Hui Zhu

Zicheng Qian

Ruxi Shi

Shengyuan Zhao

Linyuan Liu

Hua Wang

J'adresse mes sincères remerciements à Jean-François Quint, mon directeur de thèse, pour sa grande disponibilité, ses nombreux conseils et son soutien tout au long de ma thèse. Son optimisme dans la recherche m'a beaucoup influencé.

Introduction en français

Cette thèse est constituée de quatre articles de recherche. La première partie va être publiée dans Mathematische Annalen et la dernière a été soumise.

Soit V une représentation irréductible de dimension finie du groupe de Lie G = SL m+1 (R). Soit X = PV l'espace projectif réel de V , qui est l'ensemble des droites vectorielles de V . Alors, nous avons une action du groupe G sur X. Soit µ une mesure de probabilité borélienne sur G et soit Γ µ le sous-groupe engendré par le support de µ. Nous dirons que µ est Zariski dense si Γ µ est un sous-groupe Zariski dense de G. Cela signifie que la mesure µ n'est pas concentrée sur un sous-groupe algébrique de G. Nous pouvons définir sur X une marche aléatoire associée à µ. Fixons un point x de X. Lors de chaque étape, nous nous déplaçons vers un point aléatoire gx, où g est un élément aléatoire de G, de loi µ. D'après un théorème de Furstenberg, cette marche aléatoire admet une unique mesure stationnaire ν sur X, appelée la mesure de Furstenberg ou la mesure µstationnaire. Cela signifie que la mesure ν vérifie ν = µ * ν := G g * νdµ(g), où g * ν est l'image de ν par l'action de g sur X. Cette mesure a été introduite par Furstenberg pour établir la loi des grands nombres pour les produits de matrices aléatoires. Les propriétés de la mesure µ-stationnaire sont importantes dans d'autres théorèmes limites pour les produits de matrices aléatoires.

Il y a beaucoup d'exemples intéressants de mesures µ-stationnaires. Nous nous restreignons à la basse dimension, c'est-à-dire au cas où G = SL 2 (R) et X = P(R 2 ), la droite projective réelle. Soit Γ un réseau de SL 2 (R) (par exemple SL 2 (Z)). Furstenberg a construit des exemples où µ est une mesure portée par Γ et la mesure stationnaire ν est exactement la mesure de Lebesgue de X. Cette construction a été utilisée dans le travail de Furstenberg sur le bord de Poisson, qui a donné des propriétés de rigidité des réseaux. Récemment, les gens se sont intéressés aux propriétés de dimension et d'absolue continuité de la mesure µ-stationnaire ν quand µ a un support fini. Voir [START_REF] Bárány | Stationary measures for projective transformations: the Blackwell and Furstenberg measures[END_REF] et [START_REF] Bourgain | Finitely supported measures on SL 2 (R) which are absolutely continuous at infinity[END_REF] pour des exemples de mesures stationnaires absolument continues et [START_REF] Hochman | On the dimension of Furstenberg measure for SL 2 (R) random matrix products[END_REF] pour des exemples de mesure stationnaires de dimension totale.

Mentionnons aussi une autre classe de mesures, les convolutions de Bernoulli. Soient X 0 , X 1 , . . . des variables aléatoires i.i.d telles que P(X 0 = 1) = P(X 0 = -1) = 1/2. Soit ν λ la convolution de Bernoulli de paramètre λ ∈ (0, 1), qui est définie comme la distribution de la variable aléatoire j≥0 X j λ j . Elle peut être vue comme une mesure stationnaire sur R pour l'action d'un groupe résoluble. Voir l'exemple 1.1.6. Des auteurs se sont intéressés à la dimension et la régularité des convolutions de Bernoulli. Il y a beaucoup de travaux récents dans ce cadre. Voir par exemple [START_REF] Shmerkin | Absolute continuity of self-similar measures, their projections and convolutions[END_REF], [START_REF] Hochman | Self similar sets, entropy and additive combinatorics[END_REF] et [START_REF] Péter | Recent progress on Bernoulli convolutions[END_REF].

Avant d'énoncer notre question principale, nous introduisons une autre propriété de la mesure stationnaire. Nous aurons besoin d'une hypothèse de moment exponentiel fini, c'est-à-dire qu'il existe strictement positif tel que G g dµ(g) < +∞. Dorénavant, nous supposons toujours que notre mesure µ est Zariski dense avec un moment exponentiel fini. Rappelons que X = PV . Guivarc'h a établi la régularité höldérienne des mesures stationnaires. Cela signifie qu'il existe des nombres C, c > 0 tels que, pour tout r > 0, le r-voisinage de tout hyperplan de X a une ν-mesure inférieure à Cr c . Cela implique que la mesure stationnaire ν a une dimension positive. Cela implique aussi que ν n'est pas concentrée sur un hyperplan, ce qui est raisonnable vue l'hypothèse de Zariski densité de µ.

Décroissance de Fourier

Notre problème principal ici est la décroissance de Fourier de la mesure stationnaire. Considérons d'abord l'exemple G = SL 2 (R) et X = P(R 2 ). Fixons l'identification de P(R 2 ) avec le cercle T R/πZ, donnée par l'action transitive du groupe PSO 2 . Nous pouvons alors définir les coefficients de Fourier de la mesure stationnaire ν par ν(k) = (0.0.1) En d'autres termes, les coefficients de Fourier de la mesure stationnaire ont une décroissance polynomiale. Par un argument général, la décroissance polynomiale des coefficients de Fourier implique la régularité de Guivarc'h. En réalité, la régularité est un ingrédient essentiel de la démonstration. La décroissance de Fourier pour des mesures reliées à l'algorithme des fractions continues a été étudies par Kaufman [START_REF] Kaufman | Continued fractions and Fourier transforms[END_REF], Queffélec-Ramaré [START_REF] Queffélec | Analyse de fourier des fractions continues à quotients restreints[END_REF] et Jordan-Sahlsten [START_REF] Jordan | Fourier transforms of gibbs measures for the gauss map[END_REF]. Récemment, la décroissance de Fourier des mesures de Patterson-Sullivan a été démontrée par Bourgain-Dyatlov [START_REF] Bourgain | Fourier dimension and spectral gaps for hyperbolic surfaces[END_REF]. Notre deuxième approche est inspirée par leurs méthodes.

Nous avons deux approches pour ce problème. La première (Chapitre 1) est plus élémentaire, nous utilisons le théorème de renouvellement pour les processus stochastiques. Mais le résultat est plus faible, nous pouvons seulement établir une version qualitative, à savoir |ν(k)| → 0 quand |k| → +∞. Car la décroissance exponentielle dans le théorème de renouvellement n'est pas encore connue. Par la suite, nous établissons l'existence de ce terme d'erreur grâce à notre seconde approche.

La seconde approche (Chapitre 2) est inspirée par la méthode de Bourgain et Dyatlov. L'ingrédient principal, l'estimées sommes-produits, vient de la combinatoire additive. Nous expliquerons cette approche plus loin.

Un exemple intéressant est la mesure de Patterson-Sullivan sur l'ensemble limite d'un groupe fuchsien convexe co-compact. En combinant les méthodes de Connell-Muchnik [START_REF] Connell | Harmonicity of quasiconformal measures and Poisson boundaries of hyperbolic spaces[END_REF] et de Lalley [START_REF] Lalley | Regenerative representation for one-dimensional Gibbs states[END_REF], pour un groupe fuchsien convexe co-compact, nous pouvons trouver une mesure µ telle que la mesure de Patterson-Sullivan soit µ-stationnaire. Grâce à cette observation, nous pouvons retrouver le résultat de Bourgain-Dyatlov sur la décroissance de Fourier des mesures de Patterson-Sullivan. Mais notre vitesse de décroissance est plus faible.

En dimension plus grande, nous considérons la décroissance de la transformée de Fourier dans une carte affine. Soit v 0 un vecteur unitaire de V . Soit v ⊥ 0 le sous-espace vectoriel orthogonal de v 0 dans V . Soit U le sous-ensemble ouvert de PV qui est le complémentaire de l'hyperplan Pv ⊥ 0 . Nous prenons la carte locale affine (ψ, U ) of PV , donnée par

ψ : PV ⊃ U → v ⊥ 0 , Rv → v -v 0 , v v 0 v 0 , v ,
qui est bien définie sur U . L'inverse de ψ est donnée simplement par ψ -1 :

v ⊥ 0 → U ⊂ PV, u → R(u + v 0 ).
Théorème (Theorem 2.1.1). Soit µ une mesure de probabilité borélienne Zariski dense sur SL m+1 (R) avec un moment exponentiel fini. Soit V une representation irréductible de dimension fini de SL m+1 (R). Soit ν la mesure µ-stationnaire sur PV . Soit r une fonction C 1 dont le support est contenu dans U et qui vérifie r ∞ ≤ 1. Alors, il existe > 0 tel que, pour tout ς ∈ v ⊥ 0 de norme ς suffisamment grande, on ait v ⊥ 0 e i ς,u r(u)dν(u) ≤ ς -.

Notre méthode ne permet pas de traiter le cas des groupes de Lie non déployés comme par exemple SL 2 (C). Il serait intéressant d'établir une décroissance de Fourier analogue pour SL 2 (C). Nous pouvons considérer le groupe SL 2 (Q p ) et les mesures stationnaires sur Q p . Il serait aussi intéressant d'établir une décroissance de Fourier semblable pour ce groupe.

Le théorème de renouvellement a été introduit pour la première fois par Blackwell et dans notre situation par Kesten [START_REF] Kesten | Renewal theory for functionals of a Markov chain with general state space[END_REF]. Le résultat principal (dû à Guivarc'h et Le Page [START_REF] Guivarc | Spectral gap properties for linear random walks and Pareto's asymptotics for affine stochastic recursions[END_REF]) est que, quand le temps t tend vers l'infini, la somme de renouvellement Rf (x, t) tend vers 1 σµ f , où σ µ est la constante de Lyapunov définie par σ µ := G×X σ(g, x)dµ(g)dν(x). Par définition, la constante de Lyapunov σ µ est une moyenne de la fonction cocycle σ(g, x) pour la mesure µ ⊗ ν. Le théorème de renouvellement nous donne un phénomène d'équidistribution quand le temps t est assez grand. Dans notre première approche, le théorème de renouvellement est utilisé pour borner la somme de renouvellement Rf pour une fonction f fortement oscillante. Dans notre seconde approche, nous sommes en mesure de donner un terme d'erreur exponentiel.

Théorème (Theorem 2.1.4). Soit µ une mesure de probabilité borélienne Zariski dense sur SL m+1 (R) avec un moment exponentiel fini. Soit V une représentation irréductible de SL m+1 (R). Il existe > 0 tel que, pour

f ∈ C ∞ c (R) et t ∈ R, on ait Rf (x, t) = 1 σ µ ∞ -t f (u)dLeb(u) + O f (e -|t| ),
où O f dépend du support de f et de sa norme de Sobolev.

Ce théorème est à comparer avec le théorème de renouvellement dans R (le cas commutatif). Si µ est une mesure sur R dont le support est fini, le terme d'erreur dans le théorème de renouvellement n'est jamais exponentiel.

Nous espérons que ce type de résultat peut permettre d'obtenir un terme d'erreur exponentiel dans le comptage orbital en rang supérieur. Étant donné un sous-groupe discret Γ de G, nous cherchons une asymptotique pour la croissance de #{γ ∈ Γ| d(γo, o) ≤ R}, où o est le point base de SL m+1 (R)/SO(m + 1). Voir par exemple Lalley [START_REF] Steven P Lalley | Renewal theorems in symbolic dynamics, with applications to geodesic flows, noneuclidean tessellations and their fractal limits[END_REF], Quint [START_REF] Quint | Groupes de Schottky et comptage[END_REF] et Sambarino [START_REF] Sambarino | The orbital counting problem for hyperconvex representations[END_REF]. Ce type de terme d'erreur est toujours relié à une propriété de trou spectral.

Trou spectral

Munissons PV d'une distance riemannienne. Pour γ > 0, soit C γ (PV ) l'espace des fonctions γ-höldériennes. Nous introduisons l'opérateur de transfert qui est un analogue de la fonction caractéristique dans notre cas.

Définition. Pour z dans C avec une partie réelle | z| suffisamment petite, soit P z l'opérateur sur l'espace des fonctions continues donné par (g,x) f (gx)dµ(g), pour x ∈ PV.

P z f (x) = G e zσ
Nous conservons l'hypothèse que µ est une mesure de probabilité borélienne Zariski dense sur SL m+1 (R) avec un moment exponentiel fini. L'utilisation de cet opérateur de transfert pour l'étude des produits de matrices aléatoires a été introduite par Guivarc'h et Le Page. En raison de la propriété de moment exponentiel, l'opérateur P z préserve l'espace de Banach C γ (PV ) pour γ > 0 suffisamment petit. En raison des propriétés de contraction de l'action de G dans X, pour z dans une petite boule centrée en 0, le rayon spectral de P z dans C γ (PV ) est < 1 sauf pour z = 0. En raison de la non-arithméticité de Γ µ , sur l'axe imaginaire, l'opérateur P z a aussi un rayon spectral < 1, sauf en 0. Ces faits ont été utilisé par Le Page et Guivarc'h pour donner des théorèmes limites pour des produits de matrices aléatoires (voir [START_REF] Le | Théorèmes limites pour les produits de matrices aléatoires[END_REF] et [START_REF] Benoist | Random Walks on Reductive Groups[END_REF]).

Théorème (Theorem 2.1.5). Soit µ une mesure de probabilité borélienne Zariski dense sur SL m+1 (R) avec un moment exponentiel fini. Soit V une représentation irréductible de SL m+1 (R). Pour tout γ > 0 suffisamment petit, il existe δ > 0 tel que, pour tous |b| > 1 et |a| suffisamment petit, le rayon spectral de P a+ib agissant dans C γ (PV ) vérifie ρ(P a+ib ) < 1 -δ.

