convex combinations y = p Σ 1 α i y i , z = q Σ 1 β j z j with y = z. Lemma 2.
g ∈ T prox such that y i (1 ≤ i ≤ p) and z j (1 ≤ j ≤ q) do not belong to V < g . Hence, with n ∈ 2N : lim n→+∞ g n y |g n | = lim n→∞ Σ p 1 α i g n y i |g n y i | |g n y i | |g n | = Σ p 1 α i u i v g ,
where

u i = lim n→∞ |g n y i | |g n | > 0 and v g ∈ Λ + (T ) is the unique dominant eigenvector of g in Λ + (T ).
In the same way :

lim n→∞ g n z |g n | = - q Σ 1 β j u ′ j v g ,
with u ′ j > 0. Since y = z we have a contradiction. Hence we have the required dichotomy. The last assertion follows.

We denote by νs the symmetric measure on S d-1 with projection ν s on P d-1 . In case II, we denote by ν s + (resp ν s -) the normalized restrictions of νs to

Λ + (T ) (resp Λ -(T )). For a subset X ⊂ S d-1 we recall that Co(X ) is the convex envelope of X in S d-1 . Theorem 2.16. Let µ ∈ M 1 (G), s ∈ I µ and assume T = [suppµ] satisfies i -p. If d = 1 we assume that µ is non-arithmetic. Then for any ϕ ∈ C (S d-1 ), x ∈ S d-1 , we have the uniform convergence lim n→∞ 1 n Σ n 1 ( Qs ) n ϕ(x) = π s (x)(ϕ), where, πs (x) ∈ M 1 (S d-1
) is supported on Λ(T ) and is Qs -stationary.

Furthermore there are 2 cases given by Proposition 2.14.

I Qs has a unique stationary measure πs with supp πs = Λ(T ) and πs (x) = πs for any x ∈ S d-1 . The Q s -invariant functions are constant. We have πs = e s νs and P s νs = k(s)ν s .

II Qs has two and only two extremal stationary measures π s + , π s -. We have suppπ s

+ = Λ + (T ) and π s -is symmetric of π s + . If π s + = e s ν s + , then P s ν s + = k(s)ν s + .
Also, there are 2 minimal Qs -invariant continuous functions p s + , p s -and we have πs

(x) = p s + (x)π s + + p s -(x)π s -.
Furthermore p s + (x) is equal to the entrance probability in the convex envelope Co(Λ + (T )) for the Markov chain defined by Qs . In particular p s For s = 0 we will need the following lemma, which uses results of section 3.

+ (x) = 1 (resp p s + (x) = 0) if x ∈ Λ + (T ) (resp Λ -(T ). If * ν s + ∈ M 1 (Λ + (T * ) satisfies * P s * ν s + = k(s) * ν s + , we have for u ∈ S d-1 and p(s) as in Theorem 2.6, p(s)e s + (u) = 〈u, u ′ 〉 s + d * ν s + (u ′ ) with 〈u, u ′ 〉 + = sup(0,
Lemma 2.17.

For u ∈ S d-1 ,t > 0 we denote ∆ t u = {y ∈ P d-1 ; |〈u, y〉| < t }. Then, for any ε, t > 0, x, y ∈ S d-1 lim sup n→∞ δ ε (g • x, g • y)d µ n (g ) ≤ 2 ε t ε δ ε (x, y) + 2 ε ν(∆ t x ).
In particular, for any ϕ ∈ H ε (S d-1 ) the sequence P n ϕ is equicontinuous.

Proof. We write

δ ε (g • x, g • y)d µ n (g ) = 1 [1/t,∞[ ( |S n | |S n x| ) δ ε (S n • x, S n • y)d P(ω) + 1 ]0,1/t[ ( |S n | |S n x| ) δ ε (S n • x, S n • y)d P(ω). Using Lemma 2.11 we have δ ε (S n • x, S n • y) ≤ (2 |S n | |S n x| δ(x, y)) ε .
On the other hand, using Theorem 3.2, we know that lim

n→∞ |S n | |S n x| = 1 |〈z * (ω), x〉| where z * (ω) ∈ P d-1 has law ν. Hence lim sup n→∞ δ ε (g • x, g • y)d µ n (g ) ≤ 2 ε t ε δ ε (x, y) + 2 ε P{|〈z * (ω), x〉| < t } = 2 ε t ε δ ε (x, y) + 2 ε ν(∆ t x ).
We have

| P n ϕ(x) -P n ϕ(y)| ≤ [ϕ] ε δ ε (g • x, g • y)d µ n (g ). From Theorem 3.2,
∈ C (S d-1 ) the set of functions {( Q s ) n ϕ; n ∈ N} is equicontinuous.
In view of Lemma 2.17, this is also valid if s = 0. Hence we can use here Lemma 2.12 . This gives the first convergence. Since π s (x) is Q s -stationary, its projection on P d-1 is equal to the unique Q s -stationary measure π s , hence supp π s (x) ⊂ Λ(T ). On the other hand supp π s (x) is closed and T -invariant, hence contains a T -minimal set.

In case I, Λ(T ) is the unique minimal set, hence supp

π s (x) = Λ(T ). Furthermore, if ϕ ∈ C (S d-1 ) is Q s -invariant, the sets M -= {x ; ϕ(x) = inf{ϕ(y); y ∈ P d-1 }}, M + = {x ; ϕ(x) = sup{ϕ(y); y ∈ P d-1 }},
are closed and T -invariant, hence they contain minimal sets. Since Λ(T ) is the unique minimal set,

M + ∩ M -⊃ Λ(T ) = φ, hence ϕ is constant.
Then, using Proposition 2.13 and Lemma 2.12, we get that there exists a unique stationary measure πs . It follows that πs is symmetric with projection π s on P d-1 and πs = e s νs .

In case II, the restriction to the convex envelope Co(Λ + (T )) = Φ of the projection on

P d-1 is a T -equivariant homeomorphism.
If we denote by i + its inverse, we get that i + (π s ) is the unique Qs -stationary measure supported in Φ. Hence i + (π s ) = π s + . Then π s + and π s -are extremal Qs -stationary measures. Since the projection of π s (x) on P d-1 is π s , we can write

π s (x) = (p s + (x, y)δ y + p s -(x, y)δ -y )d π s (y) = p s + (x)π s + + p s -(x)π s -,
where

p s + (x) = p s + (x, .) and p s -(x) = p s -(x, .) are Borel functions of y ∈ Λ(T ) such that p s -(x) + p s -(x) = 1. Then p s + (x)π s + is the restriction of π s (x) to Λ + (T )
, hence is a Qsinvariant measure. In view of the uniqueness of the stationary measure of Qs restricted to Λ + (T ), we get that p s

+ (x)π s + is proportional to π s + , i.e. p s + (x) is independent of y, π s + -a.e. .
Hence, the first assertion of the theorem implies that the only extremal Qs -stationary measures are π s + and π s -. The corresponding facts for ν s + and ν s -follow. From the mean ergodic theorem in C (S d-1 ) and the equicontinuity property of (Q s ) n we know that the operator defined by lim If we restrict the convergence of 1

n n-1 Σ 0 ( Qs ) k (δ x ) to x ∈ Φ,
in view of the fact that the restriction to Φ of the projection on P d-1 is a homeomorphism onto its image, we get

p s + (x) = 1, p s -(x) = 0 if x ∈ Φ. Let us denote by τ the entrance time of S n (ω) • x in Φ ∪ -Φ and by a E s
x the expectation symbol associated with the Markov chain S n (ω)• x defined by Q s . Using theorem 2.6 we

get a E s x (1 Φ∪-Φ (S τ • x)) = 1. Since p s + (x) is a Qs -invariant function p s + (S n • x) is a martin- gale, hence p s + (x) = a E s x (p s + (S τ • x)). Since p s + (x) = 1 on Φ and p s + (x) = 0 on -Φ we get p s + (x) = a E s x (1 Φ (S τ • x))
, hence the stated interpretation for p s + (x). As in Lemma 2.10, we verify that the function ϕ

(u) = p(s) 〈u, u ′ 〉 s + d * ν s + (u ′ ) on S d-1 satisfies P s ϕ = k(s)ϕ, hence the function ϕ e s satisfies Q s ( ϕ e s ) =
Λ + (T ) = {u ∈ S d-1 : 〈u, u ′ 〉 ≥ 0 for any u ′ ∈ Λ + (T * )}
is non trivial, closed, T -invariant and has non zero interior, hence Λ + (T ) contains either Λ + (T ) or Λ -(T ) and has trivial intersection with one of then. We can assume Λ + (T ) ⊃ Λ + (T ). Then, for u ∈ Λ + (T ) and any u ′ ∈ Λ + (T * ), we have 〈u,

u ′ 〉 + = 〈u, u ′ 〉, hence, ϕ(u) = p(s) |〈u, u ′ 〉| s d * ν s + (u ′ ) = e s (u), i.e., ϕ e s = 1 on Λ + (T ). Also we have 〈u, u ′ 〉 + = 0 for u ∈ Λ -(T ), u ′ ∈ Λ + (T * ). Since Q( ϕ e s ) =
(u) = p(s) 〈u, x〉 s + d * ν s -(x) = p s -(u)e s (u). From above we know that if s ≥ 0 and ϕ ∈ C (S d-1 ) the sequence ( Q s ) n ϕ is equicontinu- ous. Lemma 2.
12 reduces the discussion of the behaviour of ( Q s ) n ϕ to the existence of eigenvalues z of Q s with |z| = 1. In this direction we have the following Corollary 2.18. For s ∈ I µ , the equation Q s ϕ = e i θ ϕ with e i θ = 1, ϕ ∈ C (S d-1 ) has a non trivial solution only in case I. In that case e i θ = -1, ϕ is antisymmetric, satisfies ϕ(g • x) = -ϕ(x) on suppµ × Λ(T ) and is uniquely defined up to a coefficient.

Proof. We observe that, since ϕ satisfies Q s ϕ = e i θ ϕ, the function ϕ ′ defined by ϕ ′ (x) = ϕ(-x) satisfies also Q s ϕ ′ = e i θ ϕ ′ . Then ϕ + ϕ ′ is symmetric and defines a function ϕ in

C (P d-1 ) with Q s - ϕ = e i θ - ϕ . If e i θ = 1, Theorem 2.6 gives - ϕ = 0, i.e ϕ is antisymmetric.
Furthermore, in case II, the restriction of ϕ to Λ + (T ) satisfies the same equation and the projection of Λ + (T ) on Λ(T ) is an equivariant homeomorphism. Then Theorem 2.6 gives a contradiction. Hence if ϕ ∈ C (S d-1 ) satisfies Q s ϕ = e i θ ϕ, then we are in case I. Also, passing to absolute values as in the proof of Theorem 2.6 we get Q s |ϕ| = |ϕ|, |ϕ| =cte. Furthermore by strict convexity we have on suppµ × Λ(T ), ϕ(g

• x) = e i θ ϕ(x), hence ϕ 2 (g • x) = e 2i θ ϕ 2 (x). Since ϕ 2 is symmetric and satisfies Q s ϕ 2 = e 2i θ ϕ 2 , we get e 2i θ = 1, i.e e i θ = -1; in particular ϕ(g • x) = -ϕ(x) on suppµ × Λ(T ). If ϕ ′ ∈ C (S d-1 ) satisfies also Q s ϕ ′ = -ϕ ′ , we get from above Q s ϕ ′ ϕ = ϕ ′ ϕ , ϕ ′ ϕ (-x) = ϕ ′ ϕ (x), hence ϕ ′ is pro- portional to ϕ.

Laws of large numbers and spectral gaps

Here we develop section 2 in a quantitative direction. A martingale construction plays an essential role in this study. The renewal theorems of section 4 are based on Theorem 3.2 below. Also the Doeblin-Fortet inequality in Corollary 3.21 is used in section 5 to show the homogeneity at infinity of the stationary measure for an affine random walk on V . Theorem A in the introduction follows directly from Corollaries 3.19, 3.20.

Notation

As in section 2, we assume that condition i-p is valid for the semigroup T = [suppµ]. If d = 1 we assume that T = [suppµ] is non arithmetic. For s ∈ I µ we consider the functions q s and q s n (n > 0) on P d-1 ×G, defined by

q s (x, g ) = 1 k(s) e s (g • x) e s (x) |g x| s = q s 1 (x, g ), q s n (x, g ) = 1 k n (s) e s (g • x) e s (x) |g x| s ,
hence by the definition of e s we have q s n (x, g ) d µ n (g ) = 1. We denote by P d-1 2 the flag manifold of planes and by P d- 1 1,2 the manifold of contact elements on P d-1 . Such a plane is defined up to normalisation by a 2-vector x ∧ y ∈ ∧ 2 V and we can assume |x ∧ y| = 1. We write g (x ∧ y) = g x ∧ g y. Also a contact element ξ is defined by its origin x ∈ P d-1 and a line through x. Hence we can write ξ = (x, x ∧ y)

where |x| = |x ∧ y| = 1. The following additive cocycles of the actions of G on P d-1 , P d- 12 , P d-1 1,2 will play an essential role:

σ 1 (g , x) = log |g x|, σ 2 (g , x ∧ y) = log|g (x ∧ y)|, σ(g , ξ) = log |g (x ∧ y)| -2 log |g x|.
In addition to the norm of g we will need to use the quantity

γ(g ) = sup(|g |, |g -1 |) ≥ 1.
Clearly, for any x ∈ V , with |x| = 1, we havelog γ(g

) ≤ log|g x| ≤ log γ(g ). For a finite sequence ω = (g 1 , g 2 , • • • , g n ) we write S n (ω) = g n • • • g 1 ∈ G, q s n (x, ω) = n Π k=1 q s (S k-1 • x, g k ).
We denote by Ω n the space of finite sequences ω = (g 1 , g 2 , • • • , g n ) and we write Ω = G N .

We observe that θ s (x, g ) = |g x| s e s (g •x) e s (x) satisfies the cocycle relation

θ s (x, g g ′ ) = θ s (g ′ • x, g )θ s (x, g ′ ), hence q s n (x, ω) = 1 k n (s) θ s (x, S n (ω)). Definition 3.1. We denote Q s
x ∈ M 1 (Ω) the limit of the projective system of probability measures q s k (x, .)µ ⊗k on Ω k . We write

Q s = Q s x d π s (x) where π s is the unique Q s - stationary measure on P d-1 .
We recall that Theorem 2.6 implies that π s is not supported by an hyperplane.

The corresponding expectation symbol will be written E s

x and the shift on Ω will be denoted by θ. We write also E s (ϕ) = E s x (ϕ)d π s (x). The path space of the Markov chain defined by Q s is a factor space of a Ω = P d-1 × Ω, and the corresponding shift on a Ω will be written a θ with a θ(x, ω) = (g 1 (ω) • x, θω). Hence (P d-1 × Ω, a θ) is a skew product over (Ω, θ). The projection on Ω of the Markov measure a Q s

x = δ x ⊗ Q s x is Q s x , hence a Q s = δ x ⊗ Q s x d π s (x) projects on Q s .
The uniqueness of the Q s -stationary measure π s implies the ergodicity of the a θ-invariant measure a Q s , hence Q s is also θ-invariant and ergodic.

If s = 0, the random variables g k (ω) are i.i.d with law µ and Q • = P = µ ⊗N . Here, under condition i-p, we extend the results of [START_REF] Guivarc | Products of random matrices and convergence theorems[END_REF] to the case s ≥ 0, in particular we construct a suitable kernel-valued martingale with contraction properties as in [START_REF] Furstenberg | Boundary theory and stochastic processes on homogeneous spaces[END_REF]. This will allow us to prove strong forms of the law of large numbers for S n (ω) and to compare the measures Q s x when x varies. Then we can deduce the simplicity of the dominant Lyapunov exponent of S n (ω) under the θ-invariant probability Q s for s ≥ 0. Spectral gap properties for twisted convolution operators on the projective space and on the unit sphere will follow.

A martingale and the equivalence of

Q s x to Q s
When convenient we identify x ∈ P d-1 with one of its representatives x in S d-1 . We recall that the Markov kernel * Q s is defined by * Q s ϕ = 1 k(s) * e s * P s (ϕ * e s ) where * P s ϕ(x) = ϕ(g • x)|g x| s d µ * (g ), * P s ( * e s ) = k(s)( * e s ) and * Q s has a unique stationary measure * π s . Furthermore we have * π s = * e s * ν s where * ν s ∈ M 1 (P d-1 ) is the unique solution of * P s ( * ν s ) = k(s)( * ν s ). We denote by m the unique rotation invariant probability measure on P d-1 .

Theorem 3.2. Let Ω ′ ⊂ Ω be the (shift-invariant) Borel subset of elements ω ∈ Ω such that S * n (ω).m converges to a Dirac measure δ z * (ω). Then g * 1 • z * (θω) = z * (ω), Q s (Ω ′ ) = 1, the law of z * (ω) under Q s is * π s and * π s is proper. In particular if ω ∈ Ω ′ and |〈x, z * (ω)〉〈y, z * (ω)〉| = 0, then lim n→∞ δ(S n (ω) • x, S n (ω) • y) = 0. If ω ∈ Ω ′ and ξ = (x, x ∧ y) ∈ P d-1 1,2 lim n→∞ |S n (ω)x| |S n (ω)| = |〈z * (ω), x〉|, lim n→∞ S * n • m = δ z * (ω) .
In particular, if 〈z * (ω),

x〉 = 0 then lim n→∞ σ(S n , ξ) = -∞.
Also, for any x The proof of Theorem 3.2 is based on the following lemmas, in particular on the study of a kernel-valued martingale.

∈ P d-1 Q s x is equivalent to Q s and
Lemma 3.3. Assume z * ∈ P d-1 and u n ∈ G is a sequence such that lim n→∞ u * n • m = δ z * . Then, for any x, y ∈ P d-1 with |〈z * , x〉〈z * , y〉| = 0 we have lim n→∞ δ(u n • x, u n • y) = 0. If ξ = (x, x ∧ y) ∈ P d-1
1,2 and 〈z * , x〉 = 0 then

lim n→∞ |u n x| |u n | = |〈z * , x〉|, lim n→∞ σ(u n , ξ) = -∞.
These convergences are uniform on any compact subset on which 〈z * , x〉 do not vanish.

Proof. We denote by e i (1 ≤ i ≤ d ) an orthonormal basis of V , by ēi the projection of e i in P d-1 , by Ā+ the set of diagonal matrices

a = diag(a 1 , a 2 , • • • , a d ) and a 1 ≥ a 2 ≥ • • • ≥ a d > 0. We write u n = k n a n k ′ n with a n ∈ Ā+ , k n , k ′ n ∈ O(d ). Then, for x ∈ P d-1 , |u n x| 2 = |a n k ′ n x| 2 = Σ d 1 (a i n ) 2 |〈k ′ n x, e i 〉| 2 . Also, u * n • m = (k ′ n ) -1 a n • m converges to z * ∈ P d-1 , which implies lim n→∞ a n • m = δ e 1 , lim n→∞ (k ′ n ) -1 . ē1 = z * .
In particular, if i > 1, we have a i n = o(a

|x ∧ y| 2 = |x| 2 |y| 2 -|〈x, y〉| 2 .
For x, y ∈ P d-1 and corresponding x, ỹ ∈ S d-1 we write |x ∧ y| = | x ∧ ỹ|. Then on P d-1 , there is an associated distance δ 1 given by δ 1 (x, y) = |x ∧ y| and we have 1

2 δ ≤ δ 1 ≤ δ. We observe that δ 1 (u n • x, u n • y) = |u n x∧u n y| |u n x| |u n y| . Also |u n x ∧ u n y| 2 = Σ i >j (a i n a j n ) 2 |〈k n ( x ∧ y), e i ∧ e j 〉| 2 ≤ d (d -1) 2 (a 1 n a 2 n | x ∧ y|) 2 .
It follows

δ 1 (u n • x, u n • y) ≤ d (d -1) 2 1/2 a 1 n a 2 n |u n x| |u n y| | x ∧ y|. Since |u n x| ∼ a 1 n |〈z * , x〉|, |u n y| ∼ a 1 n |〈z * , y〉| and a 2 n = o(a 1 n ), 〈z * , x〉〈z * , y〉 = 0, we get lim n→∞ δ 1 (u n • x, u n • y) = 0.
It follows, for any x, y

∈ P d-1 , that lim n→∞ δ(u n • x, u n • y) = 0. Also, since a 2 n = o(a 1 n ), and 〈z * , x〉 = 0 we get lim n→∞ σ(u n , ξ) = -∞.
The above calculations imply the uniformities in the convergences.

Lemma 3.4. Assume ν n ∈ M 1 (P d-1
) is a sequence such that ν n is relatively compact in variation, and each ν n is proper. Let u n ∈ G be a sequence such that u * n • ν n converges weakly to δ z * (z * ∈ P d-1 ). Then for any proper ρ ∈ M 1 (P d-1 ), u * n • ρ converges weakly to δ z * .

Proof. We can assume, in variation, lim n→∞ ν n = ν 0 where ν 0 is proper. Also we can assume, going to subsequences, that u * n converges to a quasi-projective map of the form u * , defined and continuous outside a projective subspace H ⊂ P d-1 . Let ϕ ∈ C (P d-1 ) and denote

I n = (u * n • ν n )(ϕ) -(u * • ν 0 )(ϕ) = (ν n -ν 0 )(ϕ • u * n ) + ν 0 (ϕ • u * n -ϕ • u * ).
The first term is bounded by |ϕ| ν n -ν 0 , hence it converges to zero. Since ν 0 (H) = 0 and ϕ • u * n converges to ϕ • u * outside H, we can use dominated convergence for the second term, lim

n→∞ ν 0 (ϕ • u * n -ϕ • u * ) = 0; hence lim n→∞ I n = 0. Then u * n • ν n converges to u * • ν 0 weakly. In particular u * • ν 0 = δ z * , hence u * • y = z * , ν 0 -a.e. Since ν 0 (H) = 0, we have u * • y = z * on P d-1 \ H. Since ρ is proper u * • ρ = δ z * , hence lim n→∞ u * n • ρ = δ z * .
Lemma 3.5. For x, y ∈ P d-1 the total variation measure of Q s x -Q s y is bounded by Bδ s (x, y)Q s . Furthermore, there exists c(s) > 0 such that, for any x ∈ P d-1 we have

Q s x ≤ c(s) Q s .
Proof. We write q s n (g ) = q s n (x, g )d π s (x) and we observe that for any measurable ϕ depending on the first n coordinates,

ϕ(ω)d Q s (ω) = q s n (S n (ω))ϕ(ω)d µ ⊗n (ω). Also, |(Q s x -Q s y )(ϕ)| ≤ |q s n (x, S n ) -q s n (y, S n )| |ϕ(ω)|d µ ⊗n . Using Lemma 2.11 we have for any g ∈ G |q s n (x, g ) -q s n (y, g )| ≤ b s |g | s k n (s) δ s (x, y).
Since π s is not supported by an hyperplane we can use Lemma 2.7, hence for some b > 0

q s n (g ) ≥ b |g | s k n (s) and |q s n (x, g ) -q s n (y, g )| ≤ b s b q s n (g )δ s (x, y).
It follows:

|(Q s x -Q s y )(ϕ)| ≤ b s b δ s (x, y) |ϕ(ω)|d Q s (ω), hence the first conclusion with B = b s b . Integrating with respect to π s we get, since δ(x, y) ≤ 2, Q s x ≤ (1 + B( 2) s )Q s hence the second formula with c(s) = 1 + B( 2) s .
Lemma 3.6. We consider the positive kernel κ s

x from P d-1 to P d-1 given by κ s x = |〈x,•〉| s e s (x) * ν s . Then:

g * • κ s g •x q s (x, g )d µ(g ) = κ s x , κ s x (1) = 1 e s (x) |〈x, y〉| s d * ν s (y) = p(s) ∈]0, 1]
and x → κ s x is continuous in variation. In particular S * n • κ s S n •x is a bounded martingale with respect to Q s x and the natural filtration.

Proof. We consider the s-homogeneous measure λ s on V defined by

λ s = * ν s ⊗ ℓ s . By definition of * ν s , g * λ s d µ(g ) = k(s)λ s . Then the Radon measure λ s v defined by λ s v = |〈v, •〉| s λ s satisfies g * λ s g v d µ(g ) = k(s)λ s v .
This can be written, by definition of κ s x and q s (x, g ),

g * • κ s g •x q s (x, g )d µ(g ) = κ s x .
The martingale property of S * n • κ s S n •x follows. Furthermore since p(s)e s (x) is equal to |〈x, y〉| s d * ν s (y), Lemma 2.10 gives

κ s x (1) = 1 e s (x) |〈x, y〉| s d * ν s (y) = p(s) ∈]0, 1].
The continuity in variation of x → κ s x follows from the definition.

Lemma 3.7. Let ρ ∈ M 1 (P d-1 ), H be the set of projective subspaces H of minimal dimension such that ρ(H) > 0. Then the subset of elements H ∈ H such that ρ(H) = sup{ρ(L) ; L ∈ H } is finite and non void. Furthermore, there exists ε ρ > 0 such that for any H ∈ H :

ρ(H) = c ρ or ρ(H) ≤ c ρ -ε ρ , where c ρ = sup{ρ(L) ; L ∈ H }. Proof. If H, H ′ ∈ H , H = H ′ , then dim H ∩ H ′ < dim H, hence ρ(H ∩ H ′ ) = 0.
c ρ 2 < ρ(H n ) < c ρ , lim n→∞ ρ(H n ) = c ρ , and ρ(H n ) = ρ(H m ) if n = m.
This contradicts the fact that the cardinality of the sequence H n is at most 2 c ρ .

Lemma 3.8. Assume that the Markovian kernel x → ν x ∈ M 1 (P d-1 ) is continuous in variation and satisfies

ν x = q s (x, g )g * • ν g •x d µ(g ) .
Let H p,r the set of finite unions of r distinct subspaces of dimension p and let h be the function h(x) = sup{ν x (W ); W ∈ H p,r }. Then h is continuous and the set

{x : h(x) = sup{h(y), y ∈ P d-1 }}, is closed and [suppµ]-invariant. Proof. If W ∈ H p,r is fixed the function x → ν x (W ) is continuous since |ν x (W )-ν y (W )| ≤ ν x -ν y . This implies |h(x) -h(y)| ≤ ν x -ν y , hence the continuity of h.
We have for any

W ∈ H p,r ν x (W ) = q s (x, g )ν g •x ((g * ) -1 W )d µ(g ). Hence: h(x) ≤ q s (x, g )h(g • x)d µ(g ).
Then, as in Lemma 2.9, X is [suppµ]-invariant and closed.

Lemma 3.9. Let ν x be as in Lemma 3.8. Then for any x ∈ P d-1 , ν x is proper.

Proof. We write π x = ν x ν x (1) , denote by H k the set of projective subspaces of dimension k and

H = ∪ k≥0 H k , d (x) = inf{dimH; H ∈ H , π x (H) > 0}, m(x) = sup{π x (H); H ∈ H , d i mH = d (x)}, W (x) = {H ∈ H ; π x (H) = m(x)}.
Lemma 3.7 implies that the set W (x) has finite cardinality n(x) > 0. Also we denote p = inf{d (x);

x ∈ P d-1 }, h p (x) = sup{π x (H); H ∈ H p }. Lemma 3.8 shows that h p (x) reaches its maximum β on a closed [suppµ]-invariant sub- set X ⊂ P d-1 . Hence on Λ([suppµ]) we have h p (x) = β = m(x). It follows d (x) = p on Λ([suppµ]). The relation n(x)m(x) ≤ 1 implies n(x) ≤ 1 β on Λ([suppµ]). Let r = sup{n(x); x ∈ Λ([suppµ])} and denote h p,r (x) = sup{π x (W );W ∈ H p,r }. Then Lemma 3.8 implies h p,r (x) = r β on Λ([suppµ]). Since m(x) = β, this relation implies n(x) = r on Λ([suppµ]). Let W (x) = ∪{H; H ∈ W (x)
} and let us show the local constancy of the function W (x). Using Lemma 3.7 we get

β(x) = sup{π x (H); H ∈ H p , H ∉ W (x)} < β. Let x ∈ Λ([suppµ]), U x = {y; π y -π x < β -β(x)} and H y ∈ H p with π y (H y ) = β. Then, β -π x (H y ) = π y (H y ) -π x (H y ) ≤ π y -π x < β -β(x).
Hence π x (H y ) > β(x) and, by definition of β(x) we get H y ∈ W (x) for any y ∈ U x . Since

π x is continuous in variation, U x is a neighbourbood of x, hence W (x) is locally con- stant. Since Λ([suppµ]) is compact, W = ∪W (x) x∈Λ([suppµ])
is a finite union of subspaces.

On the other hand, the relations

r β = π x (W (x)) = q s (x, g )g * • π g •x (W (x))d µ(g ) , r β ≥ (g * • π g •x )(W (x)) imply that, for any x ∈ Λ([suppµ]), r β = g * • π g •x (W (x)) µ-a.e. By definition of W (g • x), we get, (g * ) -1 W (x) = W (g • x) µ-a.e. Hence, for any g ∈ suppµ, (g * ) -1 (W (x)) = W (g • x). The relation (g * ) -1 (W ) = ∪(g * ) -1 x∈Λ([suppµ]) (W (x)) = ∪W (g • x) x∈Λ([suppµ]) shows that W is [suppµ * ]-invariant. Then condition i-p implies W = P d-1 , r = 1, p = d -1, d (x) = d -1, m(x) = 1 for any x ∈ P d-1 , hence the Lemma.
Proof of Theorem 3.2. We use the Markov kernel π s x defined by

π s x = κ s x ν s
x (1) with κ s x given in Lemma 3.6.

Then we have the harmonicity equation,

π s x = q s (x, g )g * • π s g •x d µ(g ),
and the continuity in variation of π s x . The above equation implies that the sequence of kernels S * n (ω) • π s S n (ω)•x is a Q s x -martingale with respect to the natural filtration on Ω. Since for x ∈ P d-1 , π s x ∈ M 1 (P d-1 ) we can apply the martingale convergence theorem. Since, by Lemma 3.9, π s

x is proper and, by definition, x → π s x is continuous in variation, we can use the same method as in [START_REF] Guivarc | Products of random matrices and convergence theorems[END_REF] ; because of i-p condition, the martingale S * n (ω).π s S n (ω)•x converges Q s x -a.e to a Dirac measure. Then, using Lemma 3.4, S * n •m converges Q s x -a.e. to a Dirac measure δ z * (ω) . Then, from above, for any

x ∈ P d-1 , Q s x (Ω ′ ) = 1 , z * (Q s x ) = π s x .
Hence the law of z * (ω) under Q s x is π s x . It follows, by integration,

Q s (Ω ′ ) = 1 , z * (Q s ) = z * (Q s x )d π s (x) = π s x d π s (x).
In view of the formulae for ν s x , π s x and the relation π s = e s ν s ν s (e s ) , we define * π s by z * (Q s ) = * π s . Then Lemma 3.9 and the definition of π s x give that * π s is proper. The relations:

lim n→∞ δ(S n (ω) • x, S n (ω) • y) = 0, lim n→∞ |S n (ω)x| |S n (ω)| = |〈z * (ω), x〉|, lim n→∞ σ(S n , ξ) = -∞, follow
from the geometrical Lemma 3.3, since S * n • m converges to δ z * (ω) . Using Lemma 3.5 we know that Q s

x is absolutely continuous with respect to Q s . We calculate

dQ s x dQ s (ω) as follows. By definition of Q s x and Q s , E s x ( d Q s x d Q s (ω)|g 1 , • • • , g n ) = q s n (x, S n (ω)) q s n (y, S n (ω))d π s (y) . Furthermore q s n (x, S n (ω)) q s n (y, S n (ω)) = |S n (ω)x| s |S n (ω)y| s e s (S n (ω) • x) e s (S n (ω) • y)
e s (y) e s (x) .

The martingale convergence theorem gives

d Q s x d Q s (ω) = lim n→∞ q s n (x, S n (ω)) q s n (y, S n (ω))d π s (y)
.

Using the relation lim d π s (y) -1 . Since, from above π s is proper and the

n→∞ |S n (ω)x| |S n (ω)| = |〈z(ω), x〉|, if ω ∈ Ω ′ ,
Q s -law of z * (ω) is π s , we have for any x ∈ P d-1 , |〈z * (ω), x〉| > 0, Q s -a.e
We conclude that Q s x is equivalent to Q s because dQ s

x dQ s (ω) > 0, Q s -a.e. Also, using the formulae above we have

d Q s x d Q s y (ω) = 〈z * (ω), x〉 〈z * (ω), y〉 s e s (y)
e s (x) .

The law of large numbers for log |S n (ω)x| with respect to Q s

Here, by derivatives of a function ϕ at the boundaries of an interval [a, b] we will mean finite half derivatives i.e. we write

ϕ ′ (a) = ϕ ′ (a + ) ∈ R , ϕ ′ (b) = ϕ ′ (b -) ∈ R . Theorem 3.10. Let µ ∈ M 1 (G), s ∈ I µ . Assume [suppµ] satisfies condition i-p, and log γ(g ) is µ-integrable. Assume also that |g | s log γ(g ) is µ-integrable and write L µ (s) = log |g x|q s (x, g )d π s (x) d µ(g ).
Then, for any x ∈ P d-1 , Q s -a.e, we have

lim n→∞ 1 n log |S n (ω)x| = lim n→∞ 1 n log|S n (ω)| = L µ (s).
This convergence is valid in

L 1 (Q s ) and in L 1 (Q s x ) for any x ∈ P d-1 . Furthermore k(t ) has a continuous derivative on [0, s] and if t ∈ [0, s], x ∈ P d-1 , L µ (t ) = k ′ (t ) k(t ) = lim n→∞ 1 nk n (t ) |g x| t log |g x|d µ n (g ) = lim n→∞ 1 n |g | t log |g |d µ n (g ) |g | t d µ n (g ) . In particular if α > 0 satisfies k(α) = 1, then k ′ (α) > 0.
Proof. We consider the function

f (x, ω) on a Ω defined by f (x, ω) = log |g 1 (ω)x|. If |x| = 1, we have -log|g -1 | ≤ log |g x| ≤ log |g |, hence f (x, ω) is a Q s -integrable. Moreover f (x, ω)d a Q(x, ω) = q s (x, g ) log |g x|d ν s (x)d µ(g ) = L µ (s),
and

n-1 0 ( f • a θ k )(x, ω) = log|S n (ω)x|.
As mentioned above a Q s is a θ-ergodic, hence we get using Birkhoff's theorem,

lim n→∞ 1 n log |S n (ω)x| = L µ (s), a Q s -a.e.
On the other hand, we can apply the subadditive ergodic theorem to the sequence log |S n (ω)| and to the ergodic system (Ω, θ, Q s ).

This implies that there exists L(s) ∈ R such that, Q s -a.e. and in L 1 (Q s ), the sequence Using Lemma 3.6, since Q s x ≤ c(s) Q s , this convergence is also valid Q s x -a.e. Hence by definition of a Q s , we have L(s) = L µ (s). The first assertion follows.

In order to get the L 1 -convergences, we observe that Fatou's Lemma gives lim inf

n→∞ 1 n log|S n (ω)x|d Q s (ω) ≥ L µ (s).
On the other hand, the subadditive ergodic theorem gives

lim n→∞ 1 n log |S n (ω)| d Q s (ω) = L(s) = L µ (s). Since |S n (ω)x| ≤ |S n (ω)| if |x| = 1, these two relations imply, for every x ∈ P d-1 , lim n→∞ 1 n log|S n (ω)x|d Q s (ω) = L µ (s). Now we write 1 n | log |S n (ω)x| -L(s)| ≤ 1 n (log |S n (ω)| -log |S n (ω)x|) + 1 n | log |S n (ω)| -L(s)|.
From the above calculation, the integral of the first term converges to zero. The subadditive ergodic theorem implies the same for the second term.

Hence lim

n→+∞ | log |S n (ω)x| -L(s)|d Q s (ω) = 0. Since Q s x ≤ c(s)Q s , this convergence is also valid in L 1 (Q s
x ) for any fixed x. This gives the second assertion, in particular

L µ (s) = lim n→∞ 1 n log |S n (ω)x|d Q s x (ω) = lim n→+∞ 1 n E s x (log |S n (ω)x|).
The above limit can be expressed as follows.

Let ϕ be a continuous function on P d-1 . Then Theorem 2.6 implies

lim n→∞ E s x (ϕ(S n (ω) • x)) = lim n→∞ (Q s ) n ϕ(x) = π s (ϕ), uniformly in x ∈ P d-1 . Hence L µ (s)π s (ϕ) = lim n→∞ 1 n E s x (ϕ(S n (ω) • x) log |S n (ω)x|).
In particular with ϕ = 1 e s and any x we have e s (x)L µ (s) = lim Using Theorem 2.6, we get lim

n→∞ v n (s) k n (s)e s (x) = π s 1 e s = 1.
Then the above formula for L µ (s) reduces to

L µ (s) = lim n→∞ 1 n v ′ n (s) v n (s)
.

On the other-hand, 1

n log v n (s) = 1 n s 0 v ′ n (t) v n (t) d t , lim n→∞ 1 n log v n (s) = log k(s).
The convexity of log v n on [0, s] gives (s) .

v ′ n (0) v n (0) ≤ v ′ n (t) v n (t) ≤ v ′ n (s) v n
Then the convergence of the sequences 1 (s) implies that the sequence v n (t) , to L(t ). Then dominated convergence gives the convergence of 1 n log v n (s) to log k(s) = s 0 L(t )d t . We have L(t ) = log|g x|q t (x, g )d µ(g )d π t (x), and the continuity of q s , π s given by Theorem 2.6. Then the bound of |g | t | log |g ||d µ(g ) given above implies the continuity of L(t ) on [0, s]. The integral expression of log k(s) in terms of L(t ) implies that k has a derivative and k ′ (t)

n v ′ n (0) v n (0) and 1 n v ′ n (s) v n
1 n v ′ n (t) v n (t
k(t) = L(t ) if t ∈ [0, s].
This gives the first part of the last relation in the theorem. In order to get the rest, we consider u n (t ) = |g | t d µ n (g ) and write for t

∈ [0, s] u ′ n (t ) u n (t ) = |g | t log |g |d µ n (g ) |g | t d µ n (g ) .
The convergence of 1 n log u n (t ) to log k(t ) and the convexity of the functions logu n (t ), log k(t ) give for t ∈ [0, s],

k ′ (t -) k(t ) ≤ lim inf n→∞ 1 n u ′ n (t ) u n (t ) ≤ lim sup n→∞ u ′ n (t ) u n (t ) ≤ k ′ (t + ) k(t ) . Since if t ∈]0, s[, we have k ′ (t -) = k ′ (t + ) = k ′ (t ), we get lim n→∞ 1 n u ′ n (t ) u n (t ) = k ′ (t ) k(t ) if t < s.
Furthermore, by continuity we have lim

n→∞ 1 n u ′ n (s) u n (s) = k ′ (s -) k(s)
. Now the rest of the formula follows from the expression of

u ′ n (t)
u n (t) given above. The relation L µ (α) > 0 follows from the formula L µ (t ) = k ′ (t) k(t) and the strict convexity of log k(t ).

Lyapunov exponents and spectral gaps

We begin with a more general situation than above. As special cases, it contains the Markov chains on P d-1 considered in section 2. In particular, simplicity of the dominant Lyapunov exponent given by Theorem 3.17 below will be a simple consequence of their special properties and of the general Proposition 3.11 below. For s = 0, this result was shown in [START_REF] Guivarc | Products of random matrices and convergence theorems[END_REF] under condition i-p. For the use of the Zariski closure as a tool to show condition i-p see [START_REF] Goldsheid | Lyapunov exponents of a product of random matrices[END_REF], [START_REF] Prasad | Regular elements in Zariski dense subgroups[END_REF]. We give corresponding notation.

Let X be a compact metric space, C (X , X ) the semigroup of continuous maps of X into itself which is endowed with the topology of uniform convergence. We denote by g • x the action of g ∈ C (X , X ) on x ∈ X and we consider a closed subsemigroup Σ of C (X , X ).

Let µ be a probability measure on Σ and let q(x, g ) be a continuous non negative function on X × suppµ such that q(x, g )d µ(g ) = 1. We will denote by (X , q ⊗ µ, Σ) this set of data and we will say that (X , q ⊗ µ, Σ) is a Markov system on (X , Σ). We write Ω = Σ N , we denote by Ω n the set of finite sequences of length n on Σ and for ω = (g 1 ,

g 2 , • • • , g n ) in Ω n , x ∈ X , we write q n (x, ω) = n Π 1 q(S k-1 • x, g k ) where S n = g n • • • g 1 , S 0 = I d .
We define a probability measure Q n x on Ω n by Q n x = q n (x, •)µ ⊗n and we denote by Q x the probability measure on Ω which is the projective limit of this system. If ν is a probability measure on X we set Q ν = Q x d ν(x). We will consider the extended shift a θ on a Ω = X × Ω which is defined by a θ(x, ω) = (g 1 • x, θω), and also the Markov chain on X with kernel Q defined by Qϕ(x) = ϕ(g • x)q(x, g )d µ(g ). Clearly, when endowed with the corresponding shift, the space of paths of this Markov chain is a factor system of (

X × Ω, a θ). If π is a Q-stationary measure on X , the measure Q π on Ω is shift-invariant and a Q π = δ x ⊗Q x d π(x) is a θ-invariant.
In this situation we will say that (X , q ⊗µ, Σ, π) is a

stationary Markov system. If π is Q-ergodic, then a Q π is a θ-ergodic and Q π is θ-ergodic.
We will denote by E x , E π the corresponding expectations symbols.

In particular we will consider below Markov systems of the form (X , q ⊗ µ, T ) where

Σ = T ⊂ GL(d , R) (d > 1)
, and also extensions of them. We can extend the action of

g ∈ T to X × P d-1 by g (x, v ) = (g • x, g • v ) and define a new Markov chain with kernel Q 1 by Q 1 ϕ(x, v ) = ϕ(g • x, g • v )q(x, g )d µ(g ).
Given a Q-stationary probability measure π, we will denote by C 1 the compact convex set of probability measures on X ×P d-1 which have projection π on X . The same considerations apply if P d-1 is replaced by P d- 1 2 , the manifold of 2-planes or P d- 1 1,2 , the manifold of contact elements in P d-1 . Then we define similarly the kernels Q 2 ,Q 1,2 and the convex sets C 2 ,C 1,2 .

Since S n = g n • • • g 1 and the g k are Q π -stationary random variables where Q π is θ-invariant and ergodic, the Lyapunov exponents of the product S n are well defined as soon as

log |g 1 (ω)|d Q π (ω) and log |g -1 1 (ω)|d Q π (ω)
are finite (see [START_REF] Ragunathan | A proof of Oseledec's multiplicative ergodic theorem[END_REF]). In particular the two largest ones γ 1 and γ 2 are given by

γ 1 = lim n→∞ 1 n log |S n (ω)|d Q π (ω), γ 1 + γ 2 = lim n→∞ 1 n log| ∧ 2 S n (ω)|d Q π (ω).
In order to study the values of γ 1 , γ 2 we need to consider the above Markov operators

Q 1 ,Q 2 ,Q 1,2
and the convex sets C 1 ,C 2 ,C 1,2 of corresponding stationary measures. We denote q(g ) = sup

x∈P d -1
|q(x, g )| and we assume log γ(g )q(g )d µ(g ) < ∞. For η 1 ∈ C 1 , we will write

I 1 (η 1 ) = σ 1 (g , v )d η 1 (x, v )d µ(g ), and similarly with η 2 ∈ C 2 , η 1,2 ∈ C 1,2 we define I 2 (η 2 ), I 1,2 (η 1,2 )
where we use the cocycles σ 1 , σ 2 , σ defined at the beginming of this section. The following result will be a basic tool in this subsection.

Proposition 3.11. With the above notation, let T be a closed subsemigroup of GL(d , R), (X , q ⊗µ, T, π) a stationary and uniquely ergodic Markov system. Assume that S * n •m converges Q π -a.e to a Dirac measure δ z * (ω) such that for any v ∈ P d-1 , 〈v, z * (ω)〉 = 0 Q π -a.e. Assume that log γ(g )q(g )d µ(g ) is finite. Then we have γ 2 -γ 1 = sup{I 1,2 (η); η ∈ C 1,2 } < 0, and the sequence

1 n sup x,v,v ′ E x (log δ(S n (ω) • v, S n (ω) • v ′ ) δ(v, v ′ ) ) converges to γ 2 -γ 1 < 0.
The proof uses the same arguments as in [START_REF] Guivarc | Simplicité de spectres de Lyapunov et propriété d'isolation spectrale pour une famille d'opérateurs de transfert sur l'espace projectif[END_REF]. The condition lim

n→∞ S * n • m = δ z * (ω) is satisfied in the examples of subsection 3.2 (see Theorem 3.2).

Lemma 3.12. Let m p be the natural SO(d )-invariant probability measure on the submanifold of p-decomposable unit multivectors x

= v 1 ∧ v 2 ∧ • • • ∧ v p .
Then there exists c > 0 such that for any u ∈ EndV :

0 < log | ∧ p u| -log |(∧ p u)(x)|d m p (x) ≤ c. Proof. We write u in polar form u = kak ′ with k, k ′ ∈ SO(d ), a = diag(a 1 , a 2 • • • , a d ) and a 1 ≥ a 2 ≥ • • • ≥ a d > 0. We write also x = k ′′ ε p with k ′′ ∈ SO(d ), ε p = e 1 ∧ e 2 ∧ • • • ∧ e p , hence, log | ∧ p ux|d m p (x) = log | ∧ p ak ′ k ′′ ε p |d m(k ′′ ) = log | ∧ p akε p |d m(k),
where m is the normalized Haar measure on SO(d ). Furthermore

| ∧ p akε p | ≥ 〈∧ p akε p , ε p 〉| = | ∧ p a||〈kε p , ε p 〉|, log| ∧ p ux|d m p (x) ≥ log| ∧ p u| + log |〈kε p , ε p 〉|d m(k) = log| ∧ p u| + log |〈x, ε p 〉|d m p (x).
Hence it suffices to show the finiteness of the integral in the right hand side. But the set of unit decomposable p-vectors is an algebraic submanifold of the unit sphere of ∧ p V and m p is its natural Riemannian measure. Since the map x → 〈x, ε p 〉 2 is polynomial, there exists q ∈ N, c > 0 such that :

ct q ≤ m p {x; 〈x, ε p 〉 2 ≤ t } ≤ 1.
The push forward σ of m p by this map is an absolutely continuous probability measure on [0,1] which satisfies σ(0, t ) ≥ ct q/2 . Then

log |〈x, ε p 〉|d m p (x) = 1 0 log t d σ(t ) > -∞,
since the integral 1 0 t q/2-1 d t is finite for q > 0.

We recall from [START_REF] Kesten | Sums of stationary sequences cannot grow slowler than linearly[END_REF] the following Lemma 3.13. Let (E , θ, ν) be a dynamical system where ν is a θ-invariant probability measure, f a ν-integrable function.

If n Σ 1 f •θ k converges ν-a.e to -∞, then one has ν( f ) < 0.
Lemma 3.14. Let E be a compact metric space, P a Markov kernel on E which preserves the space of continuous functions on E , C (P ) the compact convex set of P -stationary measures. Then, for every continuous function f on E , the sequence sup

x∈E 1 n n Σ 1 P k f (x) con- verges to sup{η( f ); η ∈ C (P )}. In particular, if for all η, η ′ ∈ C (P ) we have η( f ) = η ′ ( f ), then we have the uniform convergence, lim n→∞ 1 n n Σ 1 P k f (x) = η( f ).
Proof. Let J ⊂ R be the set of cluster values of the sequences 1

n n-1 Σ 0 (P k f )(x n ) with x n ∈ E .
We will show that the convex envelope of J is equal to {η( f ) ; η ∈ C (P )}. If the se-

quence 1 n k n k Σ 0 (P i f )(x n k ) converges to c ∈ R, we can consider the sequence of probability measures η k = 1 n k n k -1 Σ 0 P i δ x n k
and extract a convergent subsequence with limit η ∈ C (P ).

Then, since f is continuous we have

η( f ) = lim k→∞ 1 n k n k -1 Σ 0 (P i f )(x n k ) = c.
Conversely, if η ∈ C (P ) is ergodic, Birkhoff's theorem applied to the sequence 1

n n-1 Σ 0 (P i f )(x)
gives η-a.e,

lim n→∞ 1 n n-1 Σ 0 (P i f )(x) = η( f ), hence there exists x ∈ E such that η( f ) is the limit of 1 n n-1 Σ 0 (P i f )(x). If η is not ergodic, η
is a barycenter of ergodic measures, hence η( f ) belongs to the convex envelope of J . In view of the above, this shows the first claim. Since J is closed, the convex envelope of J

is a closed interval [a, b], hence b = lim n→∞ 1 n sup x∈E n-1 Σ 0 (P i f )(x) = sup η∈C (P) η( f ).
Lemma 3.15. We have the formulae

γ 1 = lim n→∞ 1 n sup x,v log |S n (ω)v |d Q x (ω) = sup η∈C 1 I 1 (η), γ 1 + γ 2 = lim n→∞ 1 n sup x,v,v ′ log |S n (ω)v ∧ S n (ω)v ′ |d Q x (ω) = sup η∈C 2 I 2 (η).
Proof. We consider the Markov chain on X × P d-1 with kernel Q 1 defined by

Q 1 ϕ(x, v ) = ϕ(g • x, g • v )q(x, g )d µ(g ),
and the function ψ(x, v ) = σ 1 (g , v )q(x, g )d µ(g ). We observe that

σ 1 (S n (ω), v )d Q x (ω) = n-1 Σ 0 Q k 1 ψ(x, v ),
and ψ is continuous since log γ(g )q(g )d µ(g ) < ∞. Also, since π is the unique Qstationary measure, any Q 1 -stationary measure has projection π on X . Then, using Lemma 3.14,we have sup

η∈C 1 I 1 (η) = lim n→∞ 1 n sup x,v σ 1 (S n (ω), v )d Q x (ω),
which gives the second part of the first formula. In order to show the first part we consider η ∈ C 1 which is Q 1 -ergodic, the extended shift θ on X × P d-1 × Ω and the function

f (x, v, ω) = σ 1 (g 1 (ω), v ). Then θ(x, v, ω) = (g 1 • x, g 1 • v, θω) and σ 1 (S n (ω), v ) = n-1 Σ 0 f • θ k (x, v, ω). Also, f (x, v, ω) is Q η -integrable where Q η = δ (x,v ) ⊗ Q x d η(x, v ).
Using the subadditive ergodic theorem, we have

I 1 (η) = 1 n σ 1 (S n (ω), v )d Q x (ω)d η(x, v ) ≤ lim n→∞ 1 n log |S n (ω)|d Q π (ω) = γ 1 .
We show now that for some η ∈ C 1 we have γ 1 = I 1 (η). Using Lemma 3.12, we know that

0 ≤ log |S n (ω)| -log |S n (ω)v |d m(v ) ≤ c, hence, integrating with respect to Q π , we have 0 ≤ d m(v ) (log |S n (ω)| -log |S n (ω)v |d Q π (ω) ≤ c.
Then the sequence of non negative functions h n (v ) on P d-1 given by

h n (v ) = 1 n (log |S n (ω)| -log |S n (ω)v |)d Q π (ω) satisfies 0 ≤ h n (v ) ≤ c
n with c given by Lemma 3.12, lim n→∞ h n (v )d m(v ) = 0. It follows that we can find a subsequence h n j such that h n j (v ) converges to zero m-a.e, hence

γ 1 = lim j →∞ 1 n j σ 1 (S n j (ω), v )d Q π (ω), m -a.e.,
in particular this convergence is valid for some v ∈ P d-1 . The sequence of probability measures 1

n j n j Σ 1 Q k 1 (π ⊗ δ v
) has a weakly convergent subsequence η j with a Q 1 -invariant limit η. Furthermore, the function ψ considered above is continuous, hence

η(ψ) = lim j →∞ 1 n j σ 1 (S n j (ω)v )d Q π (ω) = γ 1 , γ 1 = I 1 (η) = sup η 1 ∈C 1 I 1 (η 1 ).
The same argument is valid for log|S n (ω)(v ∧ v ′ )| with m replaced by m 2 , hence the second formula.