Même dans le cas de SL 2 (R), ce résultat est nouveau et connu seulement dans des cas particuliers. Quand µ est portée par un nombre fini d'éléments de SL 2 (R) et que ces éléments engendrent un sous-semi-groupe de Schottky, ce résultat est dû à Naud [START_REF] Naud | Expanding maps on Cantor sets and analytic continuation of zeta functions[END_REF]. Quand µ est absolument continue par rapport à la mesure de Haar de SL 2 (R), ce résultat peut être obtenu directement en utilisant des propriétés d'oscillation forte.

Il est intéressant de constater que ces trois propriétés, la décroissance de Fourier, le théorème de renouvellement et le trou spectral sont essentiellement équivalentes. Dans la première approche, nous utilisons le théorème de renouvellement pour prouver la décroissance de Fourier. Dans la seconde approche, nous utilisons la décroissance de Fourier pour démontrer le trou spectral, puis utilisons le trou spectral pour établir le théorème de renouvellement. Ces propriétés sont analogues à des phénomènes de la théorie des surfaces convexes co-compactes. Dans ce contexte plus géométrique, la décroissance de Fourier a été étudiée récemment par Bourgain-Dyatlov ; le trou spectral peut être interprété comme l'existence d'une zone sans zéro pour la fonction zêta de Selberg ou le trou dans les valeurs propres de l'opérateur de Laplace de la surface ; le théorème de renouvellement est remplacé par le problème de comptage des orbites du groupe fondamental ou celui des géodésiques fermées primitives. Voir [START_REF] Borthwick | Spectral theory of infinite-area hyperbolic surfaces[END_REF] et les références qui y sont données.

Estimées sommes-produits

Nous expliquons à présent notre deuxième approche, qui est inspirée par le travail de Bourgain et Dyatlov. L'ingrédient essentiel est une propriété de décroissance de Fourier des convolutions multiplicatives de mesures sur R, qui découle de l'estimées sommesproduits discrétisée de Bourgain. Grossièrement, l'estimées sommes-produits nous dit que la taille d'un sous-ensemble de Z doit grandir rapidement par des sommes ou des produits. Cette propriété vient de la combinatoire additive. La propriété discrétisée a été introduite par Katz et Tao. Bourgain a démontré l'estimées sommes-produits discrétisée dans [START_REF] Bourgain | On the Erdős-Volkmann and Katz-Tao ring conjectures[END_REF]. Une de ses nombreuses applications est un résultat de trou spectral dans SU(2) [START_REF] Bourgain | On the spectral gap for finitely-generated subgroups of su (2)[END_REF], qui a été ensuite généralisé dans [START_REF] Benoist | A spectral gap theorem in simple lie groups[END_REF] et [START_REF] Boutonnet | Local spectral gap in simple lie groups and applications[END_REF]. Notre seconde approche peut être vue comme un analogue dans SL m+1 (R).

Pour des groupes de Lie de rang supérieur, nous avons besoin de généraliser la décroissance de Fourier des convolutions multiplicatives à R n . He-de Saxcé ont démontré une version de l'estimées sommes-produits discrétisée dans R n . En utilisant leur résultat, nous pouvons démontrer le Théorème (Theorem 3.1.1). Étant donné κ 0 > 0, il existe , 1 > 0 et k ∈ N tels que la propriété suivante est vérifiée pour δ > 0 suffisamment petit. Soit µ une mesure de probabilité sur [1/2, 1] n ⊂ R n qui vérifie la condition de (δ, κ 0 , ) non concentration projective, à savoir ∀ρ ≥ δ, sup a∈R,v∈S n-1

(π v ) * µ(B R (a, ρ)) = sup a,v µ{x| v, x ∈ B R (a, ρ)} ≤ δ -ρ κ 0 . (0.0.2) Alors, pour tout ξ ∈ R n avec ξ ∈ [δ -1 /2, δ -1 ], exp(2iπ ξ, x 1 • • • x k )dµ(x 1 ) • • • dµ(x k ) ≤ δ 1 . (0.0.3)
Ce théorème est utilisé dans la démonstration de la décroissance de Fourier pour SL m+1 (R). Comme nous l'avons déjà dit, le cas de R est dû à Bourgain [START_REF] Bourgain | The discretized sum-product and projection theorems[END_REF].

Finitude des petites valeurs propres

Notre dernier résultat concerne une classe de variétés qui contient les surfaces hyperboliques convexes cocompactes. Soit H 3 la variété hyperbolique de dimension 3 simplement connexe. La notion de finitude géométrique a été introduite par Ahlfors lors de l'étude du problème suivant: soit Γ un sous-groupe discret de type fini du groupe des isométries directes de H 3 (isomorphe à PSL 2 (C)). Le problème consiste à savoir si l'ensemble limite de Γ est constitué de la sphère entière ou s'il a mesure de de Lebesgue nulle. On l'appelle la conjecture mesurée d'Ahlfors et c'est maintenant un théorème grâce à des progrès en géométrie hyperbolique. La définition originale de la finitude géométrique est qu'il existe un domaine fondamental pour l'action de Γ sur H 3 qui est un polyèdre possédant un nombre fini de côtés. Mais cette définition en termes de polyèdres est difficile à généraliser. Nous utiliserons la définition de Bowditch [START_REF] Bowditch | Geometrical finiteness with variable negative curvature[END_REF], qui demande que la partie épaisse de l'enveloppe convexe de l'ensemble limite soit compacte modulo Γ.

Nous nous donnons une variété géométriquement finie localement symétrique de rang un, ce qui signifie que son revêtement universel est

X = H n F pour F = R, C, H ou H n F = H 2 O .
Soit δ(X) l'exposant de croissance, qui vaut (n + 1) dim R F -2. Le spectre de l'opérateur de Laplace est relié à de nombreuses quantités, comme le taux de croissance des orbites du groupe fondamental et la dimension de Hausdorff de l'ensemble limite. Nous généralisons un résultat de Lax et Phillips pour X = H n R [START_REF] Lax | The Asymptotic Distribution of Lattice Points in Euclidean and Non-Euclidean Spaces[END_REF] et un résultat de Hamenstädt dans le cas général [START_REF] Hamenstädt | Small eigenvalues of geometrically finite manifolds[END_REF].

Introduction

This thesis consists of four research papers. The first part will appear in Mathematische Annalen and the last part has been submitted.

Let V be a finite dimensional irreducible representation of the Lie group G = SL m+1 (R). Let X = PV be the real projective space of V , which is the set of lines of V . Then we have a group action of G on X. Let µ be a Borel probability measure on G and let Γ µ be the subgroup generated by the support of µ. We call µ Zariski dense if Γ µ is a Zariski dense subgroup of G. This means that the measure µ does not concentrate on any algebraic subgroup of G. We can give a random walk on X induced by µ. Fix a point x in X. At each step, we go to a random point gx, where g is a random element in G with the law of µ. By a theorem of Furstenberg, this random walk has a unique stationary measure ν on X, called the Furstenberg measure or the µ-stationary measure. That is to say, the measure ν satisfies ν = µ * ν := G g * νdµ(g), where g * ν is the pushforward of ν by the action of g on X. This measure was introduced by Furstenberg when he established the law of large numbers for products of random matrices. The properties of the µ-stationary measure is important in other limit theorems for products of random matrices.

There are many interesting examples of µ-stationary measures. We restrict our attention to low dimension, that is G = SL 2 (R) and X = P(R 2 ), the real projective line. Let Γ be a lattice in SL 2 (R) (for example SL 2 (Z)). Furstenberg constructed examples where µ is a measure supported on Γ and the µ-stationary measure ν is exactly the Lebesgue measure on X. This construction was used in Furstenberg's work on the Poisson boundary, which gave some rigidity properties of lattices. Recently, people have been interested in the dimension and the absolute continuity of the µ-stationary measure ν when µ has finite support. Please see [START_REF] Bárány | Stationary measures for projective transformations: the Blackwell and Furstenberg measures[END_REF] and [START_REF] Bourgain | Finitely supported measures on SL 2 (R) which are absolutely continuous at infinity[END_REF] for examples of absolutely continuous stationary measures, and [START_REF] Hochman | On the dimension of Furstenberg measure for SL 2 (R) random matrix products[END_REF] for examples of stationary measures with full dimension.

We also mention another class of measures, the Bernoulli convolutions. Let X 0 , X 1 , . . . be i.i.d. random variables such that P(X 0 = 1) = P(X 0 = -1) = 1/2. Let ν λ be the Bernoulli convolution with parameter λ ∈ (0, 1), defined to be the distribution of the random variable j≥0 X j λ j . This can be seen as a stationary measure on R for the action of a solvable group. Please see Example 1.1.6. People are interested in the dimension and the regularity of the Bernoulli convolutions. There are many recent works on this setting. See for instance [START_REF] Shmerkin | Absolute continuity of self-similar measures, their projections and convolutions[END_REF], [START_REF] Hochman | Self similar sets, entropy and additive combinatorics[END_REF] and [START_REF] Péter | Recent progress on Bernoulli convolutions[END_REF].

Introduction

Before stating our main question, we introduce another property of the stationary measure. We need the hypothesis of finite exponential moment, that is there exists positive such that G g dµ(g) < +∞.

From now on, we always suppose that our measure µ is Zariski dense with a finite exponential moment. Recall that X = PV . Guivarc'h proved Hölder regularity of stationary measures. This means that there exist C, c positive such that for every r positive, the r neighbourhood of any hyperplane in X has ν measure less than Cr c . This implies that the stationary measure ν has positive dimension. This also says that ν does not concentrate on some hyperplane, which is reasonable due to the hypothesis of Zariski density of µ.

Fourier decay

Our main problem here is the Fourier decay of the stationary measure. Let us first see the example G = SL 2 (R) and X = P(R 2 ). Fix the identification of P(R 2 ) with the circle T R/πZ, given by the transitive action of the group PSO 2 . We can define the Fourier coefficients of the stationary measure ν by ν(k) = T e 2ikx dν(x), for k ∈ Z.

Theorem (Theorem 1.1.1, Theorem 2.1.2). Let µ be a Zariski dense Borel probability measure on SL 2 (R) with a finite exponential moment. Let ν be the µ-stationary measure on T. Then there exists positive such that |ν(k)| = O(|k| -).

(0.0.4)

In other words, the Fourier coefficients of the stationary measure have polynomial decay. By a general argument, the polynomial decay of Fourier coefficients implies Guivarc'h's regularity. In fact, the regularity is a crucial ingredient in the proof. The Fourier decay of similar measures related to continued fractions have been studied by Kaufman [START_REF] Kaufman | Continued fractions and Fourier transforms[END_REF], Queffélec-Ramaré [START_REF] Queffélec | Analyse de fourier des fractions continues à quotients restreints[END_REF] and Jordan-Sahlsten [START_REF] Jordan | Fourier transforms of gibbs measures for the gauss map[END_REF]. Recently, the Fourier decay of Patterson-Sullivan measures was proved by Bourgain-Dyatlov [START_REF] Bourgain | Fourier dimension and spectral gaps for hyperbolic surfaces[END_REF]. Our second approach is inspired by their methods.

We have two different approaches for this problem. The first approach (Chapter 1) is more elementary, we use the renewal theorem from the theory of stochastic processes. But the result is weaker, we can only prove a qualitative version, that is |ν(k)| → 0 as |k| → +∞. Because the exponential error term in the renewal theorem was not yet known. Later, the exponential error term is proved by our second approach through the Fourier decay.

The second approach (Chapter 2) is inspired by the method of Bourgain and Dyatlov. The main ingredient comes from additive combinatorics, the sum-product estimate. We will explain this approach later.

Lyapunov constant σ µ is an average of the cocycle function σ(g, x) with respect to the measure µ⊗ν. The renewal theorem gives us a phenomenon of equidistribution when the time t is large enough. In our first approach, the renewal theorem is used to bound the renewal sum Rf when f is a function with high oscillation. With our second approach, we are able to give an exponential error term.

Theorem (Theorem 2.1.4). Let µ be a Zariski dense Borel probability measure on SL m+1 (R) with finite exponential moment. Let V be an irreducible representation of SL m+1 (R). There exists > 0 such that for f ∈ C ∞ c (R) and t ∈ R, we have

Rf (x, t) = 1 σ µ ∞ -t f (u)dLeb(u) + O f (e -|t| ),
where O f depends on the support and some Sobolev norm of f .

We should compare this result with the renewal theorem on R (the commutative case). If µ is a measure on R whose support is finite, then the error term in the renewal theorem is never exponential.

Our result improves a result of Boyer, where the error term is polynomial on t. We hope this type of result can give some exponential error terms in the orbital counting problem of higher rank. Given a discrete subgroup Γ of SL m+1 (R), we are interested in the asymptotic for the growth of #{γ ∈ Γ| d(γo, o) ≤ R}, where o is the base point in SL m+1 (R)/SO(m + 1). See for instance Lalley [START_REF] Steven P Lalley | Renewal theorems in symbolic dynamics, with applications to geodesic flows, noneuclidean tessellations and their fractal limits[END_REF], Quint [START_REF] Quint | Groupes de Schottky et comptage[END_REF] and Sambarino [START_REF] Sambarino | The orbital counting problem for hyperconvex representations[END_REF]. This type of error term is always connected with some spectral gap property.

Spectral gap

Equip PV with a Riemannian distance. For γ positive, let C γ (PV ) be the space of γ-Hölder functions. We introduce the transfer operator, which is an analogue of the characteristic function in our case.

Definition. For z in C with the real part | z| small enough, let P z be the operator on the space of continuous functions, which is given by (g,x) f (gx)dµ(g), for x ∈ PV.

P z f (x) = G e zσ
We keep the assumption that µ is a Zariski dense Borel probability measure on SL m+1 (R) with a finite exponential moment. The use of this transfer operator on the products of random matrices has been introduced by Guivarc'h and Le Page. Due to the property of exponential moment, when | z| is small enough, the operator P z preserves the Banach space C γ (PV ) for γ > 0 small enough. Due to the contracting action of G on X, for z in a small ball centred at 0, the spectral radius of P z on C γ (PV ) is less than 1 except at 0. Due to the non-arithmeticity of Γ µ , on the imaginary line, the operator P z also has spectral radius less than 1 except at 0. These were used to give limit theorems for products of random matrices by Le Page and Guivarc'h (Please see [START_REF] Le | Théorèmes limites pour les produits de matrices aléatoires[END_REF] and [START_REF] Benoist | Random Walks on Reductive Groups[END_REF]).

Theorem (Theorem 2.1.5). Let µ be a Zariski dense Borel probability measure on SL m+1 (R) with finite exponential moment. Let V be an irreducible representation of SL m+1 (R). For every γ > 0 small enough, there exists δ > 0 such that for all |b| > 1 and |a| small enough the spectral radius of P a+ib acting on C γ (PV ) satisfies ρ(P a+ib ) < 1 -δ.

Even in the case of SL 2 (R), the result is new and only known in some special case. When µ is supported on a finite number of elements of SL 2 (R) and these elements generate a Schottky semi group, this result is due to Naud [START_REF] Naud | Expanding maps on Cantor sets and analytic continuation of zeta functions[END_REF]. When µ is absolutely continuous with respect to the Haar measure on SL 2 (R), this result can be obtained directly using high oscillations.

It is interesting that the three objects, the Fourier decay, the Renewal theorem and the spectral gap are roughly equivalent. In the first approach, we use the Renewal theorem to prove the Fourier decay. In the second approach, we use the Fourier decay to prove the spectral gap, and then use the spectral gap to prove the Renewal theorem. They are analogue with similar objects for convex cocompact surfaces. In this more geometric setting, the Fourier decay was recently studied by Bourgain-Dyatlov; the spectral gap can be interpreted as the zero free region of the Selberg zeta function or the gap of the eigenvalues of the Laplace operator on the surface; the renewal theorem is replaced by the counting problem of the lattice points or the primitive closed geodesics. Please see Borthwick [START_REF] Borthwick | Spectral theory of infinite-area hyperbolic surfaces[END_REF] and the references there.

Sum-product estimates

Now we explain our second approach, which is inspired by the work of Bourgain and Dyatlov. The key ingredient is a Fourier decay of multiplicative convolution of measures on R, which is a consequence of the discretized sum-product estimate of Bourgain.

The sum-product estimate roughly says that a subset of Z must grow rapidly under sum or product. This comes from additive combinatorics. The discretized setting was introduced by Katz and Tao. Bourgain proved the discretized sum-product estimate in [START_REF] Bourgain | On the Erdős-Volkmann and Katz-Tao ring conjectures[END_REF]. One of its many applications is a spectral gap result in SU(2) [START_REF] Bourgain | On the spectral gap for finitely-generated subgroups of su (2)[END_REF], which was further generalized in [START_REF] Benoist | A spectral gap theorem in simple lie groups[END_REF] and [START_REF] Boutonnet | Local spectral gap in simple lie groups and applications[END_REF]. Our second approach can be seen as an analogue in SL m+1 (R).

For Lie groups of higher rank, we need to generalize the Fourier decay of multiplicative convolutions to R n . He-de Saxcé have proved a version of discretized sum-product estimate in R n [START_REF] He | Sum-product for real Lie groups[END_REF]. Using their result, we are able to prove the following:

Introduction Then for all ξ ∈ R n with ξ ∈ [δ -1 /2, δ -1 ], exp(2iπ ξ, x 1 • • • x k )dµ(x 1 ) • • • dµ(x k ) ≤ δ 1 . (0.0.6)
This theorem is used in the proof of Fourier decay for SL m+1 (R). As we have already mentioned, the case of R is due to Bourgain [START_REF] Bourgain | The discretized sum-product and projection theorems[END_REF].

Finiteness of small eigenvalues

Our last result concerns a class of manifolds which contains convex cocompact hyperbolic surfaces. Let H 3 be the simply connected hyperbolic three manifold. The definition of geometric finiteness was first introduced by Ahlfors in studying the following problem: Let Γ be a discrete finitely generated subgroup of the oriented isometry group of H 3 (isomorphic to PSL 2 (C)). The problem is whether the limit set of Γ is a full sphere or has Lebesgue measure zero. This is called Ahlfors' measure conjecture, which is a theorem now due to progresses of hyperbolic geometry. The original definition of geometric finiteness is that there exists a fundamental domain of Γ acting on H 3 , which is a finitely sided polyhedra. But this definition of fundamental polyhedra is hard to generalize. We will use the definition of Bowditch [START_REF] Bowditch | Geometrical finiteness with variable negative curvature[END_REF], that the thick part of the convex hull is cocompact.

We are given a geometrically finite rank one locally symmetric manifold, which means the universal cover is

X = H n F for F = R, C, H or H n F = H 2 O .
Let δ(X) be the exponent of growth, which equals (n + 1) dim R F -2. The spectrum of the Laplace operator is related to many quantities, such as the growth rate of the number of lattice points and the Hausdorff dimension of the limit set. We generalize a result of Lax and Phillips on X = H n R [START_REF] Lax | The Asymptotic Distribution of Lattice Points in Euclidean and Non-Euclidean Spaces[END_REF] and a result of Hamenstädt in the general case [START_REF] Hamenstädt | Small eigenvalues of geometrically finite manifolds[END_REF]. Theorem (Theorem 4.1.1). Let M = Γ\X be a geometrically finite rank one locally symmetric manifold. Then the intersection of the spectrum of the Laplace operator and the critical interval (-δ(X) 2 /4, 0] consists of finitely many eigenvalues of finite multiplicities.

This result can be used to give a spectral gap for the spectrum of the Laplace operator. Then the spectral gap gives an exponential error in the orbital counting problem.

Introduction

Let µ be a Borel probability measure on SL 2 (R). The linear action of SL 2 (R) on R 2 induces an action on P 1 = P(R 2 ). For a Borel probability measure ν on P 1 , we define its convolution with µ by

µ * ν = SL 2 (R) g * νdµ(g),
where g * ν is the pushforward of ν by g. The measure ν is called µ-stationary if µ * ν = ν. We add the condition that the subgroup Γ µ generated by the support of µ is Zariski dense in SL 2 (R). In the case of SL 2 (R), Zariski density is equivalent to unsolvability. When Γ µ is Zariski dense in SL 2 (R), there is a unique µ-stationary measure (see [START_REF] Furstenberg | Noncommuting random products[END_REF], [START_REF] Guivarc | Frontiere de Furstenberg, propriétés de contraction et théoremes de convergence[END_REF]).

This stationary measure is also called the Furstenberg measure. It was first considered by Furstenberg in the study of the noncommutative law of large numbers. The stationary measure takes part in the subtle properties of random products of matrices. Please see [START_REF] Furstenberg | Noncommuting random products[END_REF], [START_REF] Guivarc | Frontiere de Furstenberg, propriétés de contraction et théoremes de convergence[END_REF] and [START_REF] Bougerol | Products of random matrices with applications to Schrödinger operators[END_REF].

In this paper, we are interested in the decay of the Fourier coefficients of stationary measures. The action of PSO 2 = SO 2 /{±Id} on P 1 is transitive and free. We fix the point x o = [1 : 0] in P 1 , then identify P 1 as the orbit space PSO 2 x o . As a group, PSO 2 is isomorphic to the circle T R/πZ. This is given by the map from T to PSO 2 , θ → cos θ -sin θ sin θ cos θ /{±Id}.

So we have a homeomorphism from T to P 1 , that is θ → [cos θ : sin θ]. We can define the Fourier coefficients of the stationary measure ν by the following formula

ν(k) = T e 2ikθ dν(θ).
We also demand that µ has a finite exponential moment, which means that there exists a constant 1 > 0 such that g 1 dµ(g) < ∞. We will prove Theorem 1.1.1. Let µ be a Borel probability measure on SL 2 (R) with a finite exponential moment, and assume that the subgroup Γ µ is Zariski dense. Then the µ-stationary measure ν is a Rajchman measure, in other words

ν(k) → 0 as |k| → +∞. (1.1.1)
Remark 1.1.2. Fourier decay of measures on fractal sets and its applications have been studied in [START_REF] Kaufman | Continued fractions and Fourier transforms[END_REF], [START_REF] Queffélec | Analyse de fourier des fractions continues à quotients restreints[END_REF], [START_REF] Jordan | Fourier transforms of gibbs measures for the gauss map[END_REF] and [START_REF] Bourgain | Fourier dimension and spectral gaps for hyperbolic surfaces[END_REF]. Our situation is much general and we introduce a quite different method.

Being a Rajchman measure is a local property (see [START_REF] Alexander | Descriptive set theory and the structure of sets of uniqueness[END_REF]): Indeed, let ν 1 be a Rajchman measure. If ν 2 is absolutely continuous with respect to ν 1 , then ν 2 is also a Rajchman measure. Conversely, the sum of two Rajchman measures is a Rajchman measure.

In this spirit, we have the following theorem:

Theorem 1.1.3. Let µ be a Borel probability measure on SL 2 (R) with a finite exponential moment, and assume that the subgroup Γ µ is Zariski dense. Let ν be the unique µ-stationary measure. Assume that r is a C 1 function on P 1 and φ is a C 2 function on P 1 such that |φ | ≥ 1/C 1 > 0 on the support of r and

r C 1 , φ C 2 ≤ C 1 for some constant C 1 > 0.
Then we have e iξφ(x) r(x)dν(x) → 0 as |ξ| → ∞, (1.1.2)

uniformly with respect to C 1 .
This is the main theorem of this paper. It will be proved in Section 1.3.

Corollary 1.1.4. Let µ be a Borel probability measure on SL 2 (R) with a finite exponential moment, and assume that the subgroup Γ µ is Zariski dense. Let ν be the unique µ-stationary measure. Then for a C 2 -diffeomorphism φ on P 1 , the pushforward of the stationary measure φ * ν is a Rajchman measure. In other words

φ * ν(k) → 0 as |k| → +∞.
(1.1.3) Theorem 1.1.1 is a special case of this corollary, where φ is the identity function.

Proof of Corollary 1.1.4 from Theorem 1.1.3. By the identification P 1 T, we may consider all the objects as living on T. Take a partition of unity of T: let r 1 , r 2 be non negative Lipschitz functions on T such that r 1 + r 2 = 1, and the supports of r 1 , r 2 are connected subintervals of T. For j = 1, 2, we can lift the function φ| suppr j to a function φ j from suppr j to R.

Then T e 2ikφ(θ) dν(θ) = T (r 1 (θ) + r 2 (θ))e 2ikφ(θ) dν(θ) = T e 2ikφ 1 (θ) r 1 (θ) + e 2ikφ 2 (θ) r 2 (θ) dν(θ).
Since φ is a diffeomorphism, the functions φ j , r j satisfy the conditions in Theorem 1.1.3. We use this theorem twice to conclude.

Let us use another coordinate system on P 1 . We identify P 1 with R ∪ {∞} through the map ϕ(x) = v 1 /v 2 , where x = Rv is a point in P 1 . Then the action of SL 2 (R) on P 1 reads as the Möbius action, that is for r ∈ R ∪ {∞} and g = a b c d in SL 2 (R), we have gr = ar+b cr+d . If the support of a µ-stationary measure ν does not contain [1 : 0], then ϕ * ν is a stationary measure on R. From Theorem 1.1.3, we get Corollary 1.1.5. Let µ be a Borel probability measure on SL 2 (R) with a finite exponential moment, and assume that the subgroup Γ µ is Zariski dense. Let ν be the unique µ-stationary measure. If the support of ν does not contain [1 : 0], then the µ-stationary measure ϕ * ν is a Rajchman measure on R. In other words ϕ * ν(ξ) = P 1 e iϕ(x)ξ dν(x) → 0 as |ξ| → +∞.

(1.1.4)

Example 1.1.6 (Solvable case). For stationary measures on R, consider the following

µ = 1 2 (δ g 1 + δ g 2 ) = 1 2 δ   √ λ -1/ √ λ 0 1/ √ λ   + 1 2 δ   √ λ 1/ √ λ 0 1/ √ λ   ,
where λ ∈ (0, 1). Then the actions of g 1 , g 2 are given by g 1 r = λr -1, g 2 r = λr + 1 for r ∈ R. By definition, a µ-stationary measure ν on R must satisfy the equation

ν = µ * ν = 1 2 ((g 1 ) * ν + (g 2 ) * ν). (1.1.5)
Let X 0 , X 1 , . . . be i.i.d. random variables such that P(X 0 = 1) = P(X 0 = -1) = 1/2. Let ν λ be the Bernoulli convolution with parameter λ, defined to be the distribution of j≥0 X j λ j . The measure ν λ satisfies (1.1.5), thus it is a µ-stationary measure on R. In [START_REF] Erdös | On a family of symmetric Bernoulli convolutions[END_REF], Erdös proved that when λ -1 is a Pisot number, the Fourier transform of ν λ does not converge to zero. In this example Γ µ is solvable, so the Zariski density condition is necessary in the theorem.

Remark 1.1.7. 1. A similar result for Bernoulli convolutions was obtained in [START_REF] Kaufman | Bernoulli convolutions and differentiable functions[END_REF]. Kaufman proved that for Bernoulli convolutions ν λ , if λ -1 is not a Pisot number, then it satisfies the same conclusion as in Corollary 1.1.3. That is, the pushforward measure φ * ν λ is a Rajchman measure, where φ is a C 1 function on R with φ > 0 everywhere. 2.Our result for the measure ν is stronger than being a Rajchman measure. Indeed, for a probability measure on T, being a Rajchman measure is not invariant by diffeomorphisms. We can find examples in [START_REF] Kaufman | On Bernoulli convolutions[END_REF]. A typical example is the standard 1 3 -Cantor measure ν, which is not a Rajchman measure. Let φ be the quadratic map r → r 2 . Then the pushforward measure φ * ν becomes a Rajchman measure with polynomial decay.