Lemma 3.16. For any η ∈ C 1 , we have γ 1 = I 1 (η) .

Proof. As in the proof of Lemma 3.15 we have

I 1 (η) = lim n→∞ 1 n σ 1 (S n (ω), v ), Q η -a.e., hence the existence of v ∈ P d-1 such that Q π -a.e., I 1 (η) = lim n→∞ 1 n log |S n (ω)v |.
Then, using Theorem 3.2 and Lemma 3.15 we get, Q π -a.e.

lim n→∞ 1 n log |S n (ω)v | |S n (ω)| = lim n→∞ 1 n log|〈z * (ω), v 〉| = 0, since 〈z * (ω), v 〉 = 0, Q π -a.e. Hence I 1 (η) = lim n→∞ 1 n log |S n (ω)|d Q π (ω) = γ 1 .
Proof of proposition 3.11 We have

γ 2 -γ 1 = (γ 1 +γ 2 )-2γ 1 , γ 1 = I 1 (η 1 ) for any η 1 ∈ C 1 and γ 1 + γ 2 = sup η 2 ∈C 2 I 2 (η 2 )
. Using the theorem of Markov-Kakutani for the inverse image of

η 2 ∈ C 2 in C 1,2 we know that any η 2 ∈ C 2 is the projection of some η 1,2 ∈ C 1,2 , hence γ 1 + γ 2 = sup η 1,2 ∈C 1,2 I 2 (η 1,2 ). If η ′ 1 is the projection of η 1,2 on P d-1 , we have I 1,2 (η 1,2 ) = I 2 (η 1,2 ) - 2I 1 (η ′ 1 )
and from Lemma 3.16,

γ 1 = I 1 (η ′ 1 ). It follows γ 2 -γ 1 = sup η 1,2 ∈C 1,2 I 1,2 (η 1,2 ).
Since I 1,2 (η 1,2 ) depends continuously of η 1,2 and C 1,2 is compact, in order to show that γ 2 -γ 1 is negative it suffices to prove I 1,2 (η) < 0, for any η ∈ C 1,2 . We consider the extended shift θ on

X × P d-1 1,2 × Ω defined by θ(x, ξ, ω) = (g 1 • x, g 1 .ξ, θω), the function f (ξ, ω) = σ(g 1 , ξ) and the θ-invariant measure Q η = δ (x,ξ) ⊗ Q x d η(x, ξ). Since S * n • m converges Q π -a.e. to δ z * (ω) , Lemma 3.3 implies lim n→∞ σ(S n (ω), ξ) = -∞, Q π -a.e.
if the origine v of ξ satisfies 〈v, z * (ω)〉 = 0. By hypothesis, this condition is satisfied for any ξ and Q π -a.e., hence we have lim

n→∞ n Σ 1 f • θ k = -∞ Q π -a.
e. for any ξ. It follows that this convergence is valid Q η -a.e., hence Lemma 3.13 gives η( f ) = I 1,2 (η) < 0. We consider

I n = 1 n E x log δ(S n (ω)•v,S n (ω)•v ′ ) δ(v,v ′ ) . With |v | = |v ′ | = 1, δ 1 (v, v ′ ) = |v ∧ v ′ | we have log δ 1 (S n (ω) • v, S n (ω) • v ′ ) δ 1 (v, v ′ ) = log |S n (ω)v ∧ S n (ω)v ′ | |v ∧ v ′ | -log |S n (ω)v | -log |S n (ω)v ′ |.
By Lemma 3.15, we have also

γ 1 + γ 2 = lim n→∞ 1 n sup x,v,v ′ E x log |(S n (ω)v ∧ S n (ω)v ′ | |v ∧ v ′ | .
Furthermore, by Lemmas 3.14 and 3.16, we have the convergence of sup

x,v

1 n E x (log |S n (ω)v |)
and inf

x,v

1 n E x (log |S n (ω)v |) to I 1 (η) = γ 1 . The uniform convergence of 1 n E x (log |S n (ω)v |) to γ 1 follows. Then the equivalence of δ 1 , δ implies that sup x,v,v ′ 1 n I n converges to γ 2 -γ 1 .
With the notations of paragraph 3 above we have the following corollaries, for products of random matrices.

lim n→∞ 1 n sup x,v,v ′ E s x log δ(S n (ω) • v, S n (ω) • v ′ ) δ(v, v ′ ) = L µ,2 (s) -L µ,1 (s) < 0,
where L µ,1 (s), L µ,2 (s) are the two highest Lyapunov exponents of S n (ω) with respect to Q s . In particular,

lim n→∞ 1 n sup v,v ′ E s log δ(S n (ω) • v, S n (ω) • v ′ ) δ(v, v ′ ) ≤ L µ,2 (s) -L µ,1 (s) < 0.
Proof. In view of Theorems 3.2, 2.6, the conditions of Proposition 3.11 are satisfied by X = P d-1 , q ⊗ µ = q s ⊗ µ, and π = π s . On the other hand we have Q s = Q s x d π(x), hence the second formula.

We will use Theorem 3.17 to establish certain functional inequalities for the operators Q s , Q s on P d-1 , S d-1 defined in section 2 and acting on H ε (P d-1 ) or H ε (S d-1 ). Using [START_REF] Kaneko | A generalisation of the Riesz-Schauder theory[END_REF], spectral gap properties will follow. We will say that Q satisfies a "Doeblin-Fortet inequality" on H ε (X ), where X is a compact metric space if we have for any

ϕ ∈ H ε (X ), [Q n 0 ϕ] ε ≤ ρ[ϕ] ε + D|ϕ| for some n 0 ∈ N where ρ < 1, D ≥ 0.

Corollary 3.18. For ε sufficiently small and s

∈ [0, s ∞ [, if |g | s γ τ (g )d µ(g ) < ∞ for some τ > 0, then lim n→∞ (sup x,y E s ( δ ε (S n • x, S n • y) δ ε (x, y) )) 1/n = ρ(ε) < 1. If k ′ (s) > 0, then lim n→∞ (sup x E s ( 1 |S n x| ε )) 1/n < 1.
Proof. The proof of the first formula is based on Theorem 3.17 and is given below. The proof of the second formula follows from a similar argument (see also [START_REF] Page | Théorème de renouvellement pour les produits de matrices aléatoires[END_REF], Theorem 1, for s = 0).

We denote α n (x, y, ω) = log

δ(S n (ω)•x,S n (ω)•y) δ(x,y)
and we observe that

e εα n ≤ 1 + εα n + ε 2 α 2 n e ε|α n | , |α n | ≤ 2 logγ(S n ).
Since t 2 e |t| ≤ e 3|t| , there exists ε 0 > 0 such that for ε ≤ ε 0

α 2 n e ε|α n | ≤ 1 ε 2 0 e 3ε 0 |α n | ≤ 1 ε 2 0 (γ(S n )) 6ε 0 .
We observe that

I n = 1 ε 2 0 E s (γ 6ε 0 (S n )
) is finite for s < s ∞ and ε 0 sufficiently small (see the proof of Corollary 3.20. below). It follows

E s (e εα n (x,y,ω) ) ≤ 1 + ε E s (α n (x, y, ω)) + ε 2 I n , sup x,y E s δ ε (S n • x, S n • y) δ ε (x, y) ≤ 1 + ε sup x,y E s log δ(S n • x, S n • y) δ ε (x, y) + ε 2 I n .
Also the quantity ρ n (ε) = sup

x,y

E s δ ε (S n • x, S n • y) δ ε (x, y) satisfies ρ m+n (ε) ≤ ρ m (ε)ρ n (ε), hence we have ρ(ε) = lim n→∞ ρ n (ε) 1/n = inf n∈N (ρ n (ε)) 1/n . It follows that, in order to show ρ(ε) < 1,
for ε small it suffices to show ρ n 0 (ε) < 1 for some n 0 . To do that, we choose n 0 such that sup

x,y

E s log δ(S n 0 • x, S n 0 • y)
δ(x, y) = c < 0 which is possible using Theorem 3.17, and we take ε sufficiently small so that ε 2 I n 0 + εc < 0. Then we get ρ n 0 (ε) ≤ 1 + ε 2 I n 0 + εc < 1.

Corollary 3.19. Let H ε (P d-1 ) be the Banach space of ε-Hölder functions on P d-1 with the norm ϕ → [ϕ] ε + |ϕ| and assume 0 ≤ s < s ∞ , |g | s γ τ (g )d µ(g ) for some τ > 0. Then for ε sufficiently small the operator Q s (defined in Theorem 2.6) on H ε (P d-1 ) satisfies the following Doeblin-Fortet inequality

[(Q s ) n 0 ϕ] ε ≤ ρ ′ (ε)[ϕ] ε + B|ϕ|, where B ≥ 0 n 0 ∈ N, ρ n 0 (ε) < ρ ′ (ε) < 1.
In particular the operator P s admits the following spectral decomposition in H ε (P d-1 )

P s = k(s)(ν s ⊗ e s +U s ),
where U s has spectral radius less than 1, and satisfies U s (ν s ⊗ e s ) = (ν s ⊗ e s )U s = 0.

Proof. From Lemma 3.5, we know that Q s x ≤ c(s)Q s , hence, using Corollary 3.18, for n ≥ n 0 sufficiently large and with ρ ′ (ε) ∈]ρ n 0 (ε), 1[, we have sup

x,y E s x δ ε (S n • x, S n • y) δ ε (x, y) ≤ ρ ′ (ε).
We can write

(Q s ) n ϕ(x) -(Q s ) n ϕ(y) = E s x (ϕ(S n • x)) -E s y (ϕ(S n • y)) = E s x (ϕ(S n • x)) -ϕ(S n • y)) + (E s x -E s y )(ϕ(S n • y)).
The first term in the right hand side is bounded by [ϕ] ε δ ε (x, y) sup

x,y

E s x δ ε (S n • x, S n • y) δ ε (x, y) i.e. by [ϕ] ε δ ε (x, y)ρ ′ (ε)
. Using Lemma 3.5, we know that the second term is bounded by B|ϕ|δ s (x, y). Hence with ε < s we get the required inequality.

From Theorem 2.6 we know that Q s has a unique stationary measure π s and 1 is the unique eigenvalue with modulus one. Then the above Doeblin-Fortet inequality implies (see [START_REF] Tulcea | Théorie ergodique pour des classes d'opérations non complètement continues[END_REF]) the relation Q s = π s ⊗ 1 + V s where V s commutes with the projection π s ⊗ 1 has spectral radius less then one and satisfies V s (π s ⊗ 1) = 0, hence the required formula for P s .

Corollary 3.20. With the notation and hypothesis of Corollary 3.19, the following Doeblin-Fortet inequality is valid, if

z = s + i t , 0 ≤ s < s ∞ : [(Q z ) n 0 ϕ] ε ≤ ρ ′ (ε)[ϕ] ε + (B + A n 0 (ε)|t | ε )|ϕ|,
where 0 ≤ A n 0 (ε) < ∞. For t = 0, the spectral radius of Q z is less than 1. Furthermore k(s) and the projection ν s ⊗ e s are analytic on ]0, s ∞ [, and 1 is a simple eigenvalue of Q s .

Proof. By definition of

Q z = Q s+i t we have (Q s+i t ) n ϕ(x) = E s x (|S n x| i t ϕ(S n • x)), hence |(Q z ) n ϕ(x) -(Q z ) n ϕ(y)| is bounded by the expression |(E s x -E s y )(|S n x| i t ϕ(S n • x)) + |E s y (|S n x| i t ϕ(S n • x) -|S n y| i t ϕ(S n • y))|.
Using Lemma 3.5 the first term is bounded by Bδ ε (x, y)E s (|ϕ|) i.e by B|ϕ|δ ε (x, y). The second term is dominated by

E s y (|S n x| i t -|S n y| i t |)|ϕ| + E s y (|ϕ(S n • x) -ϕ(S n • y)|).
As in the proof of Corollary 3.19, for n ≥ n 0 we write

E s y (|ϕ(S n • x) -ϕ(S n • y)|) ≤ [ϕ] ε E s y (δ ε (S n • x, S n • y)) ≤ [ϕ] ε δ ε (x, y) sup x,y E s y ( δ ε (S n • x, S n • y) δ ε (x, y) ) ≤ [ϕ] ε δ ε (x, y)ρ ′ (ε).
On the other hand, using the relation

||u| i t -|v | i t | ≤ 2|t | ε | log |u|-log |v || ε ≤ 2|t | ε sup( 1 |u| , 1 |v | ) ε ||u|- |v || ε , we get |(|S n x| i t -|S n y| i t || ≤ 2|t | ε sup |v |=1 1 |S n v | ε |S n (x -y)| ε ≤ 2|t | ε sup |v |=1 |S n | ε |S n v | ε δ ε (x, y). Since |S n v | ≥ |S -1 n | -1 we get E s y (|S n x| i t -|S n y| i t )|) ≤ 2c(s)|t | ε δ ε (x, y)E s (γ 2ε (S n )). Since γ(S m+n ) ≤ γ(S m )γ(S n • θ m ) and Q s is shift-invariant, E s (γ 2ε (S n )) ≤ (E s (γ 2ε (S 1 )) n < ∞.
Then for n fixed and ε sufficiently small the hypothesis implies that E s y (|S

n x| i t -|S n y| i t )) is bounded by A n (ε)|t | ε δ ε (x, y). Finally, for n = n 0 , [(Q z ) n 0 ϕ] ε ≤ ρ ′ (ε)|ϕ] ε + (B + A n 0 (ε)|t | ε )|ϕ|.
Then, using [START_REF] Tulcea | Théorie ergodique pour des classes d'opérations non complètement continues[END_REF], one gets that the possible unimodular spectral values of Q z are eigenvalues. Using Theorem 2.6, if t = 0, one get that no such eigenvalue exists, hence the spectral radius of Q z is less than 1.

In order to show the analyticity of k(s) and ν s ⊗ e s on ]0, s ∞ [, we consider the operator P z for z ∈ C close to s. We begin by showing the holomorphy of P z for Re z ∈]0, s ∞ [. Let γ be a loop contained in the strip Re z ∈

• I µ and ϕ ∈ H ε (P d-1 ). Then, since z → |g x| z is holomorphic γ P z ϕ(x)d z = G×γ ϕ(g • x)|g x| z d µ(g )d z = G ϕ(g • x)d µ(g ) γ |g x| z d z = 0,
On the other hand, the spectral gap property of the operator P s implies that k(s) is a simple pole of the function ζ → (ζI -P s ) -1 , hence by functional calculus if γ is a small circle of center k(s

) ∈ C k(s)ν s ⊗ e s = 1 2i π γ (ζI -P s ) -1 d ζ.
Since P z depends continuously of z, the function (ζI -P z ) -1 has a pole inside the small disk defined by γ, if z is close to s. Then by perturbation theory P z has an isolated spectral value k(z) close to k(s). The corresponding projection

ν z ⊗ e z satisfies k(z)ν z ⊗ e z = 1 2i π γ (ζI -P z ) -1 d ζ.
This formula and the holomorphy of P z shows that k(z) and ν z ⊗ e z are holomorphic in a neighbourhood of s. The analyticity of k(s) and ν s ⊗ e s follow. The fact that 1 is a simple eigenvalue of Q s follows from Theorem 2.6.

Corollary 3.21. Assume |g | s γ τ (g )d µ(g ) < ∞ for some τ > 0. Then given ε > 0 sufficiently small, for any ε 0 > 0 there exists δ 0 = δ 0 (ε 0 ),

n 0 = n 0 (ε 0 ) such that if x, y ∈ S d-1 satisfy δ(x, y) ≤ δ 0 , then E s ( δ ε (S n 0 • x, S n 0 • y)) ≤ ε 0 δ ε (x, y).
One has the following Doeblin-Fortet inequality on H ε (S d-1 ) with D ≥ 0 and

ρ 0 = ε 0 c(s) < 1, [( Q s ) n 0 ϕ] ε ≤ ρ 0 [ϕ] ε + D|ϕ|, where c(s) satisfies Q s x ≤ c(s)Q s .
In particular the spectral value 1 is isolated and the corresponding finite dimensional projector depends analytically on s ∈]0, s ∞ [.

In case I, the Q s -invariant functions are constant. In case II, the space of Q s -invariant functions is generated by p s + and p s -. If t = 0 the spectral radius of Q z is less than 1. Furthermore, 1 is the unique unimodular eigenvalue of Q s except in case I, where -1 is the unique non trivial possibility.

Proof. Assume ε is as in Corollary 3.18. We will use for any n ∈ N and t > 0, the relation

E s ( δ ε (S n • x, S n • y)) = E s ( δ ε (S n • x, S n • y)1 {γ(S n )>t} ) + E s ( δ ε (S n • x, S n • y)1 {γ(S n )≤t} ).
In view of Corollary 3.18 we have for some n 0 , any x, ȳ ∈ P d-1 and given ε 0 > 0, we have

E s (δ ε (S n 0 • x, S n 0 • ȳ)) ≤ ε 0 2 δ ε ( x, ȳ). Using Lemma 2.11 we have, for x, y ∈ S d-1 , δ(S n 0 • x, S n 0 • y) ≤ 2γ 2 (S n 0 ) δ(x, y), hence E s ( δ ε (S n 0 • x, S n 0 • y)1 {γ(S n 0 )>t} ) ≤ 2E s (γ 2ε (S n 0 )1 {γ(S n 0 )>t} ).
Since, as in the proof of Corollary 3.20, we have if ε is sufficiently small, E s (γ 2ε (S n 0 )) < ∞, we can choose t 0 > 0 so that E s ( δ ε (S n 0 •x, S n 0 • y)1 {γ(S n 0 )>t 0 } ) ≤ ε 0 2 δ ε (x, y). Then, on the set {γ(S n 0 ) ≤ t 0 } we have δ(S

n 0 • x, S n 0 • y) ≤ 2γ 2 (S n 0 ) δ(x, y) ≤ 2 t 2 0 δ(x, y). We observe that, if δ(u, v ) ≤ 2 with u, v ∈ S d-1 , then δ( ū, v ) = δ(u, v ). Hence, if 2 t 2 0 δ(x, y) ≤ 2, we get δ(S n 0 • x, S n 0 • y) = δ(S n 0 • x, S n 0 • ȳ) on the set {γ(S n 0 ) ≤ t 0 }. It follows, if δ(x, y) ≤ 2 2t 2 0 = δ 0 , E s ( δ ε (S n 0 • x, S n 0 • y)1 {γ(S n 0 )≤t 0 } ) ≤ ε 0 2 δ ε (x, y). Hence we get, if δ(x, y) ≤ δ 0 , E s ( δ ε (S n 0 • x, S n 0 • y)) ≤ ε 0 δ ε (x, y). Using Q s x < c(s)Q s we obtain sup δ(x,y)≤δ 0 E s x ( δ ε (S n 0 • x, S n 0 • y) δ ε (x, y) ) ≤ c(s)ε 0 .
On the other hand, for ϕ ∈ H ε (S d-1 ):

( Q s ) n ϕ(x) -( Q s ) n ϕ(y) = E s x (ϕ(S n • x) -ϕ(S n • y)) + (E s x -E s y )(ϕ(S n • y)).
In view of Lemma 3.5, the second term is bounded by B δ s (x, y)|ϕ|. Then, for δ(x, y| ≤ δ 0 we obtain, since ε ≤ s,

|( Q s ) n 0 ϕ(x) -( Q s ) n 0 ϕ(y)| ≤ c(s)ε 0 [ϕ] ε δ ε (x, y) + B|ϕ| δ ε (x, y). If δ(x, y) ≥ δ 0 we have trivially E x (|ϕ(S n 0 • x) -ϕ(S n 0 • y)|) ≤ 2 c(s) δ ε (x,y) δ ε 0 |ϕ| . Finally, on S d-1 , [( Q s ) n 0 ϕ] ε ≤ c(s)ε 0 [ϕ] ε + (B + 2 c(s) δ ε 0 )|ϕ|,

Renewal theorems for products of random matrices

It is well known that the potential theory for a random walk on R with positive drift is closely related to renewal theory (see [START_REF] Feller | An Introduction to Probability Theory and its Applications[END_REF]). In this context, one basic result gives the homogeneous behaviour at infinity of the potential measure; another basic result gives the convergence (t → ∞) of the entrance measure of the random walk into ]t , ∞[ towards a certain probability which has a density with respect to Lebesgue measure, with a simple expression in terms of the associated ladder random walk. In this section we extend these two results to linear random walks on the

G-spaces V \ {0} = S d-1 × R * + and V = P d-1 × R * + .
We denote by µ ∈ M 1 (GL(V )) the law of our random walk and we identify R to R * + via the exponential map. For the proofs we use the results of [START_REF] Kesten | Renewal theory for functionals of a Markov chain with general state space[END_REF] which give renewal theorems for a class of Markov walks on R, which satisfy the tameness conditions explained below. An important observation of [START_REF] Kesten | Random difference equations and renewal theory for products of random matrices[END_REF] is that, if µ is supported on the positive matrices, these tameness conditions are satisfied. Here we assume that the semigroup [suppµ] satisfies condition i-p and we use the results of sections 2 and 3 to show that the tameness conditions of [START_REF] Kesten | Renewal theory for functionals of a Markov chain with general state space[END_REF] are still valid in our generic situation. Hence we extend the results of [START_REF] Kesten | Random difference equations and renewal theory for products of random matrices[END_REF], [START_REF] Page | Théorème de renouvellement pour les produits de matrices aléatoires[END_REF] to the general case. This extension will play an essential role in section 5.

The renewal theorem for a class of fibered Markov chains

We begin by summarizing, with a few changes and comments, the basic notations of [START_REF] Kesten | Renewal theory for functionals of a Markov chain with general state space[END_REF]. Let (S, δ) be a complete separable metric space, P (resp P ) a Markov kernel on S × R (resp S) which preserves C b (S × R) (resp C b (S)). We assume that P commutes with the translations (x, t ) → (x, t + a) on S × R, and P is the factor kernel of P on S.If π is a P -invariant probability measure, and ℓ is Lebesgue measure on R, we note that the measure π ⊗ℓ is P -invariant. We will say that P is a fibered Markov kernel and defines a "fibered Markov chain over S". More generally,if P is a measurable kernel on S×R which satisfies the above commutation we will say that P is a measurable fibered kernel over S.In this situation,if π ⊗ ℓ is a P -invariant measure, we will say that P is a measurable fibered Markov kernel if P is positive and P 1 = 1, π ⊗ ℓ-a.e. These measurable fibered kernels will play an important role in subsection 4.3 below .In this section, from now on we denote by P a Markov fibered kernel.In our applications,for Markov fibered kernels, we will have S compact and S = P d-1 or S ⊂ S d-1 , hence S × R will be identified with a cone in V or V \ {0}. Here we consider paths on S × R starting from (x, 0) ∈ S × {0}. Such a path can be written as (x n ,V n ) n∈N with V 0 = 0,

x 0 = x, V n -V n-1 = U n (n ≥ 1).
The corresponding space of paths for a fibered Markov kernel P will be denoted a Ω = S × ∞ Π 1 (S ×R), the Markov measure on a Ω associated with P and starting from x ∈ S will be denoted by a P x and the expectation symbol will be written a E x . The space of bounded measurable functions on a measurable space Y will be denoted B(Y ). We observe that the Markov kernel P on S ×R is completely defined by the family of measures F (d u|x, y) (x, y ∈ S) where F (d u|x, y) is the conditional law of V 1 given x 0 = x, x 1 = y. Given a fixed P -stationary probability π on S, the number uF (d u|x, y)P (x, d y)d π(x) with be called the mean of P , if the corresponding integral |u|F (d u|x, y)P (x, d y)d π(x) is finite. In that case, we say that P has a 1-moment.

If t ∈ R + we define the hitting time N (t ) of the interval ]t , ∞[ by :

N (t ) = inf{n ≥ 1 ; V n > t }; (= +∞ if no such n exists).
On the event N (t ) < +∞ we take W (t ) = V N(t) -t , Z (t ) = x N(t) . If V 1 has a lifetime interpretation then W (t ) is the residual waiting time of the interval ]t , ∞[ (see [START_REF] Feller | An Introduction to Probability Theory and its Applications[END_REF]). In general the law of (Z (t ),W (t )) under P x is the hitting measure of S×]t , ∞[ starting from (x, 0). In particular, in analogy with [START_REF] Feller | An Introduction to Probability Theory and its Applications[END_REF], we define the "ladder kernel" H of P , starting from (x, t ) ∈ S × R, to be the law (possibly defective) of (x N(0) , t + V N(0) ) under a P x . In analogy with [START_REF] Feller | An Introduction to Probability Theory and its Applications[END_REF], N (0) (resp V N(0) ) is the first "ladder index" (resp "ladder height") of P . By definition H is a measurable fibered kernel over S; it plays an important role in the expression of various asymptotic quantities for P .

One needs some technical definitions concerning direct Riemann integrability, aperiodicity of the Markov chain defined by P on S × R, and the possibility of comparing a P x , a P y in a weak sense for different points x, y in S. We add some comments as follows.

Given a fibered Markov chain on S × R, we denote

C 0 = φ,C k = {x ∈ S ; a P x { V m m > 1 k ∀m ≥ k} ≥ 1 2 } for k ≥ 1. Definition 4.1. A Borel function ϕ ∈ B(S × R) is said to be d .R.i (for directly Riemann integrable) if ∞ 0 +∞ ℓ=-∞ (k + 1) sup{|ϕ(x, t )| ; x ∈ C k+1 \C k , ℓ ≤ t ≤ ℓ + 1} < +∞ ,
and for every fixed x ∈ S and any β > 0, the function t → ϕ(x, t ) is Riemann integrable on [-β, β].

In our setting below we will have C k = S for some k > 0 and for some ε > 0, any x ∈ S and m sufficiently large a P x (

V m m ≥ ε) ≥ 1 2 .
Then the following stronger form of the above definition will be used. Remark. Definition 4.2 corresponds to sup{|ϕ(x, t )|; x ∈ S} directly Riemann integrable in the sense of [START_REF] Feller | An Introduction to Probability Theory and its Applications[END_REF].

If C k = S for some k ∈ N, then clearly condition b.R.i implies con- dition d .R.i
The following definitions will help us to express the appropriate tameness conditions for (P, π), where π is a P -stationary probability.

Definition 4.3. The kernel P , the space (S, δ) and the measure π ∈ M 1 (S) being as above we consider a point (ζ, λ, y) ∈ R × [0, 1] × S and we say that (P, π) has distortion (ζ, λ) at y if for any ε > 0, there exists A ∈ B(S) with π(A) > 0 and m 1 , m 2 ∈ N, such that for any

x ∈ A, a P x {δ(x m 1 , y) + δ(x m 2 , y) < ε, |V m 1 -V m 2 -ζ| < λ} > 0.
For any f ∈ B( a Ω), ε > 0 we write

f ε (x 0 , x 1 , • • • , v 1 • • • ) = lim sup n→∞ { f (y 0 , y 1 , • • • , w 1 , • • • ) ; δ(x i , y i ) + |v i -w i | < ε if i ≤ n}.
Definition 4.4. We will say that the kernel P on S × R is non-expanding if for each fixed x ∈ S, ε > 0, there exists r 0 = r 0 (x, ε) such that for all real valued f ∈ B( a Ω) and for all y with δ(x, y) < r 0 , one has

a P y ( f ) ≤ a P x ( f ε ) + ε| f |, a P x ( f ) ≤ a P y ( f ε ) + ε| f |.
This condition of non expansion says that, in probability, if x, y ∈ S are close then the paths along the fibered Markov chain starting from x,y and defined by P remain close.

One can see that, if ϕ ∈ C b (S × R) is uniformly continuous, then the condition of non expansion for P implies that the set of functions {P n ϕ ; n ∈ N} is uniformly equicontinuous.

We denote by I.1-I.4 the following conditions, where π denotes a given P -stationary probability on S, One can see that conditions I.1 and I.4 imply that P admits the unique invariant probability π and that for ϕ ∈ C b (S), the set {P n ϕ ; n ∈ N} is equicontinuous. We will denote

ℓ + = 1 [0,∞[ ℓ
where ℓ is Lebesgue measure on R. For a bounded measure θ on S × R, we will write θ its projection on S. We recall that the Radon transform of a bounded measure on R is defined by θ(t

) = θ(]t , ∞[) if t ∈ R.
The ladder kernel H can be written as

H((y, t ), •) = (δ z ⊗ H z y * δ t )H(y, d z),
where H z y is the conditional law of V N(0) given x N(0) = z, starting from y ∈ S. Condition I.2 implies that for any y ∈ S, H(((y, t ), •) is a probability.

Then, the following extension of the classical renewal theorem is proved in [START_REF] Kesten | Renewal theory for functionals of a Markov chain with general state space[END_REF].

lim t→∞ a E x ϕ(Z (t ),W (t )) = 1 L ϕ(z, s)1 [s,∞[ (t ) a P y {x N(0) ∈ d z,V N(0) ∈ d t }d χ(y)d s = χ(ϕ), i.e. χ = 1 L (δ z ⊗ H z y ℓ + )H(y, d z)d χ(y). Moreover, if ϕ ∈ C b (S × R) is d .R.i , then, for any x ∈ S, lim t→∞ a E x ( ∞ 0 ϕ(x n ,V n -t )) = 1 L ϕ(y, s)d π(y)d s, lim t→-∞ ∞ 0 P k ϕ(x, t ) = 1 L (π ⊗ ℓ)(ϕ).

Furthermore χ is an invariant measure for the measurable Markov chain on S with kernel H(y, d z)

= a P y (x N(0) ∈ d z) and E y (N (0))d χ(y) < ∞, E y (V N(0) )d χ(y) < ∞,
where χ is absolutely continuous with respect to π.

Remark.

1. If S is compact, condition I.1 is a consequence of uniqueness of the Pstationary measure π. This follows from the law of large numbers for Markov chains with a unique stationary measure [START_REF] Breiman | The strong law of large numbers for a class of Markov chains[END_REF]: for any continuous function with 0 ≤ f ≤ 1, π( f ) > 0 we have a P x -a.e. for all x ∈ S, lim

n→∞ 1 n n-1 0 f (X k ) = π( f ) > 0. This implies condition I.1.
2. The construction of χ in [START_REF] Kesten | Renewal theory for functionals of a Markov chain with general state space[END_REF] is based on Kac's recurrence theorem and implies the absolute continuity of χ with respect to π, hence the measure χ is independent on x and absolutely continuous with respect to π ⊗ ℓ.

Tameness conditions I.1-I.4 are valid for linear random walks

We verify conditions I.1-I.4 in four related situations. As in ([35], Proposition 1) the basic property used for the validity of I.1, I.2, I.4 is the fact that for a product S n of random matrices, under condition i-p the lengths of colum vectors of S n are comparable to the norm of S n (see Theorem 3.2). We observe that this property plays also an important role in the general context of [START_REF] Benoist | Random walks on reductive groups[END_REF] (see Theorem 6.9). The aperiodicity property in condition I.3 is verified below using the properties of dominant eigenvalues of elements of T (see Proposition 2.5). Here R is identified with R * + via the map t → e t . If d > 1 we use condition i-p. If d = 1 we use non arithmeticity of µ.

The first (and simpler) situation corresponds to

S = P d-1 , S × R * + = V , P (v, •) = µ * δ v
where P is the operator on V denoted P in section 2. Also we write on P d-1 , P (x, •) = µ * δ x if x ∈ P d-1 . We will begin the verifications by this case and show how to modify the arguments in the other cases corresponding to s = α or S ⊂ S d-1 ⊂ V \ {0}. In the case, of V \ {0}, S is a compact subset of S d-1 and P (resp P ) will be the restriction to S ×R * + (resp S) of the kernel already denoted P (resp P ) in section 2. Since, for any t ∈ R * + and g ∈ G, we have g (t v ) = t g (v ), the kernels P and P define fibered Markov chains on S × R * + . As shown at the end of section 2, (Theorem 2.16) we need to consider two cases for P , depending of the fact that P preserves a proper convex cone (case II) or not (case I). In case I (resp II) we will have S = S d-1 (resp S = Co(Λ + ([suppµ]). With these choices, there exists a unique P -stationary measure on S, as follows from Theorem 2.16. In paragraph 5 below we state the detailed results for V \ {0}. We denote by α ∈ I µ the positive number (if it exists) such that k

(α) = 1, where k(s) = lim n→∞ ( |g | s d µ n (g )) 1/n .
We know from section 2, that for any s ∈ I µ , there are two Markov kernels Q s on P d-1 and Qs on S d-1 , naturally associated with the operator P s considered in section 2. We are here mainly interested in the cases s = α and s = 0, with Q 0 = P and Q0 = P , but we observe that our considerations are valid for any s ∈ I µ .

We denote by a Q s

x the natural Markov measure on the path space a Ω associated with µ and s ≥ 0. If s = 0 we use the notation a P x . In our linear situations we have a P

x = δ x ⊗P, a Q s x = δ x ⊗ Q s x where Q s x is defined in section 3. We write V k = log |S k x|, x k = S k • x and we denote by ∆ x the map from Ω = G N to a Ω given by (g 1 , g 2 , • • • ) → (x, x 1 ,V 1 , x 2 ,V 2 , • • • ). Clearly a Q α
x (resp a P x ) is the push-forward of Q α x (resp P) by ∆ x , hence we can translate the results of section 3 in the new setting.

The validity of condition I.1, in all cases, is a direct consequence of the remark following Theorem 4.5, since by Theorem 2.6 and Theorem 2.16 the kernels P , Q α , P , Qα have unique stationary measures on S.

In order to verify I.2 we begin by

S = P d-1 , S × R * + = V , P (x, .) = µ * δ x , P (v, .) = µ * δ v . Then F (A|x, y) = µ{g ∈ G, log |g x| ∈ A, g • x = y} for any Borel set A ⊂ R, and π = ν with µ * ν = ν. We observe that | log |g x|| ≤ logγ(g ) if |x| = 1 and γ(g ) = sup(|g |, |g -1 |). The finiteness of |u|F (d u|x, y) P (x, d y)d π(x) follows since log γ(g ) is µ-integrable. Also uF (d u|x, y) P (x, d y)d π(x) = log |g x|d µ(g )d ν(x) = L µ .
Then the relation a P x = ∆ x (P) and Theorem 3.10 imply, for every x ∈ P d-1 (case s = 0), and P-a.e.:

L µ = lim n→∞ 1 n log |S n x| = log|g x|d µ(g )d ν(x).
Except for L µ > 0, this is condition I.2 in the first case. If S ⊂ S d-1 is as above, the result is the same, since the involved quantities depend only on |g x| with x ∈ P d-1 , and P has a unique stationary measure on S.

In the cases of Q α and Q α it suffices also to consider the case S = P d-1 . The 1-moment condition in I.2 follows from |g | α | log γ(g )|d µ(g ) < +∞. The convergence part follows from Theorem 3.10 with

L µ (α) = q α (x, g ) log |g x|d π α (x)d µ(g ) = k ′ (α) k(α) > 0.
We show I.3 as follows. If d > 1, since the semigroup T = ∪ n≥0 (suppµ) n satisfies (i-p), we know using Proposition 2.5 that the set

∆ = log |λ h | ; h ∈ T prox is dense in R. The same is true of 2∆ = {log λ h 2 ; h ∈ T prox }.
If d = 1, the same properties follow from the non arithmeticity of µ.

We take for ζ i (i ∈ N) a dense countable subset of 2∆. Let

ζ i = logλ g ∈ 2∆, with λ g > 0, g = h 2 , h = u 1 • • • u n , u i ∈ suppµ (1 ≤ i ≤ n) and y = y(ζ i , λ) = vg ∈ P d-1 = S. We observe that, if ε is sufficiently small and B ε = {x ∈ P d-1 ; δ(x, vg ) ≤ ε}, then g • B ε ⊂ B ε ′ ,
with ε ′ < ε and g as above, Also, λ > 0 being fixed, and ε sufficiently small, we have

| log λ g -log|g x|| < λ if x ∈ B ε .
These relations remain valid for g ′ instead of g if g ′ is sufficiently close to g . Then we have for x ∈ B ε , and S 2n = g ∈ (suppµ) 2n as above,

a P x {δ(S 2n .x, vg ) < ε, | log |S 2n x| -log λ g | < λ} > 0.
With

ζ i = log λ g , y = vg , A = B ε , τ = 0, m 1 = 0, m 2 = 2n
, this implies condition I.3 for the probability a P x = ∆ x (P).

The definition of Q α x shows its equivalence to P on the σ-algebra of the sets depending of the first n coordinates. Then the relation

a Q α x = ∆ x (Q α x ) implies with g as above a Q α x {δ(S 2n .x, vg ) < ε, | log |S 2n x| -log λ g | < λ} > 0.
Hence condition I.3 is valid for a Q α x also. If we consider S d-1 instead of P d-1 , i.e.

S = S d-1 or S = Co(Λ + ([suppµ]
)), and the metric δ on S, the above geometrical argument remains valid with g = h 2 , y = ṽg ∈ Λ + ([suppµ]) in the second case, λ g > 0 and ε sufficient small. This shows I.3 in this setting.

Condition I.4 follows from the proof of Proposition 1 of [START_REF] Kesten | Random difference equations and renewal theory for products of random matrices[END_REF]. The proof of the corresponding part of this proposition is a consequence of the condition,

a P x {∃C > 0 with |S n x| ≥ C |S n | for all n} = 1,
for all x ∈ S, which implies that |S n x| and |S n y| are comparable if x and y are close.

For the proof of the above condition, we observe that if x ∈ S and s ∈ I µ , in particular if s = 0 or α, this condition has been proved in the stronger form

lim n→∞ |S n x| |S n | = |〈z * (ω), x〉| > 0 Q s x -a.e.,
in Theorem 3.2, hence condition I.4 is valid in all the cases under consideration.

Direct Riemann integrability

In case of the spaces S = P d-1 or S ⊂ S d-1 considered above, under condition i-p for [suppµ], the d .R.i condition takes the simple form given by Lemma ?? below, in multiplicative notation.

We assume now that the hypothesis of Theorem Hence there exists m 0 > 0 such that

P{ 1 n log|S n x| -2 n |S n | |S n x| > 1 2 L µ for all n ≥ m 0 } ≥ 1/2. In view of the inequality 1 n log |S n y| ≥ 1 n log |S n x| -2 n |S n | |S n x| , we have for any y ∈ S d-1 , P{ 1 n log |S n y| > L µ /2 for all n ≥ m 0 } > 1 2 . This implies C k = P d-1 ,C k+1 \ C k = φ if 1 k ≤ inf 1 m 0 , L µ 2 .
If s = α, the argument is the same with P replaced by Q α and the relation

Q α x ≤ c(α)Q α is used as follows. Q α { 1 n log |S n y| > L µ (α)/2 for all n ≥ m 0 } > 1 - 1 2c(α)
.

Since Q α y ≤ c(α)Q α , this gives for any y ∈ P d-1 ,

Q α y { 1 n log |S n y| > L µ (α)/2 for all n ≥ m 0 } > 1 2 .
Then also

C k = P d-1 for 1 k ≤ inf 1 m 0 , L µ (α) 2
. Hence we conclude as above.

The renewal theorems for linear random walks

We consider

V = P d-1 × R * + V \ {0} = S d-1 × R * +
, and we study the asymptotics of the potential kernels of the corresponding random walks defined by µ. We denote

V1 = {v ∈ V ; |v | > 1}, V 1 = {v ∈ V ; |v | > 1},
and we consider also the entrance measures H (v, .) or H(v, .) of S n v in V1 or V 1 , starting from v . Since conditions I are valid, their behaviour for v small are given by Theorem 4.5, and we will state them below. We denote by Λ([suppµ]) the inverse image of

Theorem 4.8. Assume that µ ∈ M 1 (G) is such that [suppµ] satisfies i-p, if d > 1 or µ is non arithmetic if d = 1. Assume L µ < 0, α > 0 exists with k(α) = 1, |g | α log γ(g )d µ(g ) < ∞ and write L µ (α) = lim n→∞ 1 n |g | α log |g |d µ n (g ) = k ′ (α) k(α) .
Then L µ (α) > 0 and for any u ∈ P d-1 , we have the vague convergence in V ,

lim t→0 t -α ∞ Σ 0 µ k * δ tu = e α (u) L µ (α) ν α ⊗ ℓ α ,
where

ν α ∈ M 1 (P d-1 ) (resp e α ∈ C (P d-1 ), ν α (e α ) = 1)
is the unique solution of the equation P α ν α = ν α (resp P α e α = e α ) and ν α has support Λ([suppµ]).

Furthermore, on C b ( V1 ) and for any u ∈ P d-1 ⊂ V , the ladder kernel H(t u, •) satisfies the vague convergence,

lim t→0 t -α H(t u, •) = e α (u) χα ,
where χα , defined by this convergence, is a positive measure supported on Λ1 ([suppµ]) and is absolutely continuous with respect to ν α ⊗ ℓ α .

In particular, for A = χα ( V1 ) > 0 and any u

∈ P d-1 , lim t→∞ t α P{M(u) > t } = Ae α (u).

The first convergence is valid on any continuous function f which satisfies

+∞ -∞ 2 -ℓα sup{| f (v )| ; 2 ℓ ≤ |v | ≤ 2 ℓ+1 } < ∞.
Proof. We observe that the function e α ⊗ h α on V satisfies P (e α ⊗ h α ) = e α ⊗ h α , hence we can consider the associated Markov operator Qα on V defined by

Qα ( f ) = 1 e α ⊗ h α P ( f e α ⊗ h α ). Then the potential kernel of Qα is given by ∞ 0 ( Qα ) k ( f ) = 1 e α ⊗ h α ∞ 0 P k ( f e α ⊗ h α ).
Clearly Qα commutes with dilations, hence defines a fibered Markov kernel on V .

Also the mean of Qα is L µ (α) > 0. Then, taking f = ϕ e α ⊗h α , since conditions I are valid, the result follows from Theorem 4.5. Cramér's estimation for P{sup |S n u| > t ; n ∈ N} follows with A = χα ( V1 ) > 0.

Proofs of Theorems B, B α and Corollary. These results are simple consequences of Theorems 4.7, 4.8.

Renewal theorems on V \ {0}

Here we extend Theorems 4.7 et 4.8 to the natural setting of V \ {0}. We note that for d = 1, we have V \ {0} = R * , then the results reduce to the classical renewal theorems ( [START_REF] Feller | An Introduction to Probability Theory and its Applications[END_REF]). Theorem 4.9. Assume µ is as in Theorem 4.7. Then there are 2 cases as in Theorem 2.16,

• Case I: No proper convex cone in V is suppµ-invariant. Then, in vague topology, lim v →0 ∞ 0 µ k * δ v = 1 L µ ν ⊗ ℓ,
where ν is the unique µ-stationary measure on S d-1 .

• Case II: Some proper convex cone in V is suppµ-invariant. Then, for any u ∈ S d-1 , in vague topology,

lim t→0 + ∞ 0 µ k * δ tu = 1 L µ (p + (u)ν + ⊗ ℓ + p -(u)ν -⊗ ℓ),
where ν + is the unique µ-stationary measure on

Λ + ([suppµ]), ν -is symmetric of ν + , p + (u) is the entrance probability of S n • u in Co(Λ + ([suppµ])), p -(u) = 1 - p + (u).
In the two cases these convergences are also valid on any bounded continuous functions

f on V \ {0} such that +∞ -∞ sup{| f (v )| ; 2 ℓ ≤ |v | < 2 ℓ+1 } < ∞.
In addition, for any u ∈ S d-1 , in weak topology,

lim t→0 + H(t u, •) = χ u ∈ M 1 (Λ 1 ([suppµ]),
where χ u is defined by this convergence and is absolutely continuous with respect to ν⊗ℓ.

• In case I, χ u = χ is independent on u. • In case II, with Λ 1,+ ([suppµ]) = {v ∈ V ; ṽ ∈ Λ + ([suppµ]), |v | ≥ 1} we have χ u = p + (u)χ + + p -(u)χ -,
where χ 1,+ ∈ M 1 (Λ + (suppµ)), and χ -is symmetric of χ + .

• Case I: ν α u = ν α has support Λ(T ),

• Case II:

ν α u = p α + (u)ν α + + p α -(u)ν α -, where p α + (u) (resp p α -(u)) denotes the entrance probability under Q α u of S n • u in the convex envelope of Λ + (T ) (resp Λ -(T )).
The above convergences are valid on any continuous function f which satisfies

ℓ=+∞ ℓ=-∞ 2 -ℓα sup{| f (v )| ; 2 ℓ ≤ |v | ≤ 2 ℓ+1 } < ∞. Furthermore, on C b (V 1 )
for any u ∈ S d-1 , we have the vague convergence,

lim t→0 + t -α H(t u, .) = e α (u) (p α + (u)χ α -+ p α -(u)χ α + ),
where χ α + , χ α -are defined by this convergence and are absolutely continuous with respect to ν α ⊗ ℓ.

In case I, χ α + = χ α -= χ α is a positive measure supported on Λ 1 ([suppµ]). In case II, χ α + is a positive measure supported on Λ + ([suppµ]) and χ α -is symmetric of χ α + .

On the asymptotics of k(s) (s → ∞)

For the existence of α > 0 such that k(α) = 1, we have the following sufficient condition where we denote by r (g ) the spectral radius of g ∈ G. In particular if some g ∈ [suppµ] satisfies r (g ) > 1, then k(s) > 1 for s sufficiently large.

The proof is based on the following elementary lemma which we state without proof.

Lemma 4.12. Let g ∈ G. Then for any ε > 0 there exists c(ε) > 0 and a neighbourhood V (ε) of g such for any sequence

g k ∈ V (ε) one has |g n • • • g 1 | ≥ c(ε) r n (g )(1 -ε) n .
Proof of Proposition 4.11. The convexity of log k(s) implies that lim s→∞ log k(s) s exists. Let g ∈ suppµ, hence given ε > 0 these exists a neighbourhood V (ε) of g as in Lemma 4.12 such that µ(V (ε)) = C (ε) > 0. From the lemma we have,

k(s) = lim n→∞ ( |g n • • • g 1 | s d P(ω)) 1/n ≥ lim n→∞ (c s (ε)r ns (g )(1 -ε) ns C n (ε)) 1/n = r s (g )(1 -ε) s C (ε). Hence log k(s) s ≥ log(1 -ε) + log r (g ) + log C (ε) s , therefore lim s→∞ log k(s) s ≥ log r (g ).
We observe that if µ is replaced by µ p , then k(s) is replaced by k p (s). Hence for g ∈ (suppµ) p we have from above the required inequality. If g ∈ [suppµ] then we can assume g ∈ (suppµ) p for some p ∈ N; since r (g ) > 1, we have log r (g ) > 0, hence lim

s→∞ log k(s) s > 0.
5 The tails of an affine stochastic recursion

Notation and main result

Let H be the affine group of the d -dimensional Euclidean space, i.e. the set of maps f of V into itself of the form f (x) = g x +b where g ∈ GL(V ) = G, b ∈ V . Let λ be a probability measure on H, µ its projection on G. We denote by Σ (resp T ) the closed subsemigroup of H (resp G) generated by suppλ (resp suppµ). We consider the affine random walk on V = R d defined by λ, i.e. the Markov chain on V described by the stochastic recursion,

X x n+1 = A n+1 X x n + B n+1 , X x 0 = x ∈ V,
where (A n , B n ) are H-valued i.i.d random variables with law λ. We denote Ω = H N and we endow Ω with the shift θ and the product measure P = λ ⊗N ; by abuse of notation the expectation symbol with respect to P will be denoted by E. We have

X x n = A n • • • A 1 x + n 1 A n • • • A k+1 B k .
We are interested in the case where

R n = n 1 A 1 • • • A k-1 B k converges P-a.e. to a random variable R and X x n converges in law to R. We observe that X x n -A n • • • A 1
x and R n have the same law. In that case we have

R = ∞ 0 A 1 • • • A k B k+1 , hence the random variable R satisfies the equation R = AR • θ + B, (S)
and the law ρ of R satisfies the convolution equation ρ = λ * ρ = hρd λ(h). Also, if R is unbounded, we will be interested in the tail of R in direction u, i.e. the asymptotics t → ∞ of P{〈R, u〉 > t } (resp P{|〈R, u〉| > t }), where u ∈ S d-1 (resp u ∈ P d-1 ). We are mainly interested in the "shape at infinity" of ρ i.e. the asymptotics (t → 0 + ) of the measure t • ρ where t • ρ is the push-forward of ρ by the dilation v → t v in V (t > 0). It turns out that this "shape at infinity" depends essentially of the semigroups T and Σ defined above. A basic role will be played by the top Lyapunov exponent L µ of the product of random matrices S n = A n • • • A 1 , and µ will be assumed to satisfy

log γ(g )d µ(g ) < ∞ where γ(g ) = sup(|g |, |g -1 |).
The main hypothesis will be on µ, which is always assumed to satisfy L µ < 0 and condition i-p of section 2 if d > 1, or µ non arithmetic if d = 1. We recall that the function 1/n and logk(s) is strictly convex (see Theorem 2.6). It is natural to assume that suppλ has no fixed point in V since otherwise the affine recursion reduces to a linear recursion. We denote by ∆ a (Σ) the set of fixed attractive points of the elements of Σ, i.e. fixed points

k(s) is defined on the interval I µ ⊂ [0, ∞[ by k(s) = lim n→∞ ( |g | s d µ n (g ))
h + ∈ V of elements h = (g , b) ∈ Σ such that lim n→∞ |g n | 1/n < 1. For v ∈ V \ {0} we denote H + v = {x ∈ V ; 〈v, x〉 > 1}
, and for a bounded measure ξ on V we consider its Radon transform ξ, i.e. the function on V \ {0} defined by

ξ(v ) = ξ(H + v ) with H + v = {x ∈ V ; 〈x, v 〉 > 1}.
We also write u = t v with u ∈ S d-1 , t > 0 and ξ(u, t ) = ξ( u t ). In particular, the directional tails of ξ are described by the function ξ(v ) (v → 0). We start with the basic

Proposition 5.1. Assume L µ < 0 and E(log |B|) < ∞. Then R n converges P -a.e to R = ∞ 1 A 1 • • • A k-1 B k , and for any x ∈ V , X x n converges in law to R. For all β ∈ I µ with k(β) < 1 and E(|B| β ) < ∞, we have E(|R| β ) < ∞.
The law ρ of R is the unique λ-stationary measure on V . The closure ∆ a (Σ) = Λ a (Σ) is the unique Σ-minimal subset in V and is equal to suppρ. If the semigroup T contains an element g with lim n→∞ |g n | 1/n > 1 and T has no fixed point then suppρ is unbounded.

If T satisfies condition i-p and suppλ has no fixed point in V , then ρ (W ) = 0 for any affine subspace W .