One of our motivations for establishing Theorem 1.1.1 comes from the theory of Bernoulli convolutions. One of the main questions of this theory is to determine for which parameter λ, the measure ν λ is absolutely continuous with respect to the Lebesgue measure. We have already mentioned that when λ -1 is a Pisot number, Erdös proved that ν λ is not a Rajchman measure. Thus, in particular, ν λ is not absolutely continuous with respect to the Lebesgue measure. Recently, people have been interested in the same problem for stationary measures for random walks on SL 2 (R), see [START_REF] Bourgain | Finitely supported measures on SL 2 (R) which are absolutely continuous at infinity[END_REF], [START_REF] Vadim | Matrix random products with singular harmonic measure[END_REF]. Our result shows that we cannot generalize the method of Erdös to the Zariski dense case.

Our other motivation is the same question for the Patterson-Sullivan measure on the limit set of Fuchsian groups. With Theorem 1.1.1, it suffices to prove that there exists a probability measure µ on SL 2 (R) such that the Patterson-Sullivan measure is µ-stationary, and µ has a finite exponential moment.

In [START_REF] Steven P Lalley | Renewal theorems in symbolic dynamics, with applications to geodesic flows, noneuclidean tessellations and their fractal limits[END_REF] and [START_REF] Lalley | Regenerative representation for one-dimensional Gibbs states[END_REF], Lalley announced the existence of such a µ for Schottky groups. But Lalley's proof only works for Schottky semigroups. In [START_REF] Connell | Harmonicity of quasiconformal measures and Poisson boundaries of hyperbolic spaces[END_REF], the authors proved the existence of such a µ without the moment condition in geometrically finite cases. Combining the methods of Connell, Muchnik and Lalley, we can prove the existence of such a measure µ for convex cocompact Fuchsian groups, see [Li]. Therefore, we have Corollary 1.1.8. Let Γ be a convex cocompact Fuchsian group. Then the Patterson-Sullivan measure associated to Γ is a Rajchman measure.

Remark 1.1.9. Corollary 1.1.8 also holds if we replace the Patterson-Sullivan measure by any Gibbs measure. In [Li], we have a similar realization for any Gibbs measure associated to a convex cocompact Fuchsian group, as it is done by Lalley for any Gibbs measure on the limit set of a Schottky semigroup in [START_REF] Lalley | Regenerative representation for one-dimensional Gibbs states[END_REF].

Remark 1.1.10. Using the uniform spectral gap proved in [START_REF] Naud | Expanding maps on Cantor sets and analytic continuation of zeta functions[END_REF], we can prove a polynomial decay in the convergence to zero of the Fourier coefficients of the µ-stationary measure, when the support of µ is the set of generators of a Schottky semigroup. In this case, the uniform spectral gap implies an exponential error term in the renewal theorem, which is the only obstacle for polynomial decay. Please see Remark 1.3.10 for more details. We believe it is true for the general case, but the question is still open.

Remark 1.1.11. Very recently, Bourgain and Dyatlov [START_REF] Bourgain | Fourier dimension and spectral gaps for hyperbolic surfaces[END_REF] have proved a polynomial decay of the Fourier coefficients of the Patterson-Sullivan measure associated to a convex cocompact Fuchsian group. Their method, which comes from additive combinatorics, is totally different from ours. They use the Fourier decay bound and the fractal uncertainty principle to obtain an essential spectral gap for a convex cocompact hyperbolic surface. We can not recover their result directly as in Remark 1.1.10. It is possible if we modifier some steps and use the uniform spectral gap in [START_REF] Naud | Expanding maps on Cantor sets and analytic continuation of zeta functions[END_REF], but we do not pursue in this direction in this work.

On the other hand, in the geometrically finite case, this approach can not work. The finite exponential moment condition is impossible for noncompact lattice Γ in SL 2 (R) (see [START_REF] Guivarc | Asymptotic winding of the geodesic flow on modular surfaces and continued fractions[END_REF], [START_REF] Deroin | On the question of ergodicity for minimal group actions on the circle[END_REF], [START_REF] Blachère | Harmonic measures versus quasiconformal measures for hyperbolic groups[END_REF]). That is, if µ is a measure on Γ with a finite first moment, then the µ-stationary measure ν is singular with respect to the Lebesgue measure. Maybe the generalization of the method of [START_REF] Jordan | Fourier transforms of gibbs measures for the gauss map[END_REF] works in this case, where they proved the Gibbs measures for the Gauss map which has dimension greater than 1/2 are Rajchman measures.

In this paper, our main idea is to obtain the convergence to zero of Fourier coefficients from a renewal type result.

The strategy of proof : To simplify, identify P 1 with T = R/πZ as before. The starting point is the relation

ν = µ * ν. Consider a random walk on SL 2 (R), X n = b 1 b 2 • • • b n ,
where b j are independent random variables with the same law µ. Let B n be the Borel σ-algebra generated by X 1 , . . . , X n . Let Y n = (X n ) * ν. They are random variables which take values in the space of Borel measures on T. By definition, we have

E(Y n+1 |B n ) = E((X n ) * (b n+1 ) * ν|B n ) = (X n ) * E((b n+1 ) * ν) = Y n .
Therefore {Y n } is a martingale. For t > 0, we define the stopping time by τ = inf{n ∈ N| log X n ≥ t}. Then the martingale property implies that

E((X τ ) * ν) = E(Y τ ) = E(Y 0 ) = ν.
(See Proposition 1.3.5). Thus for the Fourier coefficients, we have for k ∈ 2Z (since

ν(k) = ν(-k), we only consider k ≥ 0.) ν(k) = e ikx dν(x) = e ikx dE((X τ ) * ν)(x) = E(e ikXτ x )dν(x).
Recall our circle T is R/πZ. The idea is to find some cancellations in the "trigonometric series" E(e ikXτ x ). By the Cauchy-Schwarz inequality, it suffices to prove E(e ik(Xτ x-Xτ y) ) → 0 as k → ∞.

By analogy with the case of classical random walks on R, we expect that there exists a measurable density function p on R + such that for a continuous compactly supported function

f on R and t ∈ R, E(f (log X τ -t)) -→ R + f (u)p(u)du as t → +∞.
Then absolute continuity of the limit distribution would imply the convergence to zero of ν(k).

In the actual proof, we do not use this stopping time, but a residue process. Indeed, the latter is easier to treat with transfer operators and Fourier analysis. We will establish a limit theorem for the residue process, a generalization of the renewal theorem, in Section 1.4.

Notation: When f and g are functions on a set X, we write f (x) g(x), if there exists C > 0 independent of x ∈ X such that f (x) ≤ Cg(x), and f (x) = O(g(x)) means |f (x)| g(x). We also write f (x, y) = O y (g(x, y)), which means |f (x, y)| ≤ C y g(x, y), where C y is a constant only depending on y.

We introduce a notation O exp, (s). We write f ( , s) = O exp, (s) if for > 0 and s ∈ R, there exists a constant > 0 such that f ( , s) = O(e -s ), where all the constants only depend on . We write f (s) = O exp (s), if there exists a uniform constant such that f (s) = O(e -s ).

Preliminaries on random walks on P 1

Fix the norm induced by the standard inner product on

R 2 , v = v 2 1 + v 2 2 , which is SO 2 (R) invariant.
Then define a metric on P 1 . For two points x = Rv, x = Rw, we set

d(x, x ) = | det(v, w)| v w . This is a sine distance. If we write x = R cos θ sin θ and x = R cos θ sin θ , then d(x, x ) = | sin(θ -θ )|.
From now on, we write G = SL 2 (R) and X = P 1 .

Definition 1.2.1. For g in G and x = Rv in X, define the function σ :

G × X → R by σ(g, x) = log gv v .
This function σ is a cocycle, because for g, h in G we have

σ(gh, x) = log ghv v = log ghv hv + log hv v = σ(g, hx) + σ(h, x),
where we use the fact that the action is linear, hx = Rhv.

Lemma 1.2.2. For g in G and x, x in X with x = x , we have

d(gx, gx ) d(x, x ) = exp(-σ(g, x) -σ(g, x )). (1.2.1)
Proof. As in the definition of the distance d(•, •), we take two non zero vectors v and w in x and x respectively. By definition,

d(gx, gx ) d(x, x ) = det(gv, gw) det(v, w) v w gv gw = |det g| v w gv gw = exp -σ(g, x) -σ(g, x ) .
The proof is complete.

If the point x is near x , we know from the above equation that the cocycle σ is essentially the logarithm of the contracting or expanding ratio. Let µ be a Borel probability measure on G, and let b 1 , b 2 , • • • be independent random variables with the same law µ. Then the behavior of the mean value of the cocycle,

1 n σ(b n b n-1 • • • b 1 , x) = 1 n   n j=1 σ(b j , b j-1 • • • b 1 x)   ,
follows an asymptotic law similar to the law of large numbers. In particular,

Theorem 1.2.3. [Fur63][GR85]
Let µ be a Borel probability measure on G having an exponential moment. Assume that the subgroup Γ µ is Zariski dense. Then for all x in X, random variables b j defined as above, we have

lim n→∞ σ(b n b n-1 • • • b 1 , x) n = G X σ(g, y)dµ(g)dν(y) = σ µ > 0 a.s µ ⊗N * . (1.2.2)
The constant σ µ is called the Lyapunov exponent of µ.

Theorem 1.2.4 (Hölder regularity).

[Gui90][BL85, Chapter 6,Proposition 4.1] Under the assumptions of Theorem 1.1.1, there exist constants C > 0, α > 0 such that for every x in X and r > 0 we have ν(B(x, r)) ≤ Cr α .

(1.2.3)

We need the Cartan decomposition of the Lie group G, i.e. G = SO 2 A + SO 2 , where

A + = { e t 0 0 e -t , t ≥ 0}.
For g in G, we can write g = k g a g l g , where k g , l g are in SO 2 , and a g = diag{e κ(g) , e -κ(g) } is the diagonal matrix whose diagonal elements are e κ(g) and e -κ(g) with κ(g) ≥ 0. The positive number κ(g) is called the Cartan projection. Identify the two spaces X and T R/πZ. For an element x in X, associate it to the unique

element θ(x) in R/πZ satisfying x = R cos θ(x) sin θ(x)
. When there is no ambiguity, we will abbreviate θ(x) to x.

Let e 1 = R 1 0 , e 2 = R 0 1
, which mean elements in X. Let r θ = cos θ sin θ -sin θ cos θ be a rotation matrix in G. For g in G, choosing a decomposition g = k g a g l g , we define

x m g = l -1 g e 2 , x M g = k g e 1 . If κ(g) > 0, then x M g , x m
g are uniquely defined.

Proposition 1.2.5. For g in G with κ(g) > 0, we have

x m g = x M g -1 .
(1.2.4)

Proof. For a real number a = 0, we have

a 0 0 a -1 = 0 1 -1 0 a -1 0 0 a 0 -1 1 0 = r π/2 a -1 0 0 a r 3π/2 .
This implies that

g -1 = (k g a g l g ) -1 = l -1 g a -1 g k -1 g = l -1 g r π/2 a g r 3π/2 k -1 g . Therefore x M g -1 = l -1 g r π/2 e 1 = l -1 g e 2 = x m g .
Lemma 1.2.6. For g in G and x = Rv in X, we have

d(x m g , x) ≤ gv g v ≤ d(x m g , x) + e -2κ(g) .
(1.2.5)

Another form that will be used frequently is

σ(g, x) ≥ κ(g) + log d(x m g , x).
Proof. Suppose that the vector v has norm 1, then

gv v = k g a g l g v v = a g l g v l g v . Since d(x m g , x) = d(l -1 g e 2 , x) = d(e 2 , l g x),
it suffices to prove this inequality for diagonal elements, in other words g = diag{e κ(g) , e -κ(g) }. Hence

gv v = e κ(g) 0 0 e -κ(g) v 1 v 2 = |e 2κ(g) v 2 1 + e -2κ(g) v 2 2 | 1/2 . (1.2.6) The equality d(x m g , x) = d(e 2 , x) = |v 1 | implies that gv g v ≥ |v 1 | = d(x m g , x), gv g v ≤ |v 1 | + e -2κ(g) = d(x m g , x) + e -2κ(g) .
The proof is complete.

The following lemma is an important tool, which gives a precise approximation of the cocycle by the Cartan projection and distance.

Lemma 1.2.7. Let x, x be two points in X and let g be in G. Assume that

e -2κ(g) + d(x m g , g -1 x) ≤ 1 2 d(g -1 x , x), then |σ(g, x) -κ(g) -log d(g -1 x , x)| ≤ 2 e -2κ(g) + d(x m g , g -1 x ) d(g -1 x , x)
.

(1.2.7)

Proof. Inequality (1.2.5) implies that |e σ(g,x)-κ(g) -d(g -1 x , x)| ≤ max{|d(x m g , x) -d(g -1 x , x)|, |e -2κ(g) + d(x m g , x) -d(g -1 x , x)|} ≤ e -2κ(g) + d(x m g , g -1 x ).
Thus by hypothesis, we have

| exp(σ(g, x) -κ(g)) -d(g -1 x , x)| ≤ 1/2d(g -1 x , x). Since | log(1 + t)| ≤ 2|t| for t > -1/2, we obtain |σ(g, x) -κ(g) -log d(g -1 x , x)| = log |1 + exp(σ(g, x) -κ(g)) -d(g -1 x , x) d(g -1 x , x) | ≤ 2 | exp(σ(g, x) -κ(g)) -d(g -1 x , x)| d(g -1 x , x) ≤ 2 e -2κ(g) + d(x m g , g -1 x ) d(g -1 x , x) .
The proof is complete.

In the next proposition we summarize the large deviations principle for the cocycle and for the Cartan projection, Proposition 1.2.8. [BQ16, Thm13.11, Thm 13.17] Under the assumptions of Theorem 1.1.1, for every > 0 we have

µ * n {g ∈ G| |σ(g, x) -nσ µ | ≥ n } = O exp, (n),
(1.2.8)

µ * n {g ∈ G| |κ(g) -nσ µ | ≥ n } = O exp, (n), (1.2.9)
uniformly for all x in X and n ≥ 1.

Let t be a real number. Write [t] for the integer part of t.