Proof. Under the conditions L µ < 0 and E(log |B|) < ∞ the P-a.e. convergence of R n to R is well known as well as the moment condition E(|R| β ) < ∞ if k(β) < 1 and β ∈ I µ (see for example [START_REF] Bougerol | Strict stationarity of generalized autoregressive processes[END_REF]). We complete the argument by observing that, since L µ < 0, we have lim

n→∞ |A n • • • A 1 x| = 0 hence, since X x n -A n • • • A 1
x has the same law as R n , the convergence in law of X x n to R for any x follows. In particular, if

x ∈ V is distributed according to ξ ∈ M 1 (V ), the law of X x n is λ n * ξ = λ n * δ x d ξ(x), hence has limit ρ at n = ∞. If ξ is λ-stationary, we have λ n * ξ = ξ, hence ξ = ρ. Since L µ < 0, there exists h = (g , b) ∈ Σ, such that |g | < 1, hence lim n→∞ |g n | 1/n < 1. If h = (g , b) ∈ Σ satisfies lim n→∞ |g n | 1/n < 1, then I -g is invertible, hence the unique fixed point h + of h satisfies (I -g )h + = b, and for any x ∈ V , h n x-h + = g n (x-h + ), hence lim n→∞ h n x = h + . Taking x in suppρ we get h + ∈ suppρ, since suppρ is h-invariant. Furthermore, for any x ∈ V and h ′ ∈ Σ we have lim n→∞ h ′ h n x = h ′ (h + ) and h ′ h n ∈ Σ satisfies lim n→∞ |g ′ g n | = 0, hence the unique fixed point x n of h ′ h n satisfies lim n→∞ x n = h ′ (h + ). Then ∆ a (Σ) = Λ a (Σ) is a Σ-invariant non trivial, closed subset of suppρ.
On the other hand, for x ∈ ∆ a (Σ) we have lim

n→∞ λ n * δ x = ρ, (λ n * δ x )(Λ a (Σ)) = 1 for all n hence ρ (Λ a (Σ) = 1, i.e Λ a (Σ) = suppρ.
The Σ-minimality of Λ a (Σ) follows from the fact that, for any x ∈ V and h = (g , b) with |g | < 1, one has lim

t→∞ h n x = h + ∈ Λ a (Σ) hence
Σx ⊃ Λ a (Σ). This implies also the uniqueness of the Σ-minimal set.

Observe that, if suppρ is bounded, then the convex envelope Co(suppρ) is a compact subset of V . Also any h ∈ Σ preserves suppρ and Co(suppρ). Then Markov-Kakutani theorem implies that the affine map h has a fixed point h 0 in Co(suppρ

). If h = (g , b) ∈ Σ satisfies lim n→∞ |g n | 1/n > 1 we have : (I -g )h 0 = b and h n x -h 0 = g n (x -h 0 ), hence if x = h 0 , we have lim n→∞ |h n x| = ∞. Then suppρ is unbounded since if x ∈ suppρ = δ h 0 the point h n x belongs to suppρ. Let W = {W i ; i ∈ I } be the set of affine subspaces of minimal dimension with ρ(W i ) > 0. Since dim(W i ∩ W j ) < dimW i if i = j , we have ρ(W i ∩ W j ) = 0, hence i ∈I ρ(W i ) ≤ 1. It
follows that, for any ε > 0, the set {W j ; j ∈ I and ρ(W j ) ≥ ε} has cardinality at most 1 ε , hence ρ(W i ) reachs its maximum on a finite set {W j ; j ∈ J ⊂ I } of affine subspaces.

Then the stationarity equation λ * ρ = ρ gives on such a subspace W j ,

ρ(W j ) = ρ(h -1 W j )d λ(h). Since ρ(h -1 W j ) ≤ ρ(W j ) we get, for any h ∈ suppλ, j ∈ J , ρ(h -1 W j ) = ρ(W j ), i .e h -1 W j = W i , for some i ∈ J .
In other words the set {W j ; j ∈ J } is suppλ-invariant. If dimW j > 0, one gets that the set of directions W j ( j ∈ J ) is a suppµ-invariant finite set of subspaces of V , which contradicts condition i-p for the semigroup T . Hence each W j ( j ∈ J ) is reduced to a point w j . Then the barycenter of the finite set {w j ; j ∈ J } is invariant under suppλ, which contradicts the hypothesis. Hence ρ(W ) = 0 for any affine subspace W .

In order to state the main result of this section we consider the compactification V ∪ S d-1 ∞ , and the natural projection of S d-1

∞ on the unit sphere S d-1 . We denote by

Λ ∞ (T ) (resp Λ ∞ + (T ), Λ ∞ -(T )) the inverse image of Λ(T ) (resp Λ + (T ), Λ -(T )) in S d-1 ∞ (see sec- tion 2, paragraph 3). The closure Λ a (Σ) of Λ a (Σ) in the compact space V ∪ S d-1 ∞ is T - invariant hence Λ a (Σ) ∩ S d-1 ∞ = Λ ∞ a (Σ), which is non void if suppρ = Λ a (Σ) is unboun- ded and is a closed T -invariant subset of S d-1 ∞ . Then Proposition 2.14 applied to Λ ∞ a (Σ) ⊂ S d-1
∞ gives the following trichotomy, since condition i-p is satisfied by T • case I: T has no invariant proper convex cone and Λ ∞ a (Σ) ⊃ Λ ∞ (T ),

• case II': T has an invariant proper convex cone and Λ ∞ a (Σ) ⊃ Λ ∞ (T ),

• case II": T has an invariant proper convex cone and Λ ∞ a (Σ) contains only one of the sets

Λ ∞ + (T ), Λ ∞ -(T ), say Λ ∞ + (T ), hence Λ ∞ a (Σ) ∩ Λ ∞ -(T ) = .
We assume α ∈]0, s ∞ [ exists with k(α) = 1 (see Proposition 4.11 for a sufficient condition). As in Theorem 4.10, we consider the P α -invariant measures ν α , ν α + , ν α -. The following implies Theorem C of section I and describes the asymptotics of the probability measure t • ρ when t → 0 + : ρ has a Pareto distribution of index α ([47], p.74).

Theorem 5.2. With the above notation assume L

µ < 0, Σ has no fixed point in V , T satis- fies condition i-p, there exists α ∈]0, s ∞ [ such that k(α) = 1 and E(|B| α+ε ) < ∞, E(|A| α γ ε (A)) < ∞ for some ε > 0. If d = 1 assume also that µ is non arithmetic.
we show that ρ 0 is "small at infinity", and we define C ,C + ,C -, σ α . With the hypothesis of Theorem 5.2, we denote by * ν α u the positive kernel on S d-1 given by Theorem 4.10 and associated with µ * . Proposition 5.3. One has the equations on V \ {0},

ρ = ∞ Σ 0 µ k * (ρ -ρ 1 ), ρ(v ) = ∞ Σ 0 ((µ * ) k * δ v )( ρ -ρ 1 ). For u ∈ S d-1 , the function t → t α-1 ρ 0 (u, t ) is Riemann-integrable in generalized sense on ]0, ∞[ and, one has with r α (u) = ∞ 0 t α-1 ρ 0 (u, t )d t , p(α) = |〈x, y〉| α d ν α (x)d α ν α (y) lim t→∞ t α ρ(u, t ) = * e α (u) L µ (α) * ν α u (r α ) = C (σ α ⊗ ℓ α )(H + u ),
where C = 2 * ν α (r α )α L µ (α)p(α) ≥ 0 and σ α ∈ M 1 ( Λ(T )) are defined by the above convergence and

σ α ⊗ ℓ α satisfies µ * (σ α ⊗ ℓ α ) = σ α ⊗ ℓ α .
Furthermore suppρ is unbounded and,

• In case I:

σ α = ν α , • In case II: * ν α u (r α )σ α = 1 2 ( * ν α + (r α )ν α + + * ν α -(r α )ν α -) where * ν α + (r α ) ≥ 0, * ν α -(r α ) ≥ 0.
The proof will follow from a series of lemmas.

We start with the following simple Tauberian lemma.

Lemma 5.4. For a non negative and non increasing function f on R * + and s ≥ 0, we denote, f s (t )

= 1 t t 0 x s f (x)d x. Then the condition lim t→∞ f s (t ) = c implies lim t→∞ t s f (t ) = c.
Proof. Let b be a positive real number with b > 1 and let us observe that, since f is non increasing 1

t bt t x s f (x)d x ≤ f (t ) 1 t bt t x s d x = t s s + 1 (b s+1 -1) f (t ). It follows b s+1 -1 s + 1 t s f (t ) ≥ b f s (bt ) -f s (t ).
Then the hypothesis gives :

lim inf t→∞ b s+1 -1 s + 1 t s f (t ) ≥ (b -1)c. Using the relation lim b→1 b s+1 -1 (s + 1)(b -1) = 1 we get lim inf t→∞ (t s f (t )) ≥ c. An analogous argu- ment gives lim sup t→∞ (t s f (t )) ≤ c. It follows lim t→∞ t s f (t ) = c.
We will use below the multiplicative structure of the group R * + =]0, ∞[, and we recall that Haar measure ℓ on the multiplicative group R * + is given by d t t .

Lemma 5.5. Assume that the V -valued random variable R satisfies equation (S), and

E(|B| α+δ ) < ∞, with δ > 0. For u ∈ S d-1 and t , x > 0 we write r α (u, t ) = 1 t t 0 x α ρ 0 (u, x)d x.
Then

|r α (u, t )| ≤ 2 t α α+1 . For δ ′ small there exists C (δ ′ ) > 0 such that if t ≥ 1, |r α (u, t )| ≤ C (δ ′ )t -δ ′ . In particular the function r α (u, t ) is b.R.i on S d-1 × R * + .
Proof. Since ρ 0 is the difference of the laws of R and R -B we have by definition of

ρ 0 , | ρ 0 (u, t )| ≤ 2, hence |r α (u, t )| ≤ 2 t α α+1 . Also ρ 0 (u, x) = r 1 (u, x) -r 2 (u, x) where, r 1 (u, x) = P{x -〈B, u〉 < 〈R -B, u〉 ≤ x}, r 2 (u, x) = P{x < 〈R -B, u〉 ≤ x -〈B, u〉}. Furthermore r α = r α 1 -r α 2 with r α 1 (u, t ) = 1 t t 0 x α r 1 (u, x)d x and r α 2 (u, t ) = 1 t t 0 x α r 2 (u, x)d x.
In order to estimate r α 1 , we choose ε ∈]0, 1[ with ε > α α+δ and write, for t ≥ 2, r α 1 (u, t ) ≤

1 t t 2 x α P{< B, u >≥ x ε }d x + 1 t t 2 x α P{x -x ε < 〈R -B, u〉 ≤ x}d x + 2 α+1 (α+1)t . Then Markov's inequality gives P{〈B, u〉 ≥ x ε } ≤ x -(α+δ)ε E(|B| α+δ ).
Hence the first term I ε 1 (t ) in the above inequality satisfies

I ε 1 (t ) ≤ E(|B| α+δ ) 1 t t 2 x α-ε(α+δ) d x ≤ E(|B| α+δ )t α-ε(α+δ) .
For tt ε ≥ 2, the second term I ε 2 (t ) satisfies

I ε 2 (t ) ≤ 1 t t 2 x α P{< R -B, u >> x -x ε }d x - 1 t t-t ε 2 x α P{〈R -B, u〉 > x}d x.
In the second integral above we use the change of variables x → xx ε and we get

I ε 2 (t ) ≤ 1 t t 2 [x α -(x -x ε ) α (1 -ε x ε-1 )] P{< R -B, u >> x -x ε }d x + k 0 (ε) t with 0 < k 0 (ε) < ∞.
We observe that there exists k 1 (ε) < ∞ such that for any x ≥ 2,

x α -(x -x ε ) α (1 -ε x ε-1 ) ≤ k 1 (ε)x α+ε-1 .
For any β ∈]0, α[, Proposition 5.1 implies that E(|R| β ) < ∞. Also R satisfies equation (S) and A, R • θ are independent. Hence Markov's inequality gives

P{〈R -B, u〉x} ≤ x -β E(|A| β )E(|R| β ) ≤ k 2 (β)x -β with k 2 (β) < ∞.
It follows that for any t with tt ε > 2

I ε 2 (t ) ≤ k 0 (ε) t + k 1 (ε)k 2 (β) 1 t t 2 x α+ε-1 (x -x ε ) β d x ≤ k 3 (ε, β)t α-β+ε-1 .
It remains to choose δ, ε, β in order to obtain α + ε -1 -β < 0. We take δ so small that δ(α+δ) -1 < α and ε = (α+δ/p)(α+δ

) -1 with p ∈ N, p ≥ 2, hence ε ∈]α(α+δ) -1 , 1[. Also α + ε -1 = α -δ(1 -1/p)(α + δ) -1 > 0.
We take γ ∈]0, 1[ and

β = α -γδ(1 -1/p)(α + δ) -1 so that α + ε -1 -β = (γ -1)δ(1 -1/p)(α + δ) -1 .
With p = 2, γ = 1/2 we get α+ε-1-β = -δ/4(α+δ) -1 . We write δ ′ =inf (δ/4(α+δ) -1 , δ/2).

Hence, there exists k 3 < ∞ and δ ′ > 0 such that for t ≥ 1, r α 1 (u, t ) ≤ k 3 t -δ ′ . The same argument is valid for r α 2 , hence for some δ ′ > 0 and t ≥ 1, we have r α (u, t

) ≤ C (δ ′ )t -δ ′ , with C (δ ′ ) < ∞. Furthermore, for t ∈]0, 1] we have |r α (u, t )| ≤ 2 t α α+1 , hence the function r α (u, t ) is b.R.i . on S d-1 × R * + .
Lemma 5.6. We denote by r the finite measure on R * + defined by r (d x) = 1 ]0,1[ (x)x α d x and we write ρ 0 = ρρ 1 . Then the function h α on V \ {0} defined by

h α (v ) = |v | -α (r * ρ 0 )(v ) = 1 t t 0 x α ρ 0 ( ṽ, x)d x , is b.R.i and one has (δ u ⊗ ℓ α )(r * ρ 0 ) = ∞ 0 t α-1 ρ 0 (u, t )d t = r α (u) where t → t α-1 ρ 0 (u, t ) is Riemann-integrable on ]0, ∞[ in generalised sense.
The formula in the corollary follows by α-homogeneity, since it is valid for |v | = 1. We take a base

u i ∈ V (1 ≤ i ≤ d ) and write |R| ≤ d Σ i =1 |〈R, u i 〉| . For t large: P{|R| > t } ≤ d Σ i =1 P{|〈R, u i 〉| > t ≤ (C ′ +ε)t -α b ′α d Σ i =1 * e α (u i ), with ε > 0, C ′ = C p(α)
α , hence the result.

A dual Markov walk and the positivity of directional tails

The following proposition will play an essential role in the discussion of positivity for C + ,C -and C as defined in Proposition 5.3. We denote by Λ * a (Σ) the set of elements u in S d-1 such that the projection of Λ a (Σ) on the half line R + u is unbounded. Here, instead of the vector space V used in paragraph 2, duality in the context of Radon transforms lead us to consider a λ-random walk on the larger vector space V ×R and to use ideas of [START_REF] Kesten | Renewal theory for functionals of a Markov chain with general state space[END_REF], for the analysis of corresponding measurable fibered kernels (see subsection 4.1). However, the continuity hypotheses of [START_REF] Kesten | Renewal theory for functionals of a Markov chain with general state space[END_REF] are not in general satisfied by these kernels. Proposition 5.9. With the hypothesis of Theorem 5.

2 if M ⊂ S d-1 is T * -minimal and Λ * a (Σ) ⊃ M, then for any u ∈ M, C (u) = lim t→∞ t α P{〈R, u〉 > t } = lim t→∞ t α ρ(u, t ) > 0.
In particular with the above notations, we have C > 0.

We observe that R n = n-1

Σ 0 A 1 • • • A k B k+1 satisfies the relation 〈R n+1 , v 〉 = 〈R n , v 〉+〈B n+1 , S ′ n v 〉 where S ′ n = (A 1 • • • A n ) * . Also h = (g , b) ∈ H acts on E = (V \ {0}) × R according to the for- mula h(v, r ) = (g * v, r + 〈b, v 〉), hence the pair, (S ′ n v, r + 〈R n , v 〉) = (v n , r n ),
is a random walk on the right homogeneous H-space E . Actually, V × R is a vector space and the above formula for h(v, r ) defines a right linear representation of H in V × R which leaves invariant E ⊂ V × R. In particular, using the radial R * + -fibration of this vector space, we see that the radial projection (v, r ) → (u, p)

with v = |v |u, r = p|v | defines an H-equivariant projection from E to S d-1 × R, hence (S ′ n v, r + < R n , v >) is also a R * + -fibered Markov chain over S d-1 × R. Then we can write E = (S d-1 × R) × R * + ⊂ V × R.
The action of h = (g , b) on S d-1 × R is given by h(u, p) = (g * • u, h u p) with

h u p = 1 |g * u| (p + 〈b, u〉).
The proof of Proposition 5.9 is based on the relation 〈R, v 〉 = lim n→∞ < R n , v > and on the dynamics of the random walk (v n , r n ) = (S ′ n v, r + < R n , v >) on V × R. We denote by * P its corresponding fibered Markov kernel and we study an associated ladder process (x τ n ,W τ n ), defined below. In terms of this process we can give a new expression for ρ(v ) = P{〈R, u〉 > t } as a potential of a non negative function on E . Then we can use a weak renewal theorem for (x τ n ,W τ n ) and obtain Proposition 5.9.

We will consider successively the two components

u n = v n |v n | , p n = r n |v n | and finally the fibered Markov chain (v n , r n ) above S d-1 × R.
Let M be a T * -minimal subset of S d-1 , hence (see section 2, paragraphe 3), M = Λ + (T * ) (or Λ -(T * )) in cases II, or M = Λ(T ) in case I.

The following says that Λ * a (Σ) is "large".

Lemma 5.10. In cases I or II' :

Λ * a (Σ) = S d-1 . In case II" : Λ * a (Σ) ⊃ Λ + (T * ). Proof. Let Λ ∞ a (Σ) = Λ a (Σ) ∩ S d-1 ∞ and u ∈ S d-1 , u ′ ∞ ∈ Λ ∞ a (Σ) corresponds to u ′ ∈ S d-1 . If 〈u ′ , u〉 > 0, then u ∈ Λ * a (Σ). Hence the complement of Λ * a (Σ) in S d-1 is contained in the set {u ∈ S d-1 ; 〈u, u ′ 〉 ≤ 0 ∀u ′ ∞ ∈ Λ ∞ a (Σ)}.
From the discussion at the beginning of this section we know that Λ ∞ a (Σ) = φ is T -invariant and closed, hence contains Λ ∞ (T ) in cases I, II' or only Λ ∞ + (T ) in case II" with Λ ∞ -(T ) ∩ Λ ∞ a (Σ) = φ. Since Λ ∞ (T ) is symmetric and condition i-p is valid, it follows Λ * a (Σ) = S d-1 in cases I, II'. In case II", we know from the end of proof of Theorem 2.16 that the complement of

Λ * a (Σ) is contained in Λ + (T * ) = {u ∈ S d-1 ; ∀u ′ ∈ Λ + (T ), 〈u, u ′ 〉 > 0}. Since Λ + (T * ) ∩ -Λ + (T * ) = φ, we get Λ + (T * ) ⊂ Λ * a (Σ).
The random walk (v n , r n ) = (S ′ n v, r + 〈R n , v 〉) has H-equivariant projection S ′ n v on V \ {0}, the kernel * P has projection * P already defined in section 2, hence the positive homogeneous function * e α ⊗ h α , lifted to E , satisfies * P ( * e α ⊗ h α ) = * e α ⊗ h α , and we can consider the new relativized fibered Markov kernel * P α on E . If (u, p) ∈ M × R, the projection x n = (u n , p n ) of (v n , r n ) on M × R depends on the kernel * Q α given by where * q α corresponds to q α as in section 3. The important fact for the proof of Proposition 5.9 is that * Q α has a stationary probability κ with κ(M ×[t , ∞[) > 0 for any t > 0, such that p is not "too large" with respect to κ.

For the analysis of * Q α we consider on Ω the projective limit * Q α u of the system * q α n (u, •)λ ⊗n (n ∈ N) and, by abuse of notation, the corresponding expectation will be written E α u . Given a * Q α -stationary measure π α M , we write * Q α = δ u ⊗ * Q α u d π α M (u) and we denote by E α the corresponding expectation symbol. We denote by θ # the map of M × H Z into itself defined by θ # (u, ω) = (g * 1 • u, θ ω) where θ is the bilateral shift on H Z , and * Q α will again denote the natural θ # -invariant measure on M × H Z . Also we extend S ′ n (ω) as a G-valued Z-cocycle. If η is a probability measure on X = M × R, the associated Markov measure on a Ω = X × Ω, is denoted by * Q α η , and the extended shift by a θ where a θ(x, ω) = (h 1 x, θ ω). Also if η is * Q α -stationary we will consider the bilateral associated system (Ω # , a θ, η # ) where Ω # = X × H Z , a θ is the bilateral shift and η # is the unique a θ-invariant measure with projection * Q α η on X × Ω.

Lemma 5.11. Let M be a T * -minimal subset of S d-1 , π α M the unique * Q α -stationary measure on M. With the above notations, we consider the Markov chain x n = (u n , p n ), on X = M × R given by

u n+1 = g * n+1 .u n , p n+1 = p n + 〈b n+1 , u n 〉 |g * n+1 u n | , p 0 = p, u 0 = u, where (g n , b n ) is distributed according to * Q α u .
Then, for any p ∈ R, x n converges in * Q αlaw to the unique * Q α -stationary measure κ, the projection of κ on M is π α M , κ(M ×{p}) = 0 and |p| ε d κ(u, p) < ∞ for ε small. We have κ # -a.e,

lim sup n→∞ |S ′ n u||p n | = ∞, lim n→∞ |S ′ -n u||p -n | = 0. If Λ * a (Σ) ⊃ M, then κ(M×]t , ∞[) > 0 for any t > 0 and lim sup n→∞ |S ′ n u|p n = ∞, * Q α κ -a.e.
Proof. If will be convenient to use the functions a(g , u), b(h, u) defined by h u p = a(g , u)p+ b(h, u), and the random variables

a k , b k (k ∈ Z) defined by a k ( ω, u) = a(g k , S ′ k-1 • u), b k ( ω, u) = b(h k , S ′ k-1 • u).
Then we can express the action of

h n • • • h 1 ∈ H on X as, u n = S ′ n • u, y p n (u) = (h n • • • h 1 ) u p, where y p n (u) = a(S ′ n , u)p + y • n (u) and y • n (u) = Σ n 1 a n k+1 (u)b k (u), with a n k (u) = a(g n • • • g k , S ′ k-1 • u). The random variables a k , b k are * Q α -stationary and y • n has the same law as p • n ( ω, u) = n-1 Σ 0 a -1 • • • a -k b -k-1 .
We estimate E α (|p • n | ε ) for 0 < ε < τ and ε small, where p

• n ( ω, u) = n-1 Σ 0 a -1 • • • a -k b -k-1 . Since (a k • • • a 1 )(ω, u) = a(S ′ k (ω), u) we get E α (|a -1 • • • a -k | ε ) = E α (|S ′ k u| -ε ). Hence Corol- lary 3.18 gives lim k→∞ (E α (|a -1 • • • a -k | ε )) 1/k < 1 since k ′ (α) > 0 and E(|A| α+τ ) < ∞. Also for ε small, E α (|b k | ε ) = E α (| 〈B 1 , u〉 |g * u| | ε ) ≤ E α (|B 1 | ε γ ε (A)) < ∞,
using * Q α -stationarity, Hölder inequality and the condition

E(|B 1 | α+τ )+E(|A 1 | α+τ ) < ∞. Since for 0 < ε < 1 |p • n ( ω, u)| ε ≤ n-1 Σ • |a -1 • • • a -k | ε |b -k-1 | ε , we get that E α (|p • n | ε ) is bounded.
The * Q α -a.e. convergence of the partial sum p

• n ( ω, u) to p( ω, u) = ∞ Σ 0 a -1 • • • a -k b -k-
κ = δ u ⊗ δ p( ω,u) d * Q α ( ω, u) is * Q α -invariant.
As observed above, the * Q α -laws of y

( * Q α ) n (π α M ⊗δ 0 ) = κ. Since |y p n ( ω, u) -y p ′ n ( ω, u)| = a(S ′ n (ω), u)|p -p ′ | and a(S ′ n (ω), u) = |S ′ n u| -1 converges * Q α u -a.
e to zero, we get the convergence of ( * Q α ) n (π α M ⊗δ p ) to κ, for any p. On the other hand, if η ′ is a * Q α -stationary measure on M × R, its projection on M is * Q α -stationary, hence equal to π α M , since M is T * -minimal. Then, from above For the analysis of * P α , ρ we consider the optional time τ on X × Ω given for p = 0 by:

( | * Q α ) n η ′ converges to κ, hence η ′ = κ. The * Q α -ergodicity of κ implies the a θ-ergodicity of * Q α κ and κ # . If Λ * a (Σ) ⊃ M assume κ (M×]t , ∞[) = 0,
τ = inf{n > 0 ; p -1 〈R n , u〉 > 0}, τ = ∞ if p -1 〈R n , u〉 ≤ 0 for every n.
We observe that τ is independent on p as long as p > 0 or p < 0. By definition of p n :

p + 〈R n , u〉 = p n |S ′ n u|, p -1 p n |S ′ n u| = 1 + p -1 〈R n , u〉.
We note that (u n , p n ) is the radial projection of (v n , r n ) on

S d-1 × R ⊂ V × R while the projection (0, r n ) of (v n , r n ) on {0} × R satisfies r n = r (p -1 p n |S ′ n u|).
In (u, p, r ) coordinates on the set {r = 0}, the process (v n , r n ) can be written as (x n , r n ), and for any t > 0 the dilation (v, r ) → t (v, r ) reduces to (x, r ) → (x, t r ). Since κ(M ×{0}) = 0, * P is also a measurable Markov fibered kernel above S d-1 × R * (see subsection 4.1). This measurable setting will be useful below.Also,

τ = inf{n > 0 ; p -1 p n |S ′ n u| > 1}, τ = ∞ if p -1 p n |S ′ n u| ≤ 1, for every n.
In particular p -1 p τ > 0 where the notation p τ is used if τ is finite. Also we define σ n = τ • ( a θ) σ n-1 and σ 0 = 0, τ n = n Σ 1 σ k , so that τ can be seen as the first ladder index and p -1 p τ |S ′ τ u| as the first ladder height of the R * -valued measurable Z-cocycle ,

W n (u, p, ω) = p -1 p n |S ′ n u|
over the dynamical system (Ω # , a θ, κ # ), which is well defined since κ(M × {0}) = 0. The random times τ n can be seen as the successive times of increase for r -1 r k = p -1 p k |S ′ k u| along the random walk (v n , r n ). On the other hand, by Poincaré recurrence theorem we have κ # -a.e, limsup

It follows ψ

τ = ψ - * P τ ψ, ψ = n-1 Σ 0 ( * P τ ) k ψ τ + * P τ n ψ with * P τ n ψ(v, p) = P{t |S ′ τ n u| -1 < p -1 〈R τ n , u τ n 〉 ; τ n < ∞}.
For (x, ω) ∈ M × Ω we have either τ n (x, ω) = ∞ for some n hence * P τ n ψ = ( * P τ ) k ψ = 0 or lim n→∞ τ n (x, ω) = ∞. In the second case, lim n→∞ |S ′ τ n u| -1 = ∞, Pa.e. and, since R τ n converges P-a.e to R, we have lim

n→∞ * P τ n ψ = 0, ψ = ∞ Σ 0 ( * P τ ) k ψ τ .
The last relation follows from the definitions of ψ α τ and * P τ α , since ψ α τ is non negative.