Corollary 1.2.9. Under the assumptions of Theorem 1.1.1, for every > 0 we have

m≥n µ * m {g ∈ G|σ(g, x) ≤ t} = O exp, (n), m≥n µ * m {g ∈ G|κ(g) ≤ t} = O exp, (n), uniformly for all x in X, t > 0 and n ≥ [ t σµ-].
By the hypothesis of finite exponential moment and the Chebyshev inequality, we have Lemma 1.2.10. Under the assumptions of Theorem 1.1.1, let M µ be the finite exponential moment of µ defined by M µ = g 1 dµ(g). For s > 0, we have

µ{g ∈ G|κ(g) ≥ s} ≤ M µ e -1 s . (1.2.10) Chapter 1. Fourier decay of SL 2 (R)
Corollary 1.2.11. Under the assumptions of Theorem 1.1.1, for every > 0 we have

m≤n µ * m {g ∈ G|σ(g, x) ≥ t} = O exp, (t), m≤n µ * m {g ∈ G|κ(g) ≥ t} = O exp, (t),
uniformly for all x in X, t > 0 and n = [ t σµ+ ]. Proof. The inequality about the cocycle follows from the one about the Cartan projection, because κ(g) ≥ σ(g, x). It suffices to prove the second inequality:

• When m ≤ 2 t, where 2 > 0 is a small constant such that 2 ≤ 1 /(2 log M µ ), from
Chebyshev's inequality and the subadditivity of the Cartan projection, we have

[ 2 t] m=1 µ ⊗m {κ(g) ≥ t} ≤ [ 2 t] m=1 e -1 t e 1 κ(g) dµ ⊗m (g) ≤ [ 2 t] m=1 e -1 t g 1 dµ(g) m ≤ e -1 t M [ 2 t] µ /(M µ -1).
This implies that

[ 2 t] m=1 µ ⊗m {κ(g) ≥ t} e -t 1 /2 . • When m ∈ [ 2 t, t/(σ µ + )], we have κ(g) > t ≥ m(σ µ + ). Then use (1.2.9)
to deduce that the measure of this part is less than m∈

[ 2 t,t/(σµ+ )] O exp, (m) = O exp, (t).
The proof is complete.

The following proposition describes regularity properties of µ * n , which is a corollary of the large deviations principle.

Proposition 1.2.12. [BQ16, Prop14.3] Under the assumptions of Theorem 1.1.1, for every > 0 we have

µ * n {g ∈ G| d(gx, x ) ≤ e -n } = O exp, (n),
(1.2.11)

µ * n {g ∈ G| d(x M g , x) ≤ e -n } = O exp, (n), (1.2.12) µ * n {g ∈ G| d(x m g , g -1 x) ≥ e -(2σµ-)n } = O exp, (n), (1.2.13) µ * n {g ∈ G| d(x M g , gx) ≥ e -(2σµ-)n } = O exp, (n), (1.2.14)
uniformly for all x, x in X and n ≥ 1.

Corollary 1.2.13. Under the assumptions of Theorem 1.1.1, for every > 0 we have

µ * n {g ∈ G| d(gx, x ) ≤ e -t } = O exp, (t), (1.2.15)
uniformly for all x, x in X, t > 0 and n ≥ t/ . For every > 0 we have

µ * n {g ∈ G| d(x M g , gx) ≥ e -t } = O exp, (t), (1.2.16)
uniformly for all x in X, t > 0 and n ≥ t/(2σ µ -).

Proof. There exists an integer n t ≤ n such that n t < t ≤ (n t + 1). By inequality (1.2.11), we have µ * nt {d(gx, x ) ≤ e -nt } e -nt . This implies that

µ * n {g ∈ G|d(gx, x ) ≤ e -t } = G µ * nt {l ∈ G|d(l(hx), x ) ≤ e -t }dµ * (n-nt) (h) ≤ G µ * nt {l ∈ G|d(l(hx), x ) ≤ e -nt }dµ * (n-nt) (h)
e -nt e -(t/ -1) e -t/ .

The second inequality follows from the same argument.

The following lemma describes the difference between the cocycle and the Cartan projection.

Lemma 1.2.14. [BQ16, Lemma 17.8] Under the assumptions of Theorem 1.1.1, for every > 0, there exist C > 0, > 0 such that for all n ≥ l > 0 and x in X, there exists a subset S n,l,x ⊂ G × G, which satisfies

µ * (n-l) ⊗ µ * l (S c n,l,x ) ≤ Ce -l = O exp, (l),
and for all (g 1 , g 2 ) ∈ S n,l,x , we have

|κ(g 1 g 2 ) -σ(g 1 , g 2 x) -κ(g 2 )| ≤ e -l .
By the identification X T, we can work on T. Since the circle T is a quotient space of R, it has the induced orientation. For two different points x, y in T, which are not the two endpoints of a diameter, they divide the circle into two arcs. Call the arc with longer length the large arc, and the other arc the small arc x y. For a function φ on T, it can be seen as a function Φ on R with period π. Define φ (θ) as the derivative of Φ.

We introduce a sign for two different points x, y in X, where x, y are not the two endpoints of a diameter. If in the small arc x y, the point x is the start point in the orientation sense, then we define sign(x, y) = 1; otherwise, we define sign(x, y) = -1. We have a Newton-Leibniz formula on the circle

φ(y) -φ(x) = sign(x, y) x y φ (θ)dθ, (1.2.17)
where dθ is the Lebesgue measure on T induced by the Lebesgue measure on R with total mass π.

Definition 1.2.15 (Orientation). Let x, y, z be three points in X. Define

sign(x, y, z) =      0 if any two points coincide, 1 if {x, y, z} is counterclockwise, -1 otherwise.
Proposition 1.2.16. Let x, y be two different points in X, and let g be in G such that κ(g) > 2 and d(x m g , x), d(x m g , y) > e -κ(g) . Then sign(gx, gy) = sign(x, y, x m g ).

(1.2.18)

Proof. With the same argument as in the proof of Lemma 1.2.6, it suffices to prove the statement in case g = a g , that is sign(a g x, a g y) = sign(x, y, e 2 ). If x is a point in X such that d(e 2 , x ) > e -κ(g) , then d(a g x , e 1 ) = d(a g x , a g e 1 ) = d(x , e 1 ) exp(-σ(a g , x ) -σ(a g , e 1 )).

By (1.2.5), we obtain σ(a g , x ) ≥ κ(g) + log d(e 2 , x ) > 0, so

d(a g x , e 1 ) ≤ exp(-κ(g)) ≤ e -2 .
Thus the action of a g on the interval B(e 2 , e -κ(g) ) c is contracting with fixed point e 1 , and the image is in the interval B(e 1 , e -κ(g) ).

Especially, e 2 is not in B(e 1 , e -κ(g) ) and the small arc a g x a g y is contained in B(e 1 , e -κ(g) ). By definition we have sign(a g x, a g y) = sign(a g x, a g y, e 2 ).

Since the action of a g on T preserves the orientation, we have sign(a g x, a g y, e 2 ) = sign(x, y, e 2 ). The proof is complete.

Decrease of the Fourier transform

Here we give a proof of Theorem 1.1.3, by admitting the technical results that will be proved in the following two sections. Recall the notations G = SL 2 (R) and X = P 1 . Definition 1.3.1. Let Σ = n∈N G ×n be the symbol space of all finite sequences with elements in G. Let µ be a Borel probability measure on G, and let µ ⊗n be the product measure on G ×n . Then µ ⊗n can be seen as a measure on Σ which is nonzero only on G ×n . Let μ be the measure on Σ defined by μ

= δ ∅ + µ + µ ⊗2 + • • • .
Let the integer ω(g) be the length of an element g in Σ. Then an element g can be written as (g 1 , g 2 , . . . , g ω ), where ω is the abbreviation of ω(g).

Let T be the shift map on Σ, defined by T g = T (g 1 , g 2 , . . . , g ω ) = (g 1 , g 2 , . . . , g ω-1 ), when ω(g) ≥ 2, and T g = ∅, when ω(g) = 1, 0.

Let L be the left shift map on Σ, defined by Lg = L(g 1 , g 2 , . . . , g ω ) = (g 2 , . . . , g ω-1 , g ω ), when ω(g) ≥ 2, and Lg = ∅, when ω(g) = 1, 0.

When considering the action of g on X, we write gx

= g 1 • • • g ω x, σ(g, x) = σ(g 1 • • • g ω , x), x m g = l -1 g 1 ••
•gω e 2 , as well as the Cartan projection κ(g) = κ(g 1 • • • g ω ).

Remark 1.3.2. When using this definition, we may meet the convolution measure µ * n on G or the product measure µ ⊗n on G ×n . Denote F :

G ×n → G by F (g 1 , g 2 , . . . , g n ) = g 1 • • • g n , then F * (µ ⊗n ) = µ * n .
Definition 1.3.3. For t > 0, define two sets that contain all the sequences which make the value of the Cartan projection pass t,

M + t = {g ∈ Σ| κ(T g) < t ≤ κ(g)}, M - t = {g ∈ Σ| κ(T g) ≥ t > κ(g)}.
Remark 1.3.4. In some special cases, for b j in suppµ, the Cartan projection κ(b

1 b 2 • • • b n ) is increasing with respect to n. Then M - t has μ measure zero. Let X n = b 1 b 2 • • • b n
be a random walk on G, where b j are i.i.d. random variables taking values in G with the same law µ. Let τ be the stopping time defined by τ = inf{n ∈ N|κ(X n ) ≥ t}. In such special case

μ(M + t ∩ G ×n ) = P(τ = n).
So in the measure sense, M + t is a set of the steps. That is for μ-almost every g in

M + t , it is of the form g = (b 1 , b 2 , . . . , b τ ) = (X 1 , X -1 1 X 2 , . . . , X -1 τ -1 X τ
) which corresponds to the set of steps of the trajectory (X 1 , X 2 , . . . , X τ ). But this is not always true for general cases.

By Corollary 1.2.9, these two sets M + t , M - t have finite μ measure. We have a property of M + t , M - t due to the definition of stationary measures. Our proof is a generalization of the property of the stopping time for martingales.

Proposition 1.3.5. Under the assumptions of Theorem 1.1.1, for a real number t > 0 and a continuous function f on X, we have

X f (x)dν(x) = X g∈M + t f (gx)dμ(g) - g∈M - t f (gx)dμ(g) dν(x).
Proof. For a natural number N , let

F N = X g∈M + t ,ω(g)≤N f (gx)dμ(g) - g∈M - t ,ω(g)≤N f (gx)dμ(g) + ω(g)=N,κ(g)<t f (gx)dμ(g) dν(x).
Then F o = X f (x)dν(x). Since all the terms are finite, we have

F N +1 -F N = X g∈M + t ,ω(g)=N +1 f (gx)dμ(g) - g∈M - t ,ω(g)=N +1 f (gx)dμ(g) + ω(g)=N +1,κ(g)<t f (gx)dμ(g) - ω(g)=N,κ(g)<t f (gx)dμ(g) dν(x).
By the relation ν = µ * ν, the set of integration of the last term becomes {ω(g) = N + 1, κ(T g) < t}. Compare these sets of integration

{g ∈M + t , ω(g) = N + 1} ∪ {ω(g) = N + 1, κ(g) < t} ={ω(g) = N + 1, κ(T g) < t, κ(g) ≥ t} ∪ {ω(g) = N + 1, κ(g) < t} ={ω(g) = N + 1, κ(T g) ≥ t, κ(g) < t} ∪ {ω(g) = N + 1, κ(T g) < t} ={g ∈ M - t , ω(g) = N + 1} ∪ {ω(g) = N + 1, κ(T g) < t}.
Therefore,

F N +1 = F N = • • • = F o .
Corollary 1.2.9 and Inequality (1.2.9) imply that μ{g ∈ M ± t , ω(g) > N }, μ{ω(g) = N, κ(g) < t} → 0, as N → ∞. Thus

F N → X g∈M + t f (gx)dμ(g) - g∈M - t f (gx)dμ(g) dν(x) as N → ∞,
which completes the proof.

With these preparations, we start to prove Theorem 1.1.3, by admitting Lemma 1.5.2, Corollary 1.5.5 and Proposition 1.4.28.

Proof of Theorem 1.1.3. We will prove that there exist constants 0 > 0, C 0 > 0 such that for every s > 0, the Fourier transform e iξφ(θ) r(θ)dν(θ) is less than C 0 e -0 s for all |ξ| large enough depending on s.

Fix a constant 3 ≤ 1/10. Write t = (log |ξ| -s)/2, and take |ξ| large enough such that t > 10s.

Step 1: Let e ξ (x) be the function e iξφ(x) r(x). Using Proposition 1.3.5 and the Cauchy-Schwarz inequality, we have

X e ξ (x)dν(x) = g∈M + t X e ξ (gx)dν(x)dμ(g) - g∈M - t X e ξ (gx)dν(x)dμ(g) ≤ μ(M + t ) 1/2 M + t X e ξ (gx)dν(x) 2 dμ(g) 1/2 + μ(M - t ) 1/2 M - t X e ξ (gx)dν(x) 2 dμ(g) 1/2
. By Lemma 1.5.2 and Proposition 1.3.5, μ(M + t ), μ(M - t ) are uniformly bounded with t. Change the order of integration, then φ * (rdν)(ξ) X 2 M + t e iξ(φ(gx)-φ(gy)) r(gx)r(gy)dμ(g)dν(x)dν(y)

1/2 + X 2 M - t
e iξ(φ(gx)-φ(gy)) r(gx)r(gy)dμ(g)dν(x)dν(y)

1/2 . (1.3.1)
From now on, we only consider M + t . The set M - t has similar properties, and the needed changes will be discussed in remarks, which appear at the end of each section.

Step 2: The main approximation, which will be proved in Section 1.5, replaces the distance φ(gx) -φ(gy) with φ e -2κ(g) d(x, y). The intuition here is that in a large set, whose complement has exponentially small measure, the behavior is nice.