Remark. The function ψ satisfies the basic property of potentials for * P τ , * P τ ψ ≤ ψ, lim n→∞

( * P τ ) n ψ = 0.
This property is a key for understanding non triviality of the Cramer-type estimate for ψ. It is also valid for other natural functions such as ψ defined by

ψ(v, p) = P{sup n≥1 |W n | > t }.
Let τ be as above and Λ * a (Σ) ⊃ M, hence using Lemma 5.11 we have κ(X + ) > 0. Let * Q α,τ (x, .) be the law of x τ under * Q α

x . Since τ < ∞, * Q α κ -a.e. , this kernel is a measurable Markov kernel with respect to κ on X + , hence * Q α,τ is not κ-ergodic in general and it is natural to consider the first return time to X + as well as the corresponding induced Markov operator * Q α + on X + . Following the idea of ( [START_REF] Kesten | Renewal theory for functionals of a Markov chain with general state space[END_REF], Lemma 2) we interpret the ladder index τ as a first return time to a subset of Ω # , we construct a stationary measure for * Q α,τ on X + , and we show the finiteness of the corresponding expectation of τ. Lemma 5.13. Assume Λ * a (Σ) ⊃ M and write

τ = inf{n > 0 ; p -1 p n |S ′ n u| > 1} = inf{n > 0 ; p -1 〈R n , u〉 > 0}, τ = ∞ if p -1 p n |S ′ n u| ≤ 1 for any n ∈ N.
Then the stopped operator * Q α,τ preserves X + and admits a stationary ergodic probability κ τ on X + which is absolutely continuous with respect to κ. The integral

E α 0 (τ) = E α u (τ)d κ τ (u, p) is finite. With γ α τ = L µ (α)E α 0 (τ) we have lim n→∞ 1 n log(|S ′ τ n u| p τ n p ) = γ α τ ∈]0, ∞[, * Q α κ τ -
(W a τ ) = γ α τ ∈ ]0, ∞[.
In order to relate κ # 0 and the kernel * Q α,τ we consider the Markov kernel adjoint to * Q α + (x, •) with respect to κ + , and we denote by * Q α -⊗ δ x the corresponding Markov measure on H Z -× X + with Z -= -N ∪ {0}. Also we write * Q α

x = δ x ⊗ * Q α + where * Q α + is supported on H N and κ # + = * Q α -⊗ δ x ⊗ * Q α + d κ + (x), in particular κ # (Ω # 0 )κ # 0 = (1 Ω # 0 ) * Q α -⊗ δ x ⊗ * Q α + d κ + (x)
with

Ω # 0 = Ω - 0 × H N and Ω - 0 ⊂ H Z -× X + .
We denote by κ τ the projection of κ # 0 on X + , hence κ τ has density u(x) given by κ # (Ω # 0 )u(x) = ( * Q α -⊗ δ x )(Ω - 0 ) with respect to κ + . It follows that the projection of κ # 0 on X + × Ω can be expressed as

u(x)δ x ⊗ * Q α + d κ + (x) = δ x ⊗ * Q α + d κ τ (x) = * Q α κ τ .
Since κ # 0 is invariant and ergodic with respect to the bilateral shift a θ τ , the same is valid for * Q α κ τ with respect to the associated unilateral shift a θ τ . Since the kernel x → * Q α

x commutes with a θ τ and * Q α,τ , the * Q α,τ -invariance and ergodicity of κ τ follows. Also we have E α 0 (τ) = E α x (τ)d κ τ (x) and the above convergences are valid * Q α κ τ -a.e. .

Remark.

If S = X + , P = * P α,+ , π = κ + ,and in the corresponding measurable setting, the measure κ τ is closely connected with the measure χ of Theorem 4.5 . The measure κ τ can be caracterized as the unique * Q α,τ -stationary measure which is absolutely continuous with respect to κ + . However, in contrast to [START_REF] Kesten | Renewal theory for functionals of a Markov chain with general state space[END_REF], the function log |p| is not known to be κ-integrable, but we know that lim n→∞ |S ′ -n u|p -n p -1 = 0.

The following weak renewal theorem for a general fibered Markov chain will allow us to control potentials of the measurable fibered Markov kernel * P τ α on X + × R * + . We recall some notation of section 4 as follows.

Let (S, π) be a complete separable metric space, where π is a probability measure. We consider a general measurable Markov chain on S × R with kernel P , we assume that P commutes with the R-translations and we denote Lebesgue measure on R by ℓ . We assume that the measure π ⊗ ℓ is P -invariant. Here, in contrast to section 4, our setting is the measurable one;in particular the symbol sup means essential supremum. We write a path of the corresponding Markov chain as (x n ,V n ) where x n ∈ S and V n ∈ R, we denote by a P x the Markov measure on the paths starting from x ∈ S and we write a P π = a P x d π(x), a E x for the corresponding expectation symbol. In this context the following weak analogue of the renewal Theorem 4.5 holds.

P π {| V n n -γ| > ε} = 0,
with γ > 0. Then we have

lim t→∞ 1 t 0 -t d s S U ψ(x, s)d π(x) = 1 γ (π ⊗ ℓ)(ψ).
Furthermore if ψ is a non negative measurable function on S × R and lim

t→-∞ U ψ(x, t ) = 0, π-a.e. . then ψ = 0, π ⊗ ℓ-a.e. .
If ψ is a measurable function on S × R which satisfies:

|ψ| b = ℓ=∞ ℓ=-∞ sup{|ψ(x, s)| ; x ∈ S, s ∈ [ℓ, ℓ + 1[} < ∞,
then the above convergence is also valid.

Proof. We observe that the maximum principle implies |U ψ| = sup

x,t

|U ψ|(x, t ) < ∞, since U ψ is essentially locally bounded. For ε > 0, t > 0 we denote

n 1 = n 1 (t ) = [ 1 γ εt ], n 2 = n 2 (t ) = [ 1 γ (1+ε)t ]
where [t ] denotes integer part of t > 0.We write:

∞ Σ 0 P k ψ = U ψ, n-1 Σ 0 P k ψ = U n ψ, ∞ Σ n+1 P k ψ = U n ψ, m Σ n P k ψ = U m n ψ, I (t ) = 1 t 0 -t d s S (U ψ)(x, s)d π(x) = 3 Σ 1 I k (t ) -I 4 (t )
where

I 1 (t ) = 1 t S d π(x) ∞ -∞ U n 2 n 1 ψ(x, s)d s, I 2 (t ) = 1 t S d π(x) 0 -t U n 1 ψ(x, s)d s, I 3 (t ) = 1 t S d π(x) 0 -t U n 2 ψ(x, s)d s, I 4 (t ) = 1 t S d π(x) R\[-t,0] U n 2 n 1 ψ(x, s)d s.
We estimate each term I k (t ) separately. We have, since the measure π⊗ℓ is P -invariant,

I 1 (t ) = n 2 -n 1 +1 t (π ⊗ ℓ)(ψ), hence lim t→∞ I 1 (t ) = 1 γ (π ⊗ ℓ)(ψ). Furthermore, |I 4 (t )| ≤ |ψ| t (n 2 -n 1 + 1) sup n 1 ≤n≤n 2 S ( a P x {V n ≤ a} + a P x {V n ≥ t -a})d π(x).
Since n -1 V n converges to γ > 0 in probability, the above integral has limit zero, hence lim t→∞ I 4 (t ) = 0.

We have also |I 2 (t )| ≤ ε γ π ⊗ ℓ(ψ). In order to estimate I 3 (t ) we denote, for n ∈ N and s > 0, ρ s n = inf{k ≥ n ; -a ≤ V n -s ≤ a}, where ψ is supported on [-a, a], and we use the interpretation of U n ψ as the expected number of visits to ψ after time n :

U n ψ(x, s) ≤ |U ψ| a P x {ρ s n < ∞}.
Taking n = [ (1+ε)t γ ] = n 2 we get

I 3 (t ) ≤ |U ψ| S a P x {V k -t ≤ a, for some k ≥ [ (1 + ε)t γ ]}d π(x).
Since V n n converges to γ > 0 in probability, we get lim t→∞ I 3 (t ) = 0.

Since ε is arbitrary we get finally, lim t→∞

I (t ) = 1 γ (π ⊗ ℓ)(ψ).
The second conclusion follows by restriction and truncation of ψ on S × [-a, a].

For the proof of the last assertion we observe that for any ℓ ∈ Z,

∆ = |U 1 S×[0,1] | = |U 1 S×[ℓ,ℓ+1[ | < ∞. Writing ψ = ℓ=∞ Σ ℓ=-∞ ψ1 S×[ℓ,
p ) = γ α τ > 0, * Q α κ τ -a.e.
Since the canonical Markov measure associated with κ τ and * Q α,τ is a push-forward of * Q α κ τ , this convergence is also valid with respect to this canonical Markov measure. Then, using Lemma 5.12 and ψ α τ ≥ 0, we can apply Proposition 5.14 with V n = log(p -1 p τ n )|S ′ τ n u|, γ = γ α τ > 0, S = X + , to the measurable Markov kernel P = * P τ α on X + ×R * + , to the potential ∞ Σ 0 ( * P τ α ) k ψ α τ of the non negative function ψ α τ ≤ ( * e α ⊗h α ) -1 and to the * Q α,τ -stationary measure π = κ τ ; we get ψ α τ = 0, κ τ ⊗ ℓ-a.e., hence

P{t < p -1 〈R, u〉 ≤ t + p -1 〈R τ , u〉, τ < ∞} = 0.
Since p -1 〈R τ , u〉 > 0, this gives p -1 〈R, u〉 ≤ 0 κ τ ⊗ Pa.e. on {τ < ∞}, in particular for some (u, p) ∈ X + , we have p -1 〈R, u〉 ≤ 0 i.e. 〈R, u〉 ≤ 0, P-a.e. on {τ < ∞}. But, since Λ * (Σ) ⊃ M, for any u ∈ M the set {〈R, u〉 > 0 ; τ < ∞} = {〈R, u〉 > 0} is not P-negligible, hence the required contradiction. Since, using Proposition 5.3, we have

C (u) = C (σ α ⊗ ℓ α )(H + u ) it follows C > 0.
Remark. The proof of Proposition 5.9 given above uses the R * -valued multiplicative cocycle W n . The interpretation of the ladder index τ as a first return time to a subset of Ω # depends on the reduction of W n to a positively valued cocycle; hence the use of the seemingly artificial inducing procedure on X + × R * , as was done above.

A Choquet-Deny type property

Here, as in section 4, we consider a fibered Markov chain on S × R, but we reinforce the hypothesis on the Markov kernel P , using spectral gap properties instead of equicontinuity properties. Hence S is a compact metric space, P commutes with R-translations and acts continuously on C b (S ×R). We define for t ∈ R, the Fourier operator P i t on C (S) by P i t ϕ(x) = P (ϕ ⊗ e i t• )(x, 0).

For t = 0 the operator P i t = P 0 is equal to P , the factor operator on S defined by P . We assume that for each t ∈ R, P i t preserves the space H ε (S) of ε-Hölder functions and is a bounded operator therein. Moreover we assume that P •i t (t ∈ R), and P satisfies the following condition D.

A weaker Choquet-Deny type result under a similar condition was shown in ( [START_REF] Guivarc | Orbits of linear group actions, random walks on homogeneous spaces and toral automorphisms[END_REF], Proposition 3.5). The stronger form given here allow us to deal with polynomially bounded P -harmonic measures and is needed for the proof of the homogeneity at infinity of ρ in Theorem 5.2.Condition D is as follows 1. For any t ∈ R, one can find n 0 ∈ N, ρ(t ) ∈ [0, 1[ and C (t ) > 0 for which

[(P i t ) n 0 ϕ] ε ≤ ρ(t )[ϕ] ε +C (t )|ϕ|.
2. For any t ∈ R, the equation P i t ϕ = e i θ ϕ, ϕ ∈ H ε (S), ϕ = 0 has only the trivial solution e i θ = 1, t = 0, ϕ =constant.

3. For some τ > 1 sup{ |a| τ P (x, 0), d (y, a) ; x ∈ S} < ∞.

Conditions 1,2 above imply that P has a unique stationary measure π and the spectrum of P in H ε (S) is of the form {1}∪∆ where ∆ is a compact subset of the open unit disk (see [START_REF] Tulcea | Théorie ergodique pour des classes d'opérations non complètement continues[END_REF] ). They imply also that, for any t = 0, the spectral radius of P i t is less than one.

With the notations of section 4, condition 3 above will allow us to estimate a E(|V n | p ) for p ≤ δ and to show the continuity of t → |P i t |.

The following is a simple consequence of conditions 1,2,3 above. Proof. Conditions 1 and 2 for P i t (t = 0) imply that the spectral radius r t of P i t satisfies r t < 1. Hence there exists C t > 0 such that for any n ∈ N,|(P i t ) n | ≤ C t ( 1 2 + r t 2 ) n . On the other hand t → |P i t | is continuous as the following calculation shows. For a, t , t ′ ∈ R, δ ′ ∈ [0, 1], we have |e i ate i at ′ | ≤ 2|a| δ ′ |t -t ′ | δ ′ hence we have

|P i t ϕ(x) -P i t ′ ϕ(x)| ≤ 2|ϕ| |t -t ′ | δ ′ |a| δ ′ P ((x, 0), d (y, a)), |P i t -P i t ′ | ≤ 2M δ ′ |t -t ′ | δ ′ .
For each t ∈ I we fix n t ∈ N such that |(P i t ) n t | ≤ 1 3 . Then the above continuity of P i t , hence of (P i t ) n t , gives that for t ′ sufficiently chose to t , |(P

i t ′ )| ≤ 1 2 . Using compactness of I we find n 1 , • • • n k such that one of the inequalities |(P i t ) n j | ≤ 1 2 (1 ≤ j ≤ k) is valid at any given point of I . Then, since |P i t | ≤ 1 with n 0 = n 1 • • • n k we get |P i t ) n 0 | ≤ 1 2 .
Using Euclidean division of n by n 0 , we get the required inequality.

We are interested in the action of P n on functions on S × R which are of the form u ⊗ f where u ∈ C (S), f ∈ L 1 (R) and we are interested also in P -harmonic Radon measures which satisfy boundedness conditions. In some proofs, since H ε (S) is dense in C (S), it will be convenient to assume u ∈ H ε (S). Definition 5.16. We say that the Radon measure θ on S ×R is translation-bounded if for any compact subset K of S × R, any a ∈ R, there exists C (K ) > 0 such that |θ(a + K )| ≤ C (K ) where a + K is the compact subset of S × R obtained from K using translation by a ∈ R.

We are led to consider a positive function ω on R d which satisfies ω(x + y) ≤ ω(x)ω(y). For example, if p ≥ 0, such a function ω p is defined by ω p (a) = (1 + |a|) p . We denote

L 1 ω (R) = { f ∈ L 1 (R); ω f ∈ L 1 (R)} and we observe that f → ω f 1 = f 1,ω is a norm under which L 1 ω (R) is a Banach algebra. The dual space of L 1 ω (R) is the space L ∞ ω (R) of measurable functions g such that g ω -1 ∈ L ∞ (R)
and the duality is given by 〈g

, f 〉 = g (a) f (a)d a. The Fourier transform f of f ∈ L 1 ω (R) is well defined by f (t ) = f (a)e i t a d a.
We denote by J c the ideal of L 1 (R) which consists of functions f ∈ L 1 (R) such that f has a compact support not containing 0 and we write J c ω = L 1 ω (R) ∩ J c . Also we denote by L 1 0 (R) the ideal of L 1 (R) defined by the condition f (0) = 0. It is well known that J c is dense in L 1 0 (R), hence for ω = ω p the ideal J c ω 2 is dense in L 1 ω (R) (see [START_REF] Helson | Harmonic Analysis, 2 d Edition[END_REF] p. 187).

Theorem 5.17. With the above notation, assume that the family P i t (t ∈ R) satisfies conditions D and let ω = ω p with p < δ. Then for any

f ∈ L 1 ω (R) ∩ J c , u ∈ C (S), we have the convergence lim n→∞ sup x∈S P n (u ⊗ f )(x, •) 1,ω = 0.
If θ is a P -harmonic Radon measure which is translation-bounded, then θ is proportional to π ⊗ ℓ. In particular π ⊗ ℓ is a minimal P -harmonic Radon measure.

The proof follows from the above considerations and the following lemmas.

Lemma 5.18. Assume π n is a sequence of bounded measures on R, ω is a positive Borel function on R such that for any x, y ∈ R, ω(x + y) ≤ ω(x)ω(y) and assume that the total variation measures

|π n | of π n satisfy sup{|π n |(ω) ; n ∈ N} < ∞. Let f ∈ L 1 ω (R) ∩ L 2 (R) and assume A n , B n are sequences of Borel subsets of R such that, with A ′ n = R\ A n , B ′ n = R\B n , 1. lim n→∞ |π n |(ω) f 1 B ′ n 1,ω = 0, 2. lim n→∞ |π n |(ω1 A ′ n ) = 0, 3. lim n→∞ π n * f 2 ω 2 1 A n +B n 1/2 1 = 0.
Then we have lim n→∞ π n * f 1,ω = 0. Furthermore, if the measures π n depend of a parameter λ and if the convergences in 1-3 are uniform in λ, then the convergence of π n * f 1,ω is also uniform in λ.

Proof. Let η, η ′ be two bounded measures on R and let A, B be Borel subsets with complements

A ′ , B ′ in R. Observe that, since 0 ≤ ω(x + y) ≤ ω(x)ω(y), ω(x + y) ≤ ω(x + y)1 A+B (x + y) + (ω1 A )(x)(ω1 B ′ )(y) + (ω1 A ′ )(x)ω(y).
It follows

|(η * η ′ )|(ω) ≤ |η * η ′ |(ω1 A+B ) + |η|(ω)| |η ′ |(ω1 B ′ ) + |η|(ω1 A ′ )|η ′ |(ω).
Then we take η = π n , η ′ = f (a)d a, A = A n , B = B n and we get

π n * f 1,ω ≤ |π n * f |(a)|ω1 A n +B n |(a)d a + |π n |(ω1 A n ) f 1 B ′ n 1,ω + |π n |(ω1 A ′ n ) f 1,ω .
Conditions 1, 2 imply that the two last terms in the above inequality have limits zero. Using condition 3 and Schwarz inequality we see that the first term has also limit zero.

If π n depends of a parameter λ, the uniformity of the convergence of π n * f 1,ω follows directly from the bound for π n * f 1,ω given above.

The following lemma is an easy consequence of condition 3 on the Markov kernel P and of Hölder inequality.

* f = θ * r f and if f n ∈ L 1 (R) converges in L 1 - norm to f ∈ L 1 (R), then θ * f n converges to θ * f in
θ x * ω p ≤ 2 p . θ x ([0, 1])ω p . Hence . θ ≤ 2 p (1 S ⊗ ω p )( θ ⊗ ℓ) and finally θ * r ≤ 2 p c(1 S ⊗ ω p )( θ ⊗ ℓ) = (1 S ⊗ ω p )(θ ⊗ ℓ) with θ = 2 p c θ.
Proof of Theorem 5.17. We fix p ∈ [1, δ[, ω = ω p , u ∈ H ε (S), u ≥ 0 and for x ∈ S, we define the positive measure π x n on R by π x n (ϕ) = P n (u ⊗ϕ)(x, 0) where ϕ is a non negative Borel function on R. We observe that π x n (1) ≤ |u|, and for f ∈ C b (R) ∩ L 1 (R), we have

P n (u ⊗ f )(x, a) = π x n ( f * δ a ) = (π x n * f * )(a),
where f * (a) = f (-a). It follows

P n (u ⊗ f )(x, •) 1 ≤ |u| f 1 and also for f ∈ L 1 ω (R), P n (u ⊗ f )(x, •) 1,ω ≤ f 1,ω sup{π x n (ω) ; x ∈ S}. From condition 3, since p < δ we have pi x n (ω) ≤ |u| a E x (1 + |V n |) p ≤ 2 p C p n p
where we have used Lemma 5.19 in order to bound a E x (1

+ |V n |) p . We fix δ > 1, we denote B n = {a ∈ R ; |a| ≤ n 1+δ }, A n = {a ∈ R ; |a| ≤ n c+1
} with c > 0 to be defined later and we verify the conditions 1-3 of Lemma 5.18 for 1+δ) . Then, using the bound of π x n (ω) given above,

π n = π x n , uni- formly in x ∈ S for f ∈ L 1 ω 2 (R). Since f ω 2 ∈ L 1 (R), Markov's inequality gives f 1 B ′ n 1,ω ≤ f 1,ω 2 n -p(
π x n (ω) f 1 B ′ n 1,ω ≤ 2 p C p n -pδ f 1,ω 2 .
Hence condition 1 of lemma 5.18 is satisfied.

We write π x n (ω1

A ′ n ) ≤ |u| a E x (ω(V n )1 A ′ n (V n ))
and use Hölder inequality for p ′ > 1, 1 q ′ = 1 -1 p ′ and p ′ p < δ:

π x n (ω1 A ′ n ) ≤ C ′ 2 p n p P{|V n | ≥ n c+1 }) 1/q ′ ≤ C ′′ n p-cp/q ′ ,
where we have used the fact that sup In order to verify condition 3 we observe that π x n (t ) = (P i t ) n u(x). For f ∈ J c ⊂ L 2 (R) we denote Y = supp f ⊂ R \ {0} and we know from Lemma 5.17, that there exist D > 0 and σ ∈ [0, 1[ such that for any t ∈ Y , n ∈ N: |(P i t ) n | ≤ Dσ n . From Plancherel formula, we get

π x n * f 2 = ( | π x n (t )| 2 | f (t )| 2 d t ) 1/2 ≤ f 2 |u| sup t∈Y |(P i t ) n | ≤ D|u|σ n f 2 .
On the other hand, ω 2 1 A n +B n 1 is bounded by a polynomial in n. Since σ < 1, condition 3 is satisfied. Hence, Lemma 5.18 gives: lim From the first part of the proof we get θ(u ⊗ f ) = 0 for any u ∈ H ε (S), f ∈ J c ω = L 1 ω (R) ∩ J c . This relation remains valid for f in the ideal I c ω of L 1 ω (R) generated by J c ω . Using regularisation on Fourier transforms we see that the closure in L 1 (R) of J c ω contains J c , hence the unique Fourier exponential which vanishes on I c ω is 1. Then using classical Fourier Analysis (see [START_REF] Helson | Harmonic Analysis, 2 d Edition[END_REF] p.187) we get that I c ω is dense in L 1 0 (R). As observed above, since θ is translation-bounded, this implies θ(u ⊗ f ) = 0 for any f ∈ L 1 0 (R). Since H ε (S) is dense in C (S), we get that θ is invariant by R-translation. Then we have θ = θ ⊗ ℓ where θ is a positive measure on S which satisfies P θ = θ. Using parts 1,2 of condition D, this implies that θ is proportional to π, hence θ is proportional to π ⊗ ℓ.

For the final assertion we observe that, if θ is a P -harmonic positive Radon measure with θ ≤ c π ⊗ ℓ, for some c > 0, then θ is translation bounded. Hence as above, θ is proportional to π ⊗ ℓ.

Homogeneity at infinity of the stationary measure

For the proof of Theorem 5.2 we prepare the following propositions and lemmas. If α ∉ N, it follows from [START_REF] Boman | Support theorems for the Radon transform and Cramer-Wold theorems[END_REF] that Theorem 5.2 is a consequence of Propositions 5.3 and 5.9. If α ∈ N, as follows from [START_REF] Weiss | Measures that vanish on half-spaces[END_REF], the situation is different in general. More precisely, as shown in ( [START_REF] Boman | Support theorems for the Radon transform and Cramer-Wold theorems[END_REF], p 706), if η ∈ M 1 (V ) is suitably choosen and α ∈ N, convergence of t -α (t • η) (t → 0 + ) on the sets H + v for every v ∈ V does not imply vague convergence. Here, we will need to use the Choquet-Deny type results of paragraph 4. We start with an improvment of Corollary 5.8. Proof. This a trivial consequence of Proposition 5.3, Corollary 5.8, Proposition 5.9 and Lemma 5.10.