To apply replacement, some regularity conditions on x, y and g are needed. Define a subset of M + t for x, y in X by

M + t (x, y) = {g ∈ M + t ||κ(g) -κ(T g)| < 3 s, d(x m g , g -1 x) < e -t , d(g -1 x, x), d(g -1 x, y) > 2e -3 s }. (1.3.2)
For fixed x, y, set Λ 0 (g) = e iξ(φ(gx)-φ(gy)) r(gx)r(gy), Λ 1 (g) = e iξsign(g -1 x,x,y)φ (gx)d(x,y) exp(-2κ(g))/(d(g -1 x,x)d(g -1 x,y)) r(gx) 2 .

We give a control of the error, which appears in the replacement.

Proposition 1.3.6. Assume that t > 2s. We have an exponential decay for all g in M + t (x, y). That is

|Λ 0 (g) -Λ 1 (g)| = O exp (s). (1.3.3)
This property will be proved in Section 1.5. We want to use some smooth cutoffs to regularize the function Λ 1 (g, x, y). Let ρ be a smooth function on R such that ρ|

[-1,1] = 1, ρ takes values in [0, 1], suppρ ⊂ [-2, 2] and |ρ | ≤ 2. Let Λ 2 (g) = Λ 1 (g)(1-ρ(d(g -1 x, x)e 3 s ))(1-ρ(d(g -1 x, y)e 3 s ))ρ( κ(g) -κ(T g) 3 s
)ρ( κ(T g) -t 2 3 s ).

(1.3.4) When d(g -1 x, x) < e -3 s or d(g -1 x, y) < e -3 s , the function Λ 2 will be 0. With fixed x, y, sign(g -1 x, x, y) is a function of g -1 x, and the discontinuity is at x and y. Hence the discontinuity of sign(g -1 x, x, y) is removed in Λ 2 .

If g ∈ M + t (x, y), it follows from definition that |κ(T g) -t| ≤ |κ(g) -κ(T g)| ≤ 3 s. Then Λ 2 = Λ 1 . Since t > 10s, using Corollary 1.5.5, Lemma 1.5.2 and (1.3.3), we get

M + t (Λ 0 -Λ 2 )dμ(g) ≤ μ(M + t -M + t (x, y)) + M + t (x,y) (Λ 0 -Λ 2 )dμ(g) = μ(M + t -M + t (x, y)) + M + t (x,y) (Λ 0 -Λ 1 )dμ(g) = O exp (s).
(1.3.5)

Step 3: Introduce the residue process for the Cartan projection. This is inspired by the stopping time. For the stopping time, the existence of the limit distribution of the residual waiting time was proved in [START_REF] Kesten | Renewal theory for functionals of a Markov chain with general state space[END_REF], but in that paper we do not have a rate of convergence, which is necessary in our method. Here we use the transfer operator to get a uniform rate of convergence. It is difficult to treat the stopping time with transfer operators, because the operator will no longer be continuous. However, the residue process, which will be introduced here, can be routinely analyzed by the transfer operator. What's more, we will get the limit distribution of gx and g -1 y simultaneously, which is important to us.

We generalize the inverse action on Σ, letting g -1 = (g 1 , . . . , g ω ) -1 = (g -1 ω , . . . , g -1 1 ) for g in Σ. For a subset M of Σ, set ι(M ) = {g -1 |g ∈ M }. Let μ be the pushforward of µ by the inverse action. Let t be a positive number. Consider the limit of the following quantity as t → ∞ n≥0 κ(g)<t≤κ(hg) f ((hg) -1 x , hgx, κ(hg) -κ(g), κ(g) -t)dµ(h)dµ * n (g), where x, x are points in X and f is a smooth, compactly supported function on X 2 × R 2 . Our result is similar to renewal theory. By Proposition 1.4.28, when t tends to infinity, the limit is

X 2 G 0 -σ(h,y) f (y , hy, σ(h, y), u)dudµ(g)dν(y)dν(y ),
where ν is the stationary measure of μ and the integral 0 -σ(h,y 1 ) = 0 if σ(h, y 1 ) < 0. Since (T g) -1 = L(g -1 ) and κ(g -1 ) = κ(g), we can define

N + t = ι(M + t ) = {g ∈ Σ|κ(Lg) < t ≤ κ(g)}.
(1.3.6) Therefore

M + t Λ 2 (g)dμ(g) = N + t Λ 2 (g -1 )d μ(g).
Recall that x, y, ρ are fixed. For x 1 , x 2 in X and v, u in R, define

λ(x 1 , x 2 ) = d(x, y)e s sign(ξ)sign(x, y, x 2 )φ (x 1 )/(d(x 2 , x)d(x 2 , y)), ϕ(x 1 , x 2 , v, u) = r(x 1 ) 2 × (1 -ρ(d(x 2 , x)e 3 s ))(1 -ρ(d(x 2 , y)e 3 s ))ρ( v 3 s )ρ( u 2 3 s ).
By the relation ξ = sign(ξ)e 2t+s , regroup the terms and rewrite the function Λ 2 (g -1 ) = e iλ(g -1 x,gx) exp(-2(κ(g)-t)) ϕ(g -1 x, gx, κ(g) -κ(Lg), κ(Lg) -t).

(1.3.7)

Note that the function λ is not continuous, but the function ϕ will remove the discontinuity as we have discussed in Step 2. In the language of the residue process, let f be the function on X 2 × R 2 defined by

f (x 1 , x 2 , v, u) = e iλ(x 1 ,x 2 ) exp(-2(u+v)) ϕ(x 1 , x 2 , v, u). (1.3.8)
Thus the function Λ 2 (g -1 ) can be written as

Λ 2 (g -1 ) = f (g -1 x, gx, κ(g) -κ(Lg), κ(Lg) -t).
By Proposition 1.4.28, for δ > 0, t > 2(|K| + δ) (where K is the projection of suppf onto R v ), we have

M + t Λ 2 dμ(g) = N + t f d μ(g) = X 2 G 0 -σ(h,x 2 ) f (x 1 , hx 2 , σ(h, x 2 ), u)dudµ(g)dν(x 2 )dν(x 1 ) + O K (δ + O δ /t)|f | Lip .
(1.3.9)

Here |f | Lip is the Lipschitz norm defined by

|f | Lip = |f | ∞ + sup (x 1 ,x 2 ,v,u) =(x 1 ,x 2 ,v ,u ) |f (x 1 , x 2 , v, u) -f (x 1 , x 2 , v , u )| d(x 1 , x 1 ) + d(x 2 , x 2 ) + |v -v | + |u -u | .
Lemma 1.3.7. There exist constants δ 0 (s) and t(δ, s) such that if δ < δ 0 (s) and t > t(δ, s), then

O K (δ + O δ /t)|f | Lip ≤ e -s .
(1.3.10)

Proof. By the definition of ρ and f , the support of f is in the compact set X 2 × [-4 3 s, 4 3 s] 2 . The size of K, the projection of suppf onto R v , is bounded by 8 3 s.

The definition of ρ implies that f is locally Lipschitz. Together with the fact that f is compactly supported, we conclude that |f | Lip is controlled by e 2s independently of x, y. Take δ small enough according to s, then take t large enough according to δ and s. We get the inequality.

Step 4: For the major term in (1.3.9), use the following lemma.

e iλ exp(-u) du = b 2 b 1 ∂ u (e iλ exp(-u) ) -iλe -u du = e iλ exp(-u) -iλe -u b 2 b 1 + b 2 b 1 e iλ exp(-u) ∂ u 1 -iλe -u du. This implies that | b 2 b 1 e iλ exp(-u) du| ≤ e iλ exp(-u) λe -u b 2 b 1 + b 2 b 1 ∂ u ( e u |λ| )du ≤ 2(e b 1 + e b 2 ) |λ| .
The proof is complete.

Chapter 1. Fourier decay of SL 2 (R)

When d(x, y) > e -3 s , due to the definition of ρ( v 3 s ), the major term only integrates on h, x 2 such that |σ(h, x 2 )| ≤ 2 3 s. The inequality |u| ≤ |σ(h, x 2 )| ≤ 2 3 s implies that ρ( u 2 3 s ) = 1. By the hypotheses on φ, when r(x 1 ) = 0, we have |φ (x 1 )| ≥ 1/C 1 > 0. Therefore |λ(x 1 , hx 2 )| = |d(x, y)e s sign(ξ)sign(x, y, hx 2 )φ (x 1 )/(d(hx 2 , x)d(hx 2 , y))| ≥ e (1-3 s) /C 1 .

We use Lemma 1.3.8 to obtain

| 0 -σ(h,x 2 ) f du| ≤ |r| 2 ∞ σ(h,x 2 ) 0 e iλ exp(-2u) du ≤ C 1 |r| 2 ∞ 1 + e 2 3 s e 1-3 s ≤ |r| 2 ∞ 2e (3 3 -1)s C 1 .
Combined with (1.3.9), they imply that M + t Λ 2 dμ(g) = O exp (s). When d(x, y) ≤ e -3 s , the Hölder regularity of stationary measure (1.2.3) implies that

X×X 1 d(x,y)≤e -3 s dν(x)dν(y) ≤ X ν(B(x, e -3 s ))dν(x) = O exp (s).
Finally we obtain

X 2 M + t Λ 2 (x, y)dμ(g)dν(x)dν(y) ≤ X 2 1 d(x,y)>e -3 s M + t Λ 2 (x, y)dμ(g)dν(x)dν(y) + X 2 1 d(x,y)≤e -3 s M + t Λ 2 (x, y)dμ(g)dν(x)dν(y) ≤ O exp (s)(1 + μ(M + t )).
By Lemma 1.5.2, the measure μ(M + t ) is uniformly bounded. By using (1.3.1) and (1.3.5), the proof is complete.

Remark 1.3.9 (Minus case). For M - t , we have another version of Lemma 1.5.2, Corollary 1.5.5 and Proposition 1.4.28. The integral | 0 -σ(h,y 1 ) f du| is replaced by | -σ(h,y 1 ) 0 f du|.

Remark 1.3.10. When s is large and ξ is of size e Cs , all the error terms have polynomial decay except the one from Proposition 1.4.28. As we have mentioned in Remark 1.1.10, a uniform spectral gap makes Proposition 1.4.28 effective. Then we will have a polynomial decay.

The uniformity with respect to r C 1 , φ C 2 and 1/ inf suppr |φ | is due to the fact that all the terms depend only on these norms and the measure µ.

Renewal theory

We define a renewal operator R as follows. For a positive bounded Borel function f on X × R, a point x in X and a real number t, we set

Rf (x, t) = +∞ n=0 G f (gx, σ(g, x) -t)dµ * n (g).
Because of the positivity of f , this sum is well defined. In [START_REF] Kesten | Renewal theory for functionals of a Markov chain with general state space[END_REF], Kesten proved a renewal theorem for Markov chains, which is valid in our case [START_REF] Guivarc | Spectral gap properties for linear random walks and Pareto's asymptotics for affine stochastic recursions[END_REF]. But a uniform speed of convergence is needed. We will give a proof using the complex transfer operator, which fulfills our demands. The treatment of the transfer operator will be along the path in [START_REF] Boyer | The rate of convergence for the renewal theorem in R d[END_REF]. The renewal theorem will give us an equidistribution phenomenon, where the key input is non-arithmeticity.

First we give a proof of renewal theorem for good functions. Then we prove some regularity properties and independence properties for the renewal process. These will imply a version of residue process. Finally, we prove a theorem for the Cartan projection from a similar theorem for the cocycle.

Fix the constant = σ µ /4 in this section. Keep in mind that the assumptions of Theorem 1.1.1 are always satisfied.

Complex transfer operators

We introduce the complex transfer operator P (z). Let H γ (X) be the space of γ-Hölder functions on X, a Banach space with the norm |f

| γ = |f | ∞ + m γ (f ) = |f | ∞ + sup x =y |f (x)-f (y)| d(x,y) γ .
For f in H γ (X) and a complex number z, define

P (z)f (x) = G e -zσ(g,x) f (gx)dµ(g).
The main properties of P (z) are summarized as follows Proposition 1.4.1. [Boy16, Theorem 4.1, Lemma 4.7] For any γ > 0 small enough, there exists η > 0 such that when | z| < η, the transfer operator P (z) is a bounded operator on H γ (X) and depends analytically on z. Moreover there exists an analytic operator U (z) on a neighborhood of 0 ≤ z < η such that the following equality holds for 0 ≤ z < η

(I -P (z)) -1 = 1 σ µ z N 0 + U (z), (1.4.1)
where N 0 is the operator defined by N 0 f = f dν Remark 1.4.2. In Proposition 1.4.1, the non-arithmeticity is crucial to prove that (I -P (z)) -1 has only one pole in the imaginary axis, which is 0. The non-arithmeticity follows from Zariski density. See for instance [START_REF] Benoist | Propriétés asymptotiques des groupes linéaires. II. In Analysis on homogeneous spaces and representation theory of Lie groups[END_REF] and [START_REF] Dal | Topologie du feuilletage fortement stable[END_REF].

The assumption of Theorem 4.1 in [START_REF] Boyer | The rate of convergence for the renewal theorem in R d[END_REF] are complicated. It is verified, in the proof of theorem 1.4, page 8 [START_REF] Boyer | The rate of convergence for the renewal theorem in R d[END_REF], that our condition on µ is enough to apply Theorem 4.1. The idea is due to Guivarch and Le Page.

Proposition 1.4.3. [Boy16, Lemma 4.4] For any γ > 0 small enough, there exist η > 0, 0 < ρ < 1, C > 0 such that when 0 ≤ z < η, for a natural number n and a γ-Hölder function f , we have

|P (z) n f | ∞ ≤ (Cρ n ) z |f | ∞ (1.4.2)
Remark 1.4.4. For further usage, we need a bound on γ. Let , ( ) be the two constants in (1.2.11), that is µ * n {d(gx, x ) ≤ e -s } ≤ Ce -s , and 1 the constant in exponential moment. Choose a small γ such that γ ≤ 1 4 max{ σµ/4

(σµ/4) , 1 }.

Renewal theory for regular functions

We start to compute the renewal operator. A result for the renewal operator for "good" functions will be proved. Let f be a function on

X × R. Define a norm by |f | L ∞ H γ = sup ξ∈R |f (x, ξ)| H γ , which is the supremum of the Hölder norm of f (•, ξ). Define another norm |f | W 1,∞ H γ = |f | L ∞ H γ + |∂ ξ f | L ∞ H γ . Write the Fourier transform f (x, ξ) = e iξu f (x, u)du.
Proposition 1.4.5. Let f be a positive bounded continuous function in L 1 (X×R, ν⊗Leb) such that its Fourier transform satisfies f ∈ L ∞ H γ and ∂ ξ f ∈ L ∞ H γ . Assume that the projection of supp f onto R is in a compact set K. Then for all t > 0 and x in X, we have

Rf (x, t) = 1 σ µ ∞ -t f (y, u)dudν(y) + 1 t O K (| f | W 1,∞ H γ ).
Proof. Combine the following two lemmas.

Lemma 1.4.6. Under the same assumption as in Proposition 1.4.5, we have

Rf (x, t) = 1 σ µ ∞ t f (y, u)dudν(y) + 1 2π e itξ U (iξ) f (x, ξ)dξ.
Proof. Introduce a local notation: for (x, t) in X × R and s ≥ 0, write (g,x) f (gx, σ(g, x) + t)dµ(g).

B s f (x, t) = G e -sσ
When s = 0, we abbreviate the notation B 0 to B. We want to prove the following equality,

n≥0 B n (f )(x, t) = lim s→0 + n≥0 B n s (f )(x, t). (1.4.3)
By definition, one has

B n s (f )(x, t) = G e -sσ(g,x) f (gx, σ(g, x) + t)dµ * n (g) = G e -sσ(g,x) (1 σ(g,x)>0 + 1 σ(g,x)≤0 )f (gx, σ(g, x) + t)dµ * n (g).
• The part 1 σ(g,x)>0 , since f ≥ 0, use the monotone convergence theorem. When

s → 0 + then n≥0 G e -sσ(g,x) 1 σ(g,x)>0 f (gx, σ(g, x) + t)dµ * n (g) → n≥0 G
1 σ(g,x)>0 f (gx, σ(g, x) + t)dµ * n (g).

• For the part 1 σ(g,x)≤0 , take

s in [0, η/2]. Proposition 1.4.3 implies that G e -sσ(g,x) 1 σ(g,x)≤0 f (gx, σ(g, x)+t)dµ * n (g) ≤ G e -ησ(g,x)/2 |f | ∞ dµ * n (g) ≤ (Cρ n ) η/2 |f | ∞ .
Since n≥0 ρ nη/2 is finite, take e -ησ(g,x)/2 |f | ∞ as the dominant function. Then use the dominated convergence theorem to conclude.

This proves equation (1.4.3).

Using the inverse Fourier transform, we have

n≥0 B n s (f )(x, t) = n≥0 G e -sσ(g,x) f (gx, σ(g, x) + t)dµ * n (g) = n≥0 G
e -sσ(g,x) 1 2π R e iξ(σ(g,x)+t) f (gx, ξ)dξdµ * n (g).

(1.4.4)

Since f (x, ξ) has compact support, | f (x, ξ)| ≤ | f (x, ξ)| L ∞ ξ H γ and |P (s) n 1| ≤ Cρ sn for s in [0, η/2] (Proposition 1.4.3), we have n≥0 G e -sσ(g,x) R | f (gx, ξ)|dξdµ * n (g) ≤ C f n≥0 G e -sσ(g,x) dµ * n (g) = C f n≥0 P (s) n (1) < ∞,
which implies that the right hand side of (1.4.4) is absolutely convergent. Consequently, we can use the Fubini theorem to change the order of the integration. By the hypothesis f (x, ξ) ∈ H γ (X), Proposition 1.4.1 implies that n≥0

B n s (f )(x, t) = 1 2π R n≥0 G e (-s+iξ)σ(g,x) f (gx, ξ)dµ * n (g)e itξ dξ = 1 2π R n≥0 P n (s -iξ) f (x, ξ)e itξ dξ = 1 2π R (1 -P (s -iξ)) -1 f (x, ξ)e itξ dξ = 1 2π R ( N 0 σ µ (s -iξ) + U (s -iξ)) f (x, ξ)e itξ dξ.
Since 1 s-iξ =

+∞ 0 e -(s-iξ)u du for s > 0, together with the property f ∈ L 1 (R), we have

1 2π R N 0 σ µ (s -iξ) f (x, ξ)e itξ dξ = 1 2π 1 σ µ X R f (y, ξ) s -iξ e itξ dξdν(y) = 1 σ µ X ∞ 0 f (y, u + t)e -su dudν(y).
When s → 0 + , since f is integrable with respect to the product measure ν ⊗ Leb, by monotone convergence theorem, the limit is 1 σµ X ∞ t f (y, u)dudν(y). Since f (x, ξ) is compactly supported, we have

lim s→0 + R U (-s + iξ) f (x, ξ)e itξ dξ = R U (iξ) f (x, ξ)e itξ dξ.
The proof is complete.

Lemma 1.4.7. Under the same assumption as in Proposition 1.4.5, we have

| e -itξ U (iξ) f (x, ξ)dξ| ≤ 1 t O K | f | L ∞ H γ + |∂ ξ f | L ∞ H γ .
Proof. Use the fact that f (x, ξ) is compactly supported and

| f (x, ξ)| H γ , |∂ ξ f (x, ξ)| H γ < ∞.
Then applying integration by parts, we have

e -itξ U (iξ) f (x, ξ)dξ = 1 it e -itξ ∂ ξ (U (iξ) f (x, ξ))dξ = 1 it e -itξ ∂ ξ (U (iξ)) f (x, ξ) + U (iξ)∂ ξ f (x, ξ) dξ.
Since the operator norms of U (iξ) and ∂ ξ U (iξ) are uniformly bounded on compact regions, the result follows.

Regularity properties of renewal measures

We have two principles in this subsection. Principle 1: Let f be a bounded Borel function supported in X × [0, a]. When we take the renewal sum outside of the interval

I t = [ t σµ+ , t+a σµ-], n∈N-It G f (gx, σ(g, x)-t)dµ * n (g) = n∈N-It G f (gx, σ(g, x)-t)1 [0,a] (σ(g, x)-t)dµ * n (g),
this sum decays exponentially with t. This is given by the large deviations principle (Corollary 1.2.9, 1.2.11). For n in the interval I t , if some property is valid for each n with an exponential error of n, we sum up. Since the length of this interval is comparable with t, this property is also valid for the renewal sum with an exponential error of t. Principle 2: The other is independence. By Proposition 1.4.5, the limit distribution of (σ(g, x) -t, gx) is 1 σµ ν ⊗ Leb, which is a product measure. That roughly means the following: As in Remark 1.3.4, let

X n = b n • • • b 1 be a random walk on G. Let F = F 1 ×F 2 where F 1 , F 2 are Borel subsets of X, R respectively. Then n≥0 P{(X n x, σ(X n , x) -t) ∈ F 1 × F 2 } → 1 σ µ ν(F 1 ) ⊗ Leb(F 2 ) as t → +∞.
More concretely, we could expect that R(1

F 1 ×F 2 )(x, t) is almost 1 σµ ν(F 1 ) ⊗ Leb(F 2 ) when t is large.
We want to use convolution to smooth out the target function. There exists an even function ψ such that it is a probability density, and the Fourier transform ψ is compactly supported. Let

ψ δ (t) = 1 δ 2 ψ( t δ 2 ). Then δ -δ ψ δ (t)dt = 1/δ -1/δ ψ(t)dt > 1 -Cδ. Proposition 1.4.8. Let δ ≤ 1/3 and b 2 ≥ b 1 . If b 2 -b 1 ≥ 2δ, then for x in X and t > 0, we have R(1 [b 1 ,b 2 ] )(x, t) (b 2 -b 1 )(1/σ µ + C δ (1 + |b 2 | + |b 1 |)/t).
(1.4.5)

If 0 ≤ b 2 -b 1 < 2δ, then for x in X and t > 0, we have R(1 [b 1 ,b 2 ] )(x, t) δ(1/σ µ + C δ (1 + |b 1 |)/t). (1.4.6) Proof. When b 2 -b 1 ≥ 2δ, if u is in [b 1 , b 2 ], then [u -b 2 , u -b 1 ] contains at least one of [0, δ] or [-δ, 0]. Therefore ψ δ * 1 [b 1 ,b 2 ] (u) = b 2 b 1 ψ δ (u -v)dv ≥ δ 0 ψ(v)dv ≥ (1 -δ)/2.
Then

1 [b 1 ,b 2 ] ≤ 3ψ δ * 1 [b 1 ,b 2 ] .
(1.4.7)

It is sufficient to bound R(ψ δ * 1 [b 1 ,b 2 ] ). Proposition 1.4.5 implies that R(ψ δ * 1 [b 1 ,b 2 ] ) = 1 σ µ ∞ -t ψ δ * 1 [b 1 ,b 2 ] + O δ t | ψδ 1[b 1 ,b 2 ] | W 1,∞ H γ .
The first term is less than

ψ δ * 1 [b 1 ,b 2 ] = (b 2 -b 1 )
. For the second term, we have

| ψδ 1[b 1 ,b 2 ] | W 1,∞ H γ = | ψδ 1[b 1 ,b 2 ] | L ∞ H γ + |∂ ξ ψδ 1[b 1 ,b 2 ] | L ∞ H γ = | ψδ 1[b 1 ,b 2 ] | L ∞ + |∂ ξ ψδ 1[b 1 ,b 2 ] | L ∞ ≤ C δ (|1 [b 1 ,b 2 ] (u)| L 1 + |u1 [b 1 ,b 2 ] (u)| L 1 ) ≤ C δ (b 2 -b 1 )(1 + |b 1 | + |b 2 |). When b 2 -b 1 ∈ [0, 2δ], the renewal sum R(1 [b 1 ,b 2 ] ) is bounded by R(1 [b 1 ,b 1 +2δ]
). Then use the previous case.

In Proposition 1.4.5, since we do not have a good control of the spectral radius of the operator U (iξ) for large |ξ|, the estimates are effective only for large t, which means that when t is small the error term will be out of control. The following lemma combines the transfer operator and the large deviations principle to give a uniform estimate.

Lemma 1.4.9. For real numbers s, t and a point x in X, we have

R(1 [0,s] )(x, t) max{1, s}.
(1.4.8)

Proof. We can suppose that s > 1. If not, then R(1 [0,s] )(x, t) ≤ R(1 [0,1] )(x, t). When t ≥ s, this is a direct corollary of Proposition 1.4.8. Fixing δ = 1/3, we get

1 + |b 1 | + |b 2 | t ≤ 1 + s t ≤ 2.
Then R(1 [0,s] )(x, t) s(1/σ µ + 2C δ ). When t < s, let m = [max{0, (t + s)/(σ µ -)}] + 1. By Corollary 1.2.9, we have

R(1 [0,s] )(x, t) ≤ R(1 [0,2s] )(x, 0) ≤ n≤m µ ⊗n {σ(g, x) ≤ 2s} + n>m µ ⊗n {σ(g, x) ≤ 2s} m + e -m s.
The proof is complete.

In the renewal theorem, the limits of the scalar part σ(g, x) and the angle part gx are independent. Using this spirit, we give the following lemma, which quantifies this independence. In the proof, when t is large enough, using Proposition 1.4.5, the remainder term will be small. When t is small, we have another estimate from the regularity of the convolution measure µ ⊗n . Proposition 1.4.10. For s > 0, a > 0, t > 5s, and x, x in X, we have

R(1 B(x ,e -s )×[0,a] )(x, t) = (1 + a) 2 O exp (s).
(1.4.9)

Proof. Decompose the region of t into two parts: Then the measure of this part, summing up the above inequality over all n ∈ [t/(σ µ + ), (t + a)/(σ µ -)], is less than C(t + a)e -s/ (1 + a)e -γs (here we use the Remark 1.4.4, 4γ ≤ / ).

• When 5s < t ≤ e 2γs ,
• When t ≥ e 2γs , we take f = 1 [0,a] (x) where (x) is a function on X such that B(x ,e -s ) = 1, supp ⊂ B(x , 2e -s ) and | | γ ≤ e γs . As in the proof of Proposition 1.4.8, we use ψ δ to regularize this function. By (1.4.7), we have

3R(ψ δ * f )(x, t) ≥ R(1 B(x ,e -s )×[0,a] )(x, t). Proposition 1.4.5 implies R(ψ δ * f ) = 1 σ µ X ∞ -t ψ δ * f (x, u)dudν(x) + C δ t (| ψ δ * f | W 1,∞ H γ ).
Since ψ δ * f (x, ξ) = ψδ (ξ) 1[0,a] (ξ) (x), the two functions are independent. We can use the same estimate as in the proof of Proposition 1.4.8. So the rest term is less than C δ (1 + a) 2 e γs /t. The major term, due to the regularity of the stationary measure (1.2.3), is controlled by ae -αs /σ µ . The result follows from the hypothesis t > e 2γs .

The proof is complete.

We also need the independence of σ(g, x) and g -1 x o , where x, x o are two points in X. For proving this property, we pass through the Cartan projection, because the order of products in the Cartan projection can be reversed. The following proof uses Lemma 1.2.14, which is a central tool to prove a renewal type theorem for the Cartan projection from a renewal type theorem for the cocycle.

Let f be a positive bounded Borel function on X × R. For (x, t) ∈ X × R, we define

R P (f )(x, t) = n≥0 G f (gx, κ(g) -t)dµ * n (g).
Lemma 1.4.11. For s > 0, a > 0, t > 10s, and x, x in X, we have

R P (1 B(x ,e -s )×[0,a] )(x, t) = (1 + a) 2 O exp (s).
(1.4.10)

Proof. Due to Corollary 1.2.9 and Corollary 1.2.11, the sum of the integral of n ≤ t/(σ µ + ) and n ≥ (t + a)/(σ µ -) is exponentially small. If suffices to consider n in the interval I t = [t/(σ µ + ), (t + a)/(σ µ -)]. Fix l = [ 4 t/σ µ ] with 4 = 1/10. By Lemma 1.2.14, there exists S n,l,x ⊂ G ×n such that µ ⊗n S c n,l,x = O exp (l), and for (g n , . . . , g 1 ) in S n,l,x , letting g = (g n , . . . , g l+1 ) and j = (g l , . . . , g 1 ), we have

|κ(gj) -σ(g, jx) -κ(j)| ≤ e -l ≤ 1. Thus µ ⊗n {κ(gj) ∈ [t, t + a], d(gjx, x ) ≤ e -s } ≤µ ⊗n {S c n,l,x } + µ ⊗n {gj ∈ S n,l,x |κ(gj) ∈ [t, t + a], d(gjx, x ) ≤ e -s } ≤O exp (l) + µ ⊗n {σ(g, jx) + κ(j) ∈ [t -1, t + a + 1], d(gjx, x ) ≤ e -s }.
Therefore summing over n and integrating first with respect to g, we get

n∈It µ ⊗n {κ(gj) ∈ [t, t + a], d(gjx, x ) ≤ e -s } ≤ n∈It µ ⊗n {σ(g, jx) + κ(j) ∈ [t -1, t + a + 1], d(gjx, x ) ≤ e -s } + tO exp (l) ≤tO exp (l) + R(1 B(x ,e -s )×[-1,a+1] )(jx, t -κ(j))dµ * l (j). (1.4.11) Hence, it is sufficient to bound R(1 B(x ,e -s )×[-1,a+1] )(jx, t -κ(j))dµ * l (j). Let G l, = {j ∈ G ×l |κ(j) ≤ l(σ µ + )}. By the large deviations principle (Corollary 1.2.11), we have µ * l G c l, = O exp (l). • For j ∈ G l, , we have t -κ(j) ≥ t -l(σ µ + ) = t -4 (σ µ + )t/σ µ > t/2 ≥ 5s.
Hence, Proposition 1.4.10 implies that

R(1 B(x ,e -s )×[-1,a+1] )(jx, t -κ(j)) (1 + a) 2 O exp (s).
• For j ∈ G c l, , Lemma 1.4.9 implies that

R(1 B(x ,e -s )×[-1,a+1] )(jx, t -κ(j)) (1 + a).
Combining the above two inequalities, we have

R(1 B(x ,e -s )×[-1,a+1] )(jx, t -κ(j))dµ * l (j) (1 + a) 2 O exp (s) + O exp (l)(1 + a).
(1.4.12)

The proof is complete.

There is a byproduct of the above lemma. When the function f does not depend on X, abbreviate R P (f )(x, t) by R P (f )(t).

Lemma 1.4.12. For real numbers s, t, we have

R P (1 [0,s] )(t) max{1, s 2 }.
(1.4.13)

Remark 1.4.13. Here the term s 2 is not optimal. With some extra work, it can be improved to s.

Proof. Suppose that s ≥ 1. If not, then R p (1 [0,s] )(x, t) ≤ R p (1 [0,1] )(x, t). When t ≥ 10, apply Lemma 1.4.11 with a = s, e -s = e -1 , x = x j , j ∈ J, where J is a finite set such that ∪ j∈J B(x j , e -1 ) covers X. So we get R P (1 [0,s] )(t) s 2 . When t < 10 ≤ 10s, let m = [max{0, (t + s)/(σ µ -)}] + 1. By Corollary 1.2.11, we have R P (1 [0,s] )(t) ≤ R P (1 [0,2s] )(0) ≤ n≤m µ * n {κ(g) ≤ t + s} + n>m µ * n {κ(g) ≤ t + s} m + e -m s.
The proof is complete. Now we are going to prove the independence of σ(g, x) and g -1 x. Recall that μ is the pushforward of µ by the inverse action. Let f be a positive bounded Borel function on

X × R. For (x o , x, t) ∈ X 2 × R, we define R I (f )(x o , x, t) = n≥0 G f (g -1 x o , σ(g, x) -t)dµ * n (g).
Proposition 1.4.14. For s > 0, a > 0, t > max{10s, 10}, and x, x , x o in X, we have

R I (1 B(x ,e -s )×[0,a] )(x o , x, t) = (1 + a) 2 O exp (s).
(1.4.14)

Proof. Due to Corollary 1.2.9 and Corollary 1.2.11, the sums of the integral of n ≤ t/(σ µ + ) and n ≥ (t + a)/(σ µ -) is exponentially small. It suffices to consider n in the interval

I t = [t/(σ µ + ), (t + a)/(σ µ -)]. Let G ,n = {g ∈ G ×n |κ(g) ≥ n(σ µ -/2), d(g -1 x o , x) > e -n , d(g -1 x o , x m g ) ≤ e -(2σµ-)n }.
By inequalities (1.2.9), (1.2.11) and (1.2.13), we have

µ ⊗n G ,n ≥ 1 -O exp (n). Since t > 10, for n in I t , we have n ≥ t/(σ µ + ) ≥ 10/(σ µ + ). For g ∈ G ,n , we have e -2κ(g) + d(x m g , g -1 x o ) d(g -1 x o , x) ≤ 2e -(2σµ-)n e -n = 2e -(2σµ-2 )n ≤ 2e -20(σµ-)/(σµ+ ) ≤ 1/2.
Using Lemma 1.2.7 with g ∈ G ,n , we have

|σ(g, x) -κ(g) -log d(g -1 x o , x)| ≤ 2 e -2κ(g) + d(x m g , g -1 x o ) d(g -1 x o , x) ≤ 4e -(2σµ-2 )n ≤ 1.
Therefore,

µ ⊗n {σ(g, x) ∈ [t, t + a], d(g -1 x o , x ) ≤ e -s } ≤ O exp (n)+ µ ⊗n {κ(g) ∈ [t -1, t + a + 1] -log d(g -1 x o , x), d(g -1 x o , x ) ≤ e -s }.
Summing up over I t and using the definition of μ, we have

n∈It µ ⊗n {σ(g, x) ∈ [t, t + a], d(g -1 x o , x ) ≤ e -s } ≤O exp (t) + n≥0 µ ⊗n {κ(g) ∈ [t -1, t + a + 1] -log d(g -1 x o , x), d(g -1 x o , x ) ≤ e -s } =O exp (t) + n≥0 μ⊗n {κ(g) ∈ [t -1, t + a + 1] -log d(gx o , x), d(gx o , x ) ≤ e -s }. (1.4.15) Hence, it is sufficient to bound R P (1 u+log d(y,x)∈[-1,a+1],d(y,x )≤e -s )(x o , t)
, where we use (y, u) to denote the variables, and the measure µ is replaced by μ. For simplicity, we use the same notation R P . Cutting the region along {y ∈ X| log d(y, x) ≤ -t 1 } and the subsets {y ∈ X| log d(y, x) ∈ [-(k + 1)s, -ks]} for 0 ≤ k < t 1 /s, where t 1 = (t -1)/9.

• When k = 0, since t -1 > 10s, we can use Lemma 1.4.11 to obtain

R P (1 u+log d(y,x)∈[-1,a+1],d(y,x )≤e -s ,d(y,x)≥e -s )(x o , t) ≤R P (1 d(y,x )≤e -s ,u∈[-1,s+a+1] )(x o , t) (1 + s + a) 2 e -s .
• When 0 < k < t 1 /s, since t + ks -1 > 10ks, again we use Lemma 1.4.11

R P (1 u+log d(y,x)∈[-1,a+1],d(y,x )≤e -s ,d(y,x)∈[e -(k+1)s ,e -ks ] )(x o , t) ≤R P (1 d(y,x)≤e -ks ,u∈[-1+ks,a+1+(k+1)s] )(x o , t) (1 + s + a) 2 e -ks .
• In the last case, log d(y, x) ≤ -t 1 , we have

R P (1 u+log d(y,x)∈[-1,a+1],d(y,x )≤e -s ,d(y,x)≤e -t 1 )(x o , t) ≤R P (1 u+log d(y,x)∈[-1,a+1],d(y,x)≤e -t 1 )(x o , t).

This is similar to the original quantity R

P (1 u+log d(y,x)∈[-1,a+1],d(y,x )≤e -s )(x o , t).
The difference is that here t 1 is comparable with t, which is crucial in the following argument. Return to the definition of R P , and discuss on the length n = ω(g).

-When n > (t + a + 1)/(σ µ -2 ), by inequality (1.2.9) and (1.2.11), we have

μ⊗n {g ∈ G ×n |κ(g)-nσ µ ≤ n , d(gx o , x) ≥ e -n } > 1-Ce -n
. By hypothesis n > (t + a + 1)/(σ µ -2 ) , the element in this set satisfies

κ(g) ≥ (σ µ -)n > t + a + 1 + n ≥ t + a + 1 -log d(gx o , x). Thus μ⊗n {g ∈ G ×n |κ(g) ∈ [t-1, t+a+1]-log d(gx o , x)} = O exp (n).
Summing over n, we see that the measure of this part is O exp (t).

-

When n ∈ [(t -1)/(σ µ + ), (t + a + 1)/(σ µ -2 )], since n ≥ (t -1)/(σ µ + ) > (t -1)/9 = t 1 , Corollary 1.2.13 implies that μ⊗n {g ∈ G ×n |d(gx o , x) ≤ e -t 1 } = O exp (t 1 ) = O exp (t).
-When n ≤ (t -1)/(σ µ + ), Corollary 1.2.11 implies the measure of this part is O exp (t).

Therefore we have

R P (1 u+log d(y,x)∈[-1,a+1],d(y,x)≤e -t 1 )(x o , t) = O exp (t 1 ).
Combining the three cases, we have finished the proof.

Residue process

We introduce the residue process, which not only deals with σ(g

n g n-1 • • • g 1 , x) but also takes into account the next step σ(g n+1 , g n g n-1 • • • g 1 x). Let f be a positive bounded Borel function on X × R 2 . For (x, t) ∈ X × R, we define the residue operator by Ef (x, t) = n≥0 f (hgx, σ(h, gx), σ(g, x) -t)dµ * n (g)dµ(h).
(1.4.16)

Let F u f (x, v, ξ) = f (x, v, u)e iuξ du be the Fourier transform on R u . Let F be a function on X × R v × R ξ ,. Define a partial Lipschitz norm by |F | L ∞ Lip = sup ξ∈R |F (ξ)| Lip = sup ξ∈R |F (ξ)| ∞ + sup (x,v),(x ,v )∈X×R |F (x, v, ξ) -F (x , v , ξ)| d(x, x ) + |v -v | . Proposition 1.4.15 (Residue process). If f is a positive bounded continuous function on X × R 2 . Assume that the projection of suppF u (f ) onto R ξ is contained in a compact set K, and |F u (f )| L ∞ Lip , |∂ ξ F u (f )| L ∞ Lip are finite.
Then for t > 0 and x ∈ X, we have

Ef (x, t) = 1 σ µ ∞ -t G X f (hy, σ(h, y), u)dν(y)dµ(h)du + 1 t O K (|F u (f )| L ∞ Lip + |∂ ξ F u (f )| L ∞ Lip ) .
(1.4.17)

Proof. For a bounded continuous function f on X × R 2 and (x, u) ∈ X × R, we define an operator Q by Qf (x, u) = G f (hx, σ(h, x), u)dµ(h). Then Ef (x, t) = n≥0 Qf (gx, σ(g, x) -t)dµ * n (g) = R(Qf )(x, t).
We want to use Proposition 1.4.5, so we need to verify the hypotheses. The function Qf is bounded and integrable by the hypotheses on f . Then

Qf (x, ξ) = Qf (x, u)e iuξ du = f (hx, σ(h, x), u)e iuξ dudµ(h) = G F u f (hx, σ(h, x), ξ)dµ(h).
Thus Qf is also compactly supported on ξ. It remains to estimate the Hölder norm of

Qf . Since F u f (x, v, ξ) is Lipschitz on (x, v) ∈ X × R, this implies that | Qf (x, ξ)-Qf (y, ξ)| ≤ G |F u f (hx, σ(h, x), ξ) -F u f (hy, σ(h, y), ξ)|dµ(h) ≤ G |F u f | L ∞ Lip (d(hx, hy) + |σ(h, x) -σ(h, y)|)dµ(h).
Using Lipschitz property of the distance and the cocycle, and finite exponential moment, we have

| Qf (x, ξ)-Qf (y, ξ)| ≤ |F u f | L ∞ Lip d(x, y) γ G (1 + κ(h)) h 2γ dµ(h) |F u f | L ∞ Lip d(x, y) γ ,
where we use the Remark 1.4.4 that 4γ ≤ 1 . Therefore Lemma 1.4.16 (Change of norm). Under the assumptions of Proposition 1.4.15, we have

| Qf | L ∞ ξ H γ |F u (f )| L ∞ Lip , |∂ ξ Qf | L ∞ ξ H γ |∂ ξ F u f | L ∞ Lip .
Proof. The second inequality follows by the same computation.

By Proposition 1.4.5, we have

R(Qf )(x, t) = 1 σ µ X ∞ -t Qf (y, u)dudν(y) + 1 t O K | Qf | L ∞ ξ H γ + |∂ ξ Qf | L ∞ ξ H γ = 1 σ µ X ∞ -t Qf (y, u)dudν(y) + 1 t O K (|F u (f )| L ∞ Lip + |∂ ξ F u (f )| L ∞ Lip ) .
The proof is complete.

Residue process with cutoff

In this section, we restrict the residue process to the sequences (g n+1 , g n , . . . , g 1 )

such that σ(g n • • • g 1 , x) < t ≤ σ(g n+1 • • • g 1 , x). Let f be a function on X × R 2 . Define a Lipschitz norm by |f | Lip = |f | ∞ + sup (x,v,u) =(x ,v ,u ) |f (x, v, u) -f (x , v , u )| d(x, x ) + |v -v | + |u -u | . (1.4.18)
Define an operator from bounded Borel functions on X × R 2 to functions on X × R by

E C f (x, t) = n≥0 σ(g,x)<t≤σ(hg,x) f (hgx, σ(h, gx), σ(g, x) -t)dµ(h)dµ * n (g).
By Lemma 1.4.21, which will be proved later, this operator is well defined. Let K be a compact set in R. We