The following is a corollary of the proof of Proposition 5.9 and of Proposition 5.21.

Corollary 5.22. With the above notation we write

γ α τ = L µ (α)E α 0 (τ), ψ α τ (v, p) = P{t < p -1 〈R, u〉 ≤ t + p -1 〈R τ , u〉 , τ < ∞} * e α (u) -1 t α and we denote by κ τ the * Q α,τ -stationary measure on X + , given by Lemma 5.12. Then, if The final formula follows from Proposition 5.21.

Λ a (Σ) ⊃ Λ ∞ + (
Remark. We observe that, if d = 1, and A, B are positive, a formula of this type for C = C + , with equality, is given in [START_REF] Enriquez | A probabilistic representation of constants in Kesten's renewal theorem[END_REF]. We don't know if such an equality is valid in our setting.

Lemma 5.23. For any compact subset K of V \ {0}, there exists a constant C (K ) > 0 such that sup t>0 t -α (t •ρ)(K ) ≤ C (K ). In particular the family ρ t = t -α (t •ρ) is relatively compact for the topology of vague convergence and any cluster value η of the family ρ t satisfies sup Proof. For some δ > 0 we have K ⊂ {x ∈ V ; |x| > δ}, hence using Corollary 5.8, t -α P{|R| > δ t } ≤ b δ α = C (K ). The relative compactness of the family ρ t follows. Also,

(t t n ) -α (t t n • ρ)(K ) ≤ C (K ), t -α (t • η)(K ) = lim n→∞ (t t n ) -α (t t n • ρ)(K ) ≤ C (K ).
Hence sup t>0 t -α (t • η)(K ) ≤ C (K ). Since e α ⊗ h α is α-homogeneous we have t • ((e α ⊗ h α )η) = (e α ⊗ h α )(t -α (t • η)). With C K = sup v ∈K (e α ⊗ h α )(v ), we get

t • ((e α ⊗ h α )η)(K ) ≤ C K t -α (t • η)(K ) ≤ C K C (K ) = C ′ (K ).
Lemma 5.24. Assume η is the vague limit of t -α n (t n • ρ) (t n → 0 + ). Then η is µ-harmonic, i.e µ * η = η.

Proof. Let ϕ be ε-Hölder continuous on V with compact support contained in the set {x ∈ V ; |x| ≥ δ} with δ > 0, and let us show lim We write

I 1 t = t ε-α E(|B 1 | ε 1 {|tR|>δ I 2 t = t ε-α E(|B 1 | ε 1 {t|A 1
Ro θ|>δ} ) and we estimate I 1 t , I 2 t as follows. We have

I 1 t ≤ δ ε-α E(|B 1 | ε |R| α-ε 1 {|tR|>δ} |. Since |R| α-ε ≤ c(|A 1 Ro θ| α-ε +|B 1 | α-ε ), using independence of Ro θ and |B 1 | ε |A 1 | α-ε , we get E (|B 1 | ε |R| α-ε ) ≤ c E (|B 1 | α ) + c E (|A 1 | α-ε |B 1 | ε ) E (|R| α-ε .
Using Hölder inequality we get E(|A 1 | α-ε |B 1 | ε ) < ∞. Also using Proposition 5.1, we get

E(|R| α-ε ) < ∞. It follows that |B 1 | ε |R| α-ε 1 {|tR|>δ} is bounded by the integrable function |B 1 | ε |R| α-ε .
Then by dominated convergence, lim t→0 I 1 t = 0. In the same way we have

I 2 t ≤ δ ε-α E(|B 1 | ε |A 1 Ro θ| α-ε 1 {t|A 1 Ro θ|>δ} .
Also, using independence and Hölder inequality, we have

E (|B 1 | ε |A 1 Ro θ| α-ε ) ≤ E(|B 1 | ε |A 1 | α-ε )E(|R| α-ε ) < ∞.
Then by dominated convergence lim

lim t→0 + t -α (t • ρ)( f 1 B δ 1 ,δ 2 ) = Λ( f 1 B δ 1 ,δ 2 ).
Proof. From the fact that ν α gives measure zero to any projective subspace and the homogeneity of Λ = σ α ⊗ ℓ α , we know that Λ gives measure zero to any affine hyperplane, hence the boundary of B η 1 ,η 2 is Λ-negligible. Then the proof follows from the vague convergence of t -α (t • ρ) to Λ and the hypothesis of Λ-negligibility of the discontinuity set of f . Lemma 5.28. There exists C > 0 such that for any f ∈ B ε,α , t > 0 and δ 2 > e,

|t -α (t • ρ)( f 1 B δ 2 )| ≤ C K f (ε)| log δ 2 | -ε .
Furthermore there exists C (ε) > 0 such that for any f ∈ B ε,α , t > 0,

δ 1 < e -1 |t -α (t • ρ)( f 1 B δ 1 )| ≤ C (ε)K f (ε)| log δ 1 | -ε .
Proof. Let ϕ ε (x) be the function on R + \ {1} given by ϕ ε (x) = x α | log x| -1-ε . For x ≥ e we have ϕ ′ ε (x) ≤ α x α-1 | log x| -1-ε . We denote F t (x) = P(|t R| > x) and we observe that, using Proposition 5.1, the non increasing function F t is continuous. We have

|t -α (t • ρ)( f 1 B ′ δ 2 )| ≤ t -α K f (ε) ∞ δ 2 ϕ ε (x)d F t (x).
Integrating by parts, we get

|t -α (t • ρ)( f 1 B ′ δ 2 )| ≤ t -α K f (ε)[ϕ ε (x)F t (x)] ∞ δ 2 + t -α K f (ε) ∞ δ 2 ϕ ′ ε (x)F t (x)d x.
From Corollary 5.7 we know that, for some C > 0, F t (x) ≤ C t α x -α . Then, using the above estimation of ϕ ′ ε (x), we get

|t -α (t • ρ)( f 1 B ′ δ 2 )| ≤ C K ε ( f ) ∞ δ 2 α | log x| 1+ε d x x ≤ C K ε ( f )| log δ 2 | -ε .
The proof of the second assertion follows the same lines and uses the estimation of |ϕ

A An analytic approach to tail-homogeneity

Under the hypothesis of compact support for λ and density for µ, an analytic proof of tail-homogeneity of ρ is given below. The full hypothesis is only used in the study of positivity properties of C ,C + ,C -, hence we split the presentation into two parts according to the hypothesis at hand on λ. Also the argument gives analytic expressions for C ,C + ,C -. We recall Wiener-Ikehara's theorem (see [START_REF] Widder | The Laplace transform[END_REF] Proof. We write A(x) = x 1 t α ν(t , ∞)d t for x ≥ 1 and we observe that the finiteness of f (s) for s < α implies lim x→∞ x α-ε ν(x, ∞) = 0 for ε > 0.

Integrating by parts we have for s > 1

∞ 1 x -s-1 A(x)d x = s -1 ∞ 1 x α-s ν(x, ∞)d x = s -1 (α -s + 1) -1 ∞ 1
x α-s+1 d ν(x).

Since A(x) ≥ 0 is increasing, we can use Wiener-Ikehara's theorem: lim The connection with the spectral gap properties in section 3 depends on the following. The hypothesis is as in Corollary 3.21.

Lemma A.2.

There exists an open set D ⊂ C which contains the set {Rez ∈]0, α]} such that (I -P z ) -1 is meromorphic in D with a unique simple pole at z = α. We have In case I, lim z→α (αz)(I -P z ) -1 = k ′ (α) -1 ( ν α ⊗ e α ).

In cases II, lim z→α (αz)(I -P z ) -1 = k ′ (α) -1 (ν α + ⊗ e α + + ν α -⊗ e α -).

Proof. We restrict to case I, since the proof is similar in cases II. The operator P z on H ε (S d-1 ) where z = s + i t , is defined by the formula P z ϕ(x) = |g x| z ϕ(g • x)d µ(g ) and Since k ′ (α) = 0, we have k(z) = 1 for z = α and |z -α| small. Also for |z -α| small, we have in case I the decomposition P z = k(z) ν z ⊗ e z +U (z) where ν z ⊗ e z is a projector on the line Ce 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 consider also the projection p z = Iν z ⊗e z and we write I -P z = (1 -k(z))( ν z ⊗ e z ) + p z (I -U (z)). Hence, for |z -α| small (I -P z ) -1 = (1 -k(z)) -1 ( ν z ⊗ e z ) + p z (I -U (z)) -1 .

In particular, lim z→α (αz)(I -P z ) -1 = k ′ (α) -1 ( ν α ⊗ e α ), and (I -P z ) -1 is meromorphic in a disk B 0 centered at α with radius ε ′ ≤ ε, with unique pole at z = α. For z = α + i t with |t | ≥ ε ′ , we get from above that there exists a disk B t centered at α + i t such that r ( P z ) < 1 for z ∈ B t , hence (I -P z ) -1 is a bounded operator depending holomorphically on z for z ∈ B t . If Rez ∈]0, α[, then r ( P z ) ≤ r ( P s ) < 1 hence (I -P z ) is invertible and the It follows for s > 0: (k(s) * ν s ( fs )) 1/s = ( * ν s ( f 1 s )) 1/s . Since f 1 s (u) = E(〈R -B, u〉 s + ) we have for s > 1, * ν s ( f 1 s )) 1/s ≤ E(|B| s ) 1/s + ( * ν s ( fs )) 1/s , (k(s) 1/s -1)( * ν s ( fs ) 1/s ≤ E(|B| s ) 1/s .

Since λ has compact support we have lim By definition of fs it follows sup{〈R, u〉 + ; (ω, u) ∈ Ω× Λ(T * )} < ∞; using Lemma 5.10, this contradicts the fact that 〈R, u〉 + is unbounded on Ω × Λ(T * ).

In cases II, the above argument can easily be modified, * ν s replaced by * ν s + and S d-1 by supp * ν + . Then we get that 〈R, u〉 + is bounded on Ω × Λ + (T * ), which contradicts Lemma 5.10.

Remark. If suppλ is compact and µ has a density, Corollary 3.21 and the use of [START_REF] Kaneko | A generalisation of the Riesz-Schauder theory[END_REF], allow us to avoid the use of the renewal theorem of section 4 and of Kac's formula in the proof of Theorem 5.2. Furthermore, if α ∉ N, Theorem 5.2 then follows from the properties of Radon transforms of positive Radon measures (see [START_REF] Boman | Support theorems for the Radon transform and Cramer-Wold theorems[END_REF]). However, in the general case one needs to use Lemma 5.13 and Proposition 5.9. On the other hand, the density assumption on µ is not necessary for the validity of Proposition A.4 as follows from Theorem 5.2.

2 .

 2 Let Qs e be the Markov kernel on S d-1 defined by Qs e ϕ = 1 k(s)e P s (ϕe) where e still denotes the function on S d-1 corresponding to e ∈ C (P d-1

  k is the projection on the space of Q s -invariant functions and is equal top s + (x)π s + + p s -(x)π s -.The continuity and the extremality of the Q sinvariant functions p s + (x) and p s -(x) follows. The corresponding facts for e s + and e s follow as in the proof of Theorem 2.6.

ϕ

  e s . By duality, cases II for µ and µ * are the same, hence there are two minimal T * -invariant subsets Λ + (T * ) and Λ -(T * ) = -Λ + (T * ). On the other hand, the set

ϕ e s ≥ 0 and ϕ e s = 1

 1 on Λ + (T ), we conclude from above that p s + = ϕ e s , hence we get the last formula and last assertions with e s -

  = 〈z * (ω),x〉 〈z * (ω),y〉 s e s (y) e s (x) .

1 n 1 n 1 n

 111 log |S n (ω)| converges to L(s). We know, using Theorem 3.2, that for fixed x and Q sa.e.,lim n→∞ |S n (ω)x| |S n (ω)| = |〈z * (ω), x〉|,and furthermore the law of z * (ω) under Q s is proper. Hence, for fixed x we have|〈z * (ω), x〉| > 0, Q s -a.e.Then for fixed x ∈ P d-1 and Q s -a.e. lim n→∞ log |S n (ω)x| = lim n→∞ log|S n (ω)| = L µ (s).

  |g x| s log |g x|d µ n (g ). We denote v n (s) = |g x| s d µ n (g ) and we observe that v ′ n (s) = |g x| s log |g x| d µ n (g ).

Definition 4 . 2 .

 42 The function ϕ ∈ B(S × R) is said to be boundedly Riemann integrable (b.R.i ) if the following holds: ℓ=∞ Σ ℓ=-∞ sup{|ϕ(x, t )|; x ∈ S, t ∈ [ℓ, ℓ+1[} < ∞, and for any fixed x ∈ S, any β > 0, the function t → ϕ(x, t ) is Riemann integrable on [-β, β].

• I. 1 .

 1 For every open set O in S with π(O) > 0, and a P x -a.e., for each x ∈ S, we have a P x {x n ∈ O for some n} = 1.• I.2. P has a 1-moment and for all x ∈ S, a P x -a.e. we havelim n→∞ V n n = L = uF (d u|x, y)P (x, d y)d π(x) > 0. • I.3. There exists a sequence (ζ i ) i ≥1 ⊂ Rsuch that the group generated by ζ i is dense in R and such that for any i ≥ 1 and λ ∈ [0, 1], there exists y = y(i , λ) ∈ S such that (P, π) has distortion (ζ i , λ) at y. • I.4. The kernel P on S × R is non-expanding. Condition I.1 means that for any x ∈ S the trajectories of P visit any non π-negligible open set with probability one. It implies that continuous P -invariant functions are constant on S. Condition I.3 guarantees that there is no r > 0 such that, with probability one, the values of V n -V m (n, m ∈ N) are of the form k(n, m)r with k(n, m) ∈ Z.

Theorem 4 . 5 .

 45 Assume conditions I.1-I.4 are satisfied for the fibered Markov kernel P . Then there exists a positive measure χ on S × R + absolutely continuous with respect to π ⊗ ℓ + such that for any x ∈ S and ϕ ∈ C b (S×]0, ∞[),

Proposition 4 . 11 .

 411 Let µ ∈ M 1 (G) and assume that k(s) = lim n→∞ ( |g | s d µ n (g )) 1/n is finite for any s > 0. For any p ∈ N and g ∈ (suppµ) p we have lim s→∞ log k(s) s ≥ p -1 log r (g ).

1 n 1 n

 11 .e., we have 〈R, u〉 ≤ -p, P-a.e. . This implies that the support of the projection of ρ on Ru is bounded in direction u; since by Proposition 5.1 we have suppρ = Λ a (Σ) this contradicts the condition Λ * a (Σ) ⊃ M. Hence κ(M×]t , ∞[) > 0 for any t > 0. Furthermore, arguments as in the proof of Proposition 5.1, using that suppλ has no fixed point in V , show κ(M × {p}) = 0 for any p ∈ R. From Theorem 3.10 we know that limn→∞ log |S ′ n u| = L µ (α) > 0, * Q α u -a.e. .Furthermore, since κ is * Q α -ergodic and κ(M × {0}) = 0 we have limsup n→∞ |p n | > 0, Q α κ -a.e.. Then we get limsup n→∞ |S ′ n u| |p n | = ∞, * Q α κ -a.e. If Λ * a (Σ) ⊃ M, from above we have κ(M×]0, ∞[> 0, and again using ergodicity, lim sup n→∞ p n > 0. Since lim n→∞ |S ′ n u| = ∞ * Q α u -a.e, it follows lim sup n→∞ |S ′ n u|p n = ∞, * Q α κa.e. Using Theorem 3.2, we have also, for any u ∈ M and * Q α u -a.e.: lim n→∞ log |S ′ -n u| = -L µ (α) < 0. The condition |p| ε d κ(u, p) < ∞ implies lim sup n→∞ log |p n | |n| ≤ 0, κ # -a.e. Then we get, lim n→∞ |S ′ -n u||p -n | = 0, κ # -a.e.

n→∞ p - 1

 1 p n ≥ 1. Since lim n→∞ |S ′n u| = ∞,then τ, τ n are finite * Q α κ -a.e and we have p -1 p τ n > 0.

Proposition 5 . 14 .

 514 With the above notation and hypotheses, assume that ψ is a compactly supported bounded non negative measurable function on S × R, the potential U ψ = ∞ Σ 0 P k ψ is essentially bounded on S × [-c, c] for any c > 0 and we have for any ε > 0 lim n→∞ a

Lemma 5 . 15 .

 515 With the above notation, let I ⊂ R be a compact subset of R\{0}. Then there exists D > 0 and σ ∈ [0, 1[ such that for any n ∈ N, sup t∈I |(P i t ) n | ≤ Dσ n .

Lemma 5 . 19 .

 519 For any p ∈[1, δ], there exists C p > 0 such that supx a E x ((|V n |) p ) ≤ C p n p . In particular, for any L > 0, sup x,n a P x {|V n | > nL} ≤ C p L p .We leave to the reader the proof of the well known first inequality. The second one follows from Markov's inequality. For a Radon measure θ on S × R and b ∈ R, we denote by θ * δ b the Radon measure defined by (θ * δ b )(ϕ) = ϕ(x, a + b)d θ(x, a); for ϕ ∈ C c (S × R), θ translation-bounded we write |θ| ϕ = sup{|θ * δ b (ϕ)|; b ∈ R}. For such measures and any bounded measure r on R, θ * r is well defined by (θ * r )(ϕ) = (θ * δ b )(ϕ)d r (b) and we have |θ * r | ϕ ≤ |r |θ ϕ where |r | is the total variation of r . In particular, f ∈ L 1 (R) can be identified with the measure r f = f (a)d a, we can define θ

xa

  E x (|U 1 | pp ′ ) < ∞ for pp ′ < δ and Lemma 5.19. If we take c > q ′ = p ′ p ′ -1 , we see that lim n→∞ π x n (ω1 A ′ n ) = 0 uniformly, hence condition 2 is satisfied.

n→∞ π x n * f 1

 1 ,ω = 0 uniformly. By density, the same relation is valid for all f ∈ L 1 ω (R). Now, let us choose p ∈]1, δ[ and ω = ω p . Since θ is translation-bounded we can assume θ to be non negative and translation-bounded. Then Lemma 5.20 gives for any r as in the lemma,θ * r ≤ (1 S ⊗ ω)(θ ⊗ ℓ). Taking r = r n as an approximate identity we have θ = lim n→∞ θ * r n in the weak sense. Hence we can assume θ ≤ (1 S ⊗ ω)(θ ⊗ ℓ) where θ is a bounded measure on S. Let f ∈ L 1 ω (R) ∩ J c (R), u ∈ H ε (S) be as above, hence f , u satisfy for every n ∈ N the following relationsθ(u ⊗ f ) = (P n θ)(u ⊗ f ) = θ(P n (u ⊗ f )) = P n (u ⊗ f )(x, a)d θ(x, a), |θ(u ⊗ f )| ≤ d θ(x) |P n (u ⊗ f )(x,a)|ω(a)d a ≤ θ sup x P n (u ⊗ f )(x, •) 1,ω .

Proposition 5 . 21 .

 521 For any u ∈ S d-1 , lim t→∞ t α P{|〈R, u〉| > t } = C p(α) α * e α (u) > 0 .In cases I, for any u ∈ S d-1 :lim t→∞ t α P{〈R, u〉 > t } = 1 2 C p(α) α * e α (u) > 0 .In case II, for any u∈ Λ + (T * ), if Λ a (Σ) ⊃ Λ ∞ + (T ): lim t→∞ t α P{〈R, u〉 > t } = p(α) α C + * e α (u) > 0.

t>0t

  -α (t • η)(K ) ≤ C (K ). Hence sup t>0 t • ((e α ⊗ h α )η)(K ) ≤ C ′ (K ) with C ′ (K ) > 0.

  t→0 + t -α I t (ϕ) = 0 where I t (ϕ) = (t • ρt .(µ * ρ))(ϕ). By definition I t (ϕ) = E(ϕ(t R) -ϕ(t A 1 Ro θ)) with ϕ(t R) = 0 if |t R| < δ and ϕ(t A 1 Ro θ) = 0 if |t A 1 Ro θ| < δ. Hence, I t (ϕ) ≤ [ϕ] ε t ε E(|B 1 | ε 1 {|tR|>δ} + |B 1 | ε 1 {t|A 1 Ro θ|>δ} ).

t→0 + I 2 t

 2 = 0. Hence lim t→0 + t -α I t (ϕ) = 0. By definition of η we have, for any g ∈ G,{lim t n →0 + t -α n (t n • (g ρ)(ϕ) = (g η)(ϕ). Furthermore we have |ϕ(x)| ≤ |ϕ|1 { x ∈ V ; |x| ≥ δ} and, |(g η)(ϕ)| ≤ |ϕ|η{x ∈ V ; |g x| ≥ δ} ≤ |ϕ| lim n→∞ t -α n P{|R| > δ |g |t n }. Using Corollary 5.8 we get |(g η)(ϕ)| ≤ b δ α |ϕ||g | α . Since |g | α d µ(g ) < ∞ and for any g ∈ G, lim t n →0 + t -α n (t n •(g ρ)(ϕ) = g η(ϕ), we have by dominated convergence lim t n →0 + t -α n (t n •(µ * ρ)(ϕ) = (µ * η)(ϕ). Then the property lim t→0 + t -α I t = 0 implies (µ * η)(ϕ) = η(ϕ), hence µ * η = η. Then for any f ∈ B ε,α lim t→0 + t -α (t • ρ)( f ) = Λ( f ).The proof depends of two lemmas in which we will use the norm v = sup 1≤i ≤d |〈x, e i 〉| instead of |v | where e i (1 ≤ i ≤ d ) is a basis of V . Also for δ > 0 and 0 < δ 1 < δ 2 we writeB δ = {v ∈ V ; v ≤ δ}, B δ 1 ,δ 2 = B δ 2 \ B δ 1 , B ′ δ = V \ B δ . Lemma 5.27. For any f ∈ B ε,α , 0 < δ 1 < δ 2 ,

1 .||

 1 ′ ε (x)| by (α + 1 + ε)x α-1 | log x| -1-ε for x ≤ e -Proof of proposition 5.26. For δ ≥ e and with D > 0 we haveB ′ δ ϕ ε ( v )d Λ(v ) ≤ D log δ| ε hence lim δ→∞ B ′ δ ϕ ε ( v )d Λ(v ) = 0. Also for δ < 1, B ′ δ ϕ ε ( v )d Λ(v ) ≤ D log δ| ε , hence lim δ→∞ B δ ϕ ε ( v )d Λ(v ) = 0.Then the Proposition follows from the lemmas.Proof of Theorem C. Except for the last assertion, Theorem C is a direct consequence of Theorem 5.2. The last assertion is the content of Proposition 5.26.

Lemma A. 1 .

 1 , p.233). Assume A(x) is non negative, increasing on [1, ∞[, f (s) = ∞ 1 x -s-1 A(x)d x is finite for s > 1, f extends as a function f meromorphic in an open set D ⊃ {0 < Rez ≤ α} and f has only a possible unique simple pole at z = α, with lim s→α - (αs) f (s) = A; then one has lim x→∞ x -1 A(x) = A. The use of this result (see Lemma A.1 below)will give the tail-homogeneity of ρ. On the other hand, if β denotes the convergence abcissa of the Mellin transform f(s) = ∞ 0 x s d ν(x),a lemma of E. Landau (see[START_REF] Widder | The Laplace transform[END_REF] p. 58) says that f cannot be extended holomorphically to a neighbourhood of β. This will allow us to show A > 0. Let ν be a probability on[1, ∞[, α > 0 such that f (x) = ∞ 1 x s d ν(x) is finite for s < α, f(s) extends to an open set D ⊃ {0 <Rez≤ α} as a meromorphic function f which has a simple pole at z = α with residue A > 0. Then one has lim x→∞ x α ν(x, ∞) = α -1 A.

x→∞ x - 1 A

 1 (x) = α -1 A.Then one can apply the Tauberian Lemma 5.4 to the decreasing function ν(x, ∞) and gets limx→∞ x α ν(x, ∞) = α -1 A.

1 k

 1 (s) P z is conjugate to the operator Q z considered in Corollary 3.21. From this corollary we deduce that Q α satisfies a Doeblin-Fortet condition and r (P α+i t ) = r ( Q α+i t ) < 1 if t = 0.On the other hand the function z → P z is holomorphic in the set {0 < Rez < s ∞ } since for any loop γ in this set we have γ P z d z = ϕ(g •x)d µ(g ) γ |g x| z d z = 0. It follows that there exists ε > 0 such that for |z -α| < ε there exists a holomorphic function k(z) such that k(z) is a simple dominant eigenvalue of P z with k(z) = 1+k ′ (α)(z-α)+•(z-α).

s→∞E(

  |B| s ) 1/s = d < ∞; also Proposition 4.11 gives lim s→∞ k(s) 1/s = c > 1. Hence lim s→∞ ( * ν s ( fs )) 1/s ≤ (c -1) -1 d < ∞.

  1/n , which implies I µ = {s ≥ 0; |g | s d µ(g ) < +∞}. Furthermore, Hölder inequality implies that I µ is an interval of the form [0, s ∞ [ or [0, s ∞ ], and logk µ (s) is convex on I µ . Also k µ

* = k µ since |g | = |g * |. If µ and µ ′ commute and c∈ [0, 1],

  ). Below we study existence and uniqueness for eigenfunctions or eigenmeasures of P s . We show equicontinuity properties of the normalized iterates of P s and P s .If * ν s ∈ M 1 (P d-1 ) satisfies * P s ( * ν s ) = k(s) * ν s ,and e s is normalized by ν s (e s ) = 1, one has p(s)e s (x) = |〈x, y〉| s d * ν s (y) where p(s) = |〈x, y〉| s d ν s (x)d * ν s (y).

	Theorem 2.6. Assume µ ∈ M 1 (G) is such that the semigroup [suppµ] satisfies (i-p) and let s ∈ I µ . If d = 1 we assume that µ is non arithmetic. Then the equation P s ϕ = k(s)ϕ has a unique continuous solution ϕ = e s , up to normalization. The function e s is positive and s -Hölder with s = inf(1, s). Furthermore there exists a unique ν s ∈ M 1 (P d-1 ) such that P s ν s is proportional to ν s . One has P s ν s = k(s)ν s and suppν s = Λ([suppµ]). The map s → ν s (resp s → e s ) is continuous in the weak topology (resp uniform topology) and the function s → log k(s) is strictly convex.

  ). Also, by convexity, k(s) is continuous. On the other hand, the uniform continuity of (x, s) → |g x| s and the fact that |g x| s ≤ |g | s is bounded by the µ-integrable function sup(|g | s 1 , |g | s 2 ) on [s 1 , s 2 ] ⊂ I µ implies the uniform continuity of P s ϕ if ϕ is fixed. Then we consider a sequence s n ∈ I µ ,

	s 0 ∈ I µ with lim s n →s 0

  Proof of Theorem 2.16. As observed in remark 2 after Proposition 2.13, if s > 0 for any ϕ

we know that ν is proper, hence lim t→0 ν(∆ t x ) = 0. Then, for x fixed we use the above estimation of lim sup n→∞ δ ε (g •x, g •y)d µ n (g ) to choose t sufficiently small in order to get the continuity of lim sup n→∞ δ ε (g •x, g •y)d µ n (g ).This gives that | P n ϕ(x)-P n ϕ(y)| depends continuously of y, hence the equicontinuity of the sequence P n ϕ.

  It follows that |u n x| ∼ a 1 n |〈z * , x〉|. Since |u n | = a 1 n , we get lim , x〉|, as asserted. We get also, if |〈y, z * 〉| = 0, |u n y| ∼ a 1 n |〈z * , y〉|. On the exterior product space ∧ 2 V there exists an O(d )-invariant scalar product such that on any decomposable 2-vector x ∧ y :

		1 n ) and
	lim n→∞	|〈k ′ n x, e 1 〉| = |〈z n→∞	|u n x| |u n |	= |〈z

* , x〉| = 0. *

  hence the result with D = B + The structure of the space of Q s -invariant functions is given by Theorem 2.16. It follows from[START_REF] Tulcea | Théorie ergodique pour des classes d'opérations non complètement continues[END_REF] that 1 is an isolated spectral value of Q s and the corresponding projection has finite rank. The same argument as in the proof of Corollary 3.20 gives the analyticity of this projection. Doeblin-Fortet inequality implies that the possible unimodular spectral values of Q z are eigenvalues. Then, as in the end of proof of Theorem 2.7, one would have for some ϕ ∈ H

	2 c(s) δ ε 0	.

ε (S d-1

), e i θ ∈ C, and any g ∈ suppµ, |g x| i t ϕ(g • x) = e i θ ϕ(x). This would contradicts Proposition 2.5 if t = 0.

The last assertion is a direct consequence of Corollary 2.19.

Proof of Theorem A. The spectral decomposition P s = k(s)(ν s ⊗ e s +U s ) is part of Corollary 3.19. The analyticity of k(s) and ν s ⊗ e s on ]0, s ∞ [ is stated in Corollary 3.20. The strict convexity of log k(s) is stated in Theorem 2.6. The fact that the spectral radius of P z is less than k(s) follows from the corresponding assertion for Q z in Corollary 3.20.

  3.10 is satisfied, we use the corresponding notations, and C k is as in Definition 4.1 . We consider the case S = P d-1 . Using Theorem 3.10 for s = 0, we get for any fixed x limWe observe that for any x, y∈ S d-1 , S n y| -|S n x ≤ |S n | δ(x, y) ≤ 2|S n |. It follows

	Proof. |S n y| |S n x|	-1 ≤ 2	|S n | |S n x|	, log|S n y| -log |S n x| ≤ 2	|S n | |S n x|	.
	Using Theorem 3.2, we get that the sequence |S n | |S n x| converges P -a.e. to hence the sequence 1 n |S n | |S n x| converges P -a.e. to zero. It follows that 1 n log |S n x| -2 1 |〈z * (ω),x〉| < ∞, n |S n | |S n x| converges P-a.e. to L µ .
	Lemma 4.6. For k large we have C k = S.

n→+∞ log |S n (ω)x| n = L µ , Pa.e..

  1 and the finiteness of E α (|p| ε ) follows. By definition, p( ω, u) satisfies the functional equation p • a θ = ap +b where p and (a, b) are independent. It follows that the probability measure κ on M × R given by the formula

  • n and p • n are the same. Since the product of π α

	with the law of y • n is ( * Q α ) n (π α M ⊗δ 0 ) we have in weak topology : lim n→∞	M

  for some t > 0, i.e. the * Q α -invariant set suppκ is contained in M×] -∞, t ]. Then, for any (u, p) ∈ suppκ we have p+ 〈R n , u〉 ≤ t |S ′ n u| * Q α u -a.e., i.e. p + 〈R n , u〉 ≤ t |S ′ n u| * q α n (u, •)λ ⊗n -a.e. for any n ∈ N and for some p ∈ R, t , u ∈ M. It follows p + 〈R n , u〉 ≤ |S ′ n u|, λ ⊗n -a.e. and since lim

	n→∞

  the vague topology. On the other hand, if r has compact support and θ is a Radon measure on R, θ * r is well defined as a Radon measure.Proof. For simplicity of notation, assume r > 0 on [0, 1]. We denote byθ k * δ k the restriction of θ to S × [k, k + 1[ (k ∈ Z) and we write θ = Σ k∈Z θ k * δ k with supp θ k ⊂ S × [0, 1].We observe that, since θ is translation-bounded, the mass of θ k is bounded for k ∈ Z, hence. + |k|) p δ k ).But, by definition of ω p and since suppr is compact, we have r * Σ + |k|) p δ k ≤ cω p for some c > 0 and it follows, θ * r ≤ c

	Lemma 5.20. With the above notation, assume that θ is a translation-bounded non neg-ative Radon measure on S × R. Let r be a non negative continuous function on R with compact support containing 0. Then for p > 1, there exists a non negative bounded meas-k∈Z (1+|k|) -p θ k is a bounded measure supported on S ×[0, 1]. We have clearly θ k ≤ (1 + |k|) p . θ, θ ≤ . θ * ω p . We desintegrate the bounded meas-ure . θ as . θ = δ x ⊗ . θ x d θ(x) where θ is the projection of . θ on S and . θ x is a probab-ure θ on S such that θ * r ≤ (1 θ = Σ ility measure. Hence

S ⊗ ω p )(θ ⊗ ℓ). . θ * Σ k∈Z (1 + |k|) p δ k , θ * r ≤ . θ * (r * Σ k∈Z (1 k∈Z (1 . θ * ω p = δ x ⊗ ( . θ x * ω p )d θ(x). But, since .

θ x is supported on [0, 1], we have .

  Proof. With the notation of Lemma 5.12, we have ψ α = P τ α ) k ψ α τ where * P τ α is a fibered Markov kernel on X + ×R * + which satisfies the conditions of Proposition 5.14 and ψ α τ is bounded by ( * e α ⊗h α ) -1 . Hence, as in the proof of the proposition, if ψ

	function with compact support, bounded by ψ α τ ,then lim t→∞ and, using Proposition 5.21, since lim t→∞ ψ α (v, p) is constant on X + ,we get ψ α (v, p) ≥ lim sup t→∞	α τ is a Borel ∞ Σ * P τ α ψ α τ (v, p), 0
	lim t→∞	ψ α (v, p) ≥	1 γ α τ ]0,∞[×X +	ψ α τ (v ′ , p)d κ τ (u ′ , p)	d t t	.
	Hence, approximating from below ψ α τ by ψ	α τ , we have
	lim t→∞	ψ α (v, p) ≥	1 γ α τ ]0,∞[×X +	ψ α τ (v ′ , p)	d t t	d κ τ (u ′ , p),
	C (u) = lim
			γ α τ	]0,∞[×X +	ψ α τ (v
	T ) we have
	C + ≥	α p(α)γ α τ ]0,∞[×X +	ψ α τ (v, p)t -1 d κ τ (u, p)d t > 0.

∞ Σ 0 ( * t→∞ t α P{〈R, u〉 > t } = * e α (u) lim t→∞ ψ α (v, p), C (u) ≥ * e α (u) ′ , p)t -1 d t d κ τ (u ′ , p).

* Q α ϕ(u, p) = ϕ(g * • u, h u p) * q α (u, g )d λ(h)

Λ([suppµ]) in V . Also we denote

As shown below these closed sets support the limits (v → 0) of H(v, .) and H(v, •). The function e s (resp measure ν s ) defined in Theorem 2.2 plays an essential role for s = 0, α in the following theorems.

The results will take two forms according as L µ > 0 or L µ < 0.

where ν ∈ M 1 (Λ([suppµ])) is the unique P -invariant measure on P d-1 . This convergence is valid on any bounded continuous function f which satisfy on V ,

Furthermore the ladder kernel H satisfies the following weak convergence,

where χ is defined by this convergence and is absolutely continuous with respect to ν ⊗ ℓ

Proof. In view of the verifications of conditions I in subsections 2, 3 and of Lemma 4.6, this is a direct consequence of Theorem 4.5 applied in the case

If L µ < 0 the following gives the asymptotic behaviour (v → 0) of the potential measure ∞ 0 µ k * δ v ; this asymptotics allow us to obtain a Cramér estimate for the random variable

Proof. In case I, the proof is the same as for Theorem 4.7 with S = S d-1 instead of P d- 1 . In case II, we take S = Co(Λ + ([suppµ]) and we observe that S × R * + is a suppµ-invariant convex cone with non zero interior to which, as in the proof of Theorem 4.7, we can apply Theorem 4.5 .

If u ∈ S (resp u ∈ -S) we have lim We denote also by U (v, •) = ∞ 0 µ k * δ v the potential kernel of the linear random walk

Clearly the kernel p + (x, d v ) commutes with the scaling x → t x (t > 0). Then it follows from above that, on C c (Φ ∪ -Φ) :

Finally we have lim

The existence of χ u follows from the first formula in Theorem 4.5 . In particular the right hand side of this formula is independent on u ∈ S. Hence, in case I, χ u is independent on u. In case II, we use S = Co(Λ + ([suppµ])) and we argue as above in order to obtain the formula

We have also the following analogue of Theorem 4.8. The proof is a combination of the arguments in the proofs of Theorems 4.8 and 4.7.

Theorem 4.10. Assume µ and α are as in Theorem 4.8. Then for any u ∈ S d-1 we have the vague convergence : lim

There are 2 cases like in Theorem 4.9.

Then suppρ is unbounded and we have the following vague convergence on V \ {0} lim

where C > 0, σ α ∈ M 1 ( Λ(T )) are defined by the above formula and the measure

In case I, we have σ α = ν α .

In case II', there exist C

In case I the Radon measure ν α ⊗ ℓ α on V \ {0} is a minimal µ-harmonic measure, and Λ is symmetric.

In cases II, ν α

+ ⊗ ℓ α and ν α -⊗ ℓ α are minimal µ-harmonic measures on V \ {0}.

Theorem 5.2 is proved in several steps, using the function ρ on V ,

A first step, based on the renewal theorems of section 4, shows the existence of the directional tails (see Corollary 5.8) i.e. existence of the limit lim t→∞ t α ρ(t -1 u). A second step is to study the positivity of these tails (see Proposition 5.9). It is based on Kac's recurrence theorem (see [START_REF] Wolfowitz | Remarks on the notion of recurrence[END_REF]) for an associated random walk on a dual H-homogeneous space of V (see Lemma 5.13); this recurrence property allow us to express ρ as a potential of a non negative function on V ×R (see Lemma 5.12), to which a weak renewal theorem can be applied (see Proposition 5.14). The action of H on affine hyperplanes of V leads us, via Radon transforms, to consider the natural linear representation of H in the vector space V × R. The corresponding linear λ-random walk is studied in paragraph 3 below. Finally the homogeneity at infinity of ρ follows from a Choquet-Deny type result (see Theorem 5.17).

Asymptotics of directional tails

We apply Theorem 4.10 to µ * -potentials of suitable functions; we pass, using the map η → η, from the convolution equation λ * ρ = ρ to a Poisson type equation on V \ {0} which involves µ * and ρ and we note that t α (t -1 •ρ)(H + u ) = t α ρ(u, t ). The corresponding convergences will play an essential role in the proof of Theorem 5.2. We denote by ρ 1 the law of R -B and we consider the signed measure ρ 0 = ρρ 1 , hence ρ 0 (V ) = 0. Also

Proof. By definition

Lemma 5.5 implies that h α (v ) is b.R.i and has limit 0 at |v | = ∞. Integration by parts in the above formula gives

Proof. In order to show that suppρ is unbounded, in view of Proposition 5.1, it suffices to show that there exists g ∈ T with lim

If not, then the trace Tr g of g is bounded by d on T . On the other hand, condition i-p implies the irreducibility of the action of T on V ⊗ C, as shown now. Let W ⊂ V ⊗ C be a proper T -invariant subspace of V ⊗C and W its complex conjugate. Then W ∩W and W +W are complexified subspaces of subpaces of V which are also T -invariant. Using irreducibility of T we get

Since λ g is a simple eigenvalue we get w = w i.e W ∩W = {0} which gives a contradiction with condition i-p, hence T acts irreducibly on V ⊗ C. Then Burnside's density theorem implies that EndV ⊗ C) is generated as an algebra by T , i.e. there exists a base g i ∈ T (i = 1, ., d 2 ) of End V. Then the linear forms u → Tr (ug i ) (i = 1, ., d 2 ) form a basis of the dual space of End V. In particular, for some constant c > 0 we have for any

is bounded, right continuous and non increasing. Since equation (S) can be written as

Since L µ < 0 the subadditive ergodic theorem applied to log|S n (ω)| gives the convergence of S n (ω)v to 0. In particular, for ξ ∈ M 1 (V ), the sequence µ n * ξ converges in law to δ 0 , hence lim

From the above convergence on V ,we have

Since the sequence (µ n+1 * (r - * ρ))(ψ) converges to zero for any bounded Borel function

We observe that, for any bounded measure ξ, we have (

Since for fixed u, ρ(u, x) = P{〈R, u〉 > x} is non increasing, Lemma 5.4 gives

In particular, we have * ν α u (r α ) ≥ 0. In case I this gives * ν α (r α ) ≥ 0 since * ν α u = * ν α . In case II, taking u ∈ Λ + (T * ) and using * p α + (u) = 1, this gives * ν α + (r α ) ≥ 0. Also, in the same way * ν α -(r α ) ≥ 0. Furthermore, in case II, using Theorem 4.10,

If * ν α (r α ) > 0, in case II we can define a probability measure σ α on Λ(T ) by

= 0, hence we can leave σ α with projection ν α on P d-1 undefined in the above formulae. In any case σ α ⊗ ℓ α is µ-harmonic.

We get another expression for the above limit, by using the formulae for * e α (u), * p α + (u), * p α -(u), p(α) of section 2 paragraph 3, for case II as follows (see Theorem 2.16), with

From above, we get * e α (u

Hence, with C = 2 * ν α (r α )α L µ (α)p(α) , σ α as above and

In case I we get the corresponding formula. The fact that suppρ is unbounded follows from Lemma 5.7, since k ′ (α) > 0 and k(α) = 1.

Corollary 5.8. For any v ∈ V \ {0}, we have

With the above notation the λ-random walk (v n , r n ) on E ⊂ V × R can be written as

We denote X = M × R and

We consider also the submarkovian stopped operator * P τ on V \ {0} × R. In (u, p, r ) coordinates on the set {r = 0} with x = (u, p) the associated process at time n starting at (x, r

), hence the restriction of * P τ to X + × R * + is well defined and is a measurable fibered kernel on X + ×R * + ⊂ E . This restriction will be again denoted by * P τ . Since * P ( * e α ⊗ h α ) = * e α ⊗ h α and τ is finite * Q α κ -a.e. , the kernel * P τ α given by : * P τ α ϕ = ( * e α ⊗ h α ) -1 P τ ( * e α ⊗ h α ϕ), is also a measurable Markov fibered kernel on X ×R * + which satisfies * P τ α 1 = 1, κ ⊗ℓ a.e. The following lemma expresses the function ρ(p -1 v ) = P{p -1 〈R, u〉 > t } = ψ(v, p) on E as a * P τ -potential of a non negative function on E . Lemma 5.12. With t v = u ∈ S d-1 , t > 0 and p = 0, we write

Proof. We write

By definition of τ, since p -1 〈R, u〉 > t > 0 and the convergence of R n to R imply τ < ∞, we have

On the other hand, since p -1 p τ > 0,

Proof. Since Λ * a (Σ) ⊃ M, Lemma 5.11 gives κ(X + ) > 0. In order to deal only with positive values of the R * -valued Z-cocycle W n (u, p, ω) = |S ′ n u|p n p -1 under * Q α κ , it is convenient to consider the two sided Markov chain x n k (k ∈ Z, x • ∈ X + ) induced on X + by x n , hence n 1 is the first return time of x n to X + and n 1 ≤ τ since p -1 p τ > 0. We note that the normalized restriction κ + of κ is a stationary ergodic measure for x n k . Also the relativized Markov kernel * P α on E = X × R * + induces a fibered measurable Markov kernel * P α,+ on X + × R * + with projection * Q α + on X + , which satisfies * P α,+ (κ + ⊗ ℓ) = κ + ⊗ ℓ. Since p -1 p n k > 0,in(u,p,r) coordinates, the corresponding bilateral Markov chain can be written as (x n k , r n k ) with r n k = r p -1 p n k |S ′ n k u|. We denote by Ω # + the subset of Ω # defined by the conditions x ∈ X + , x n ∈ X + infinitely often for n > 0 and n < 0, by κ # + the normalized restriction of κ # to Ω # + and by a θ + the induced shift. Also let Ω # 0 be the subset of Ω # + defined by the conditions x ∈ X + , sup

. We have Ω # 0 = {ν 0 = 0} and, using Lemma 5.11, lim k→∞ V -k = 0.

It follows,using the (

where q = κ # + (Ω # 0 ), hence κ # (Ω # 0 ) > 0. Furthermore if ω # ∈ Ω # 0 ,using the positivity of V k and an observation of [START_REF] Kesten | Renewal theory for functionals of a Markov chain with general state space[END_REF],we see that τ(ω # ) is the first return time of ( a θ) n (ω # ) to Ω # 0 ; hence a θ τ is the transformation on Ω # 0 induced by a θ or a θ + on Ω # 0 and τ n (ω # ) is the sequence of return times to Ω # 0 . This allow us to proceed as in ( [START_REF] Kesten | Renewal theory for functionals of a Markov chain with general state space[END_REF] Lemma 2) with the R * -valued Z-cocycle |S ′ n u| p n p -1 . Since κ # is a θ-invariant and κ # (Ω # 0 ) > 0 we can apply Kac's recurrence theorem to Ω # 0 , a θ τ and Ω # (see [START_REF] Wolfowitz | Remarks on the notion of recurrence[END_REF]), hence the normalized restriction κ # 0 of κ # to Ω # 0 is a θ τ -ergodic and stationary, the return time τ has finite expectation E α 0 (τ) and lim n→∞ τ n n = E α 0 (τ), κ # 0 -a.e. . Since κ # 0 is absolutely continuous with respect to κ # , Theorem 3.10 gives

Proof. As in the proof of Corollary 5.8, we observe that the condition η(

Hence, as in Lemma 5.25, for any compact K ⊂ V \ {0}, with η α = (e α ⊗ h α )η, we have

It follows that η α is dilation-bounded.

We recall that P (resp P ) is the convolution operator by µ on V \ {0} (resp V ), hence P (σ ⊗ ℓ α ) = σ ⊗ ℓ α and P (e α ⊗ h α ) = e α ⊗ h α . We denote by Qα the Markov operator on V deduced from P by Doob's relativisation with respect to e α ⊗ h α . On the other hand,

We observe that the fibered Markov operator Qα satisfies condition D of subsection 5.4, in view of Corollary 3.20 and of the moment condition on A.

Then Theorem 2.6 implies σ = ν α . Also, in view of Corollary 3.20 for s = α and the above observations, we can apply the second part of Theorem 5.17 to ηα with P = Qα , hence ηα is proportional to π α ⊗ ℓ, i.e. η is proportional to ν α ⊗ ℓ α . Since σ = ν α and η(H

The projections λv and ηv on V satisfy the same equation, hence are equal. As in section 3, we get that the sequences of Radon measures

On V we get, using Theorem 3.2, for some z(ω)

Let z + (ω) and z -(ω) be opposite points on S d-1 with projection z(ω) on P d-1 . The martingale convergence on V \{0} gives that

Hence, taking expectations we get η = σ ⊗ ℓ α .

Proof of theorem 5.2. The convergence of

and the positivity properties of C ,C + ,C -follows from Corollary 5.21. For the vague convergence of ρ t we observe that Lemma 5.25 gives the vague compactness of ρ t . If η = lim

hence the vague convergence of ρ t to Λ. The detailed form of Λ follows from Proposition 5.3.

For the final minimality assertions one uses the second part of Theorem 5.17 and we replace η by (e α ⊗ h α )η. We verify condition D for Q α or operators associated with Q α as follows. Part 1 of condition D follows directly from Corollary 3.21. In case I, Q α satisfies part 2 of condition D by Corollary 3.21, 1 is a simple eigenvalue of Q α and, if θ is positive Radon measure with µ * θ = θ, θ ≤ ν α ⊗ℓ α , hence θ is proportional to ν α ⊗ℓ α . In case II, one restricts Q α to the convex cone generated to Λ + (T ), so as to achieve the simplicity of 1 as an eigenvalue of Q α and the absence of other unimodular eigenvalue. Then Corollary 3.21 shows that part 2 of condition D is satisfied for the corresponding operator, hence the above argument is also valid for ν α + ⊗ ℓ α . Part 3 of condition D follows from the moment conditions assumed on λ.

Remark. In general supp(σ

∞ is smaller than Λ ∞ a (Σ) and suppσ α has a fractal structure.

In the context of extreme value theory for the process X n , the convergence stated in the theorem plays a basic role and implies that ρ has multivariate regular variation.

Actually, using the properties of Λ, [START_REF] Boman | Support theorems for the Radon transform and Cramer-Wold theorems[END_REF] gives also the weak convergence for any α (resp α ∉ 2N) in case II" (resp case I). This is valid too for α ∉ 2N if C + = C -in case II', for example if the law of B 1 is symmetric (see [START_REF] Kluppelberg | Extremal behaviour of models with multivariate random recurrence representation[END_REF]).

For the last assertion in Theorem C we need the following. Proposition 5.26. Let B ε,α be the set of locally bounded Borel functions on V \ {0} such that the set of discontinuities of f is Λ-negligible and for ε > 0,

where C ≥ 0 and

In case I , σ α is symmetric, suppσ α = Λ(T ) and C (u) = (αk ′ (α)) -1 * ν α ( dα ) * e α (u).

Proof. We write equation (S) of section 5 in the form: R -B = AR • θ.

Then equation (S) implies: 〈R -B, v

. We write a continuous z-homogeneous function f on V \ {0} as f = f ⊗ h z with f ∈ C (S d-1 ), and we recall that, as in section 2, * P f = * P z f ⊗ h z . Then, since f z and d z are z-homogeneous and continuous, equation (S) gives, (I - * P z ) fz = dz .

For u ∈ S d-1 and ε(z

Hence using Hölder inequality and the moment hypothesis we get that for u fixed, dz In case II, lim

Using the expressions of * e α (u), * e α + (u), * e α -(u) given by Theorem 2.16 we obtain in the two cases, lim

In case I, we see that σ α is symmetric and suppσ α = Λ(T ).

In cases II, the detailed expression of σ α shows that

Here, the estimation of dz in the proof of Proposition A.3 gives the analyticity of * ν z ( dz ) in an open set containing ]0, α + δ[ where νz is defined in the proof of Lemma A.2 by perturbation. Since C (u) = 0, the function fz considered in the proof of Proposition A.3 is meromorphic with no pole at α, hence Landau's lemma mentioned above gives that the convergence abcissa of x s d ρ u (x) is larger than α and fs (u) < ∞ if 0 < s < α + δ.

Now we consider the case s ≥ α + δ. We observe that in case I, the density hypothesis on µ implies that supp * ν s = supp * ν = S d-1 and the compactness hypothesis of suppλ implies that * P z defines a compact operator on C (S d-1 ). Let ∆ ⊂ {Rez > 0} be the set where I - * P z is not invertible and observe that, since r ( * P z ) < 1 if Rez ∈]0, α[, we have ∆ ∩ {0 < Rez < α} = φ. Since the function z → * P z is holomorphic, the extension of Riesz-Schauder theory given in [START_REF] Kaneko | A generalisation of the Riesz-Schauder theory[END_REF] implies that ∆ is discrete without any accumulation point and (I - * P z ) -1 is meromorphic in the domain {0 < Rez}, with possible poles in From above we know that this limit is zero, hence * ν β (ϕ m ) = 0. Since ϕ m is non negative we get ϕ m (u) = 0 * ν βa.e and the continuity of ϕ m implies ϕ m (u) = 0 for u ∈ supp * ν β = S d-1 . By induction we get ϕ j = 0 for any j ≥ 1, hence the function z → fz is holomorphic in a domain D ε which contains ]0, β + ε[ for some ε > 0 depending on the possible poles of (I - * P z ) -1 ( dz ), in R * + . Then, as above, Landau's lemma gives that fs (u) is finite for s < β + ε and u ∈ S d-1 . Hence E(〈R, u〉 s + ) is finite for u ∈ S d-1 , s < β+ε and this gives the required contradiction. Then we have for s > 0: (1 -k(s)) * ν s ( fs ) = * ν s ( ds ). We observe also that (I - * P z ) -1 ( dz ) is well defined and holomorphic in a domain which contains ]0, ∞[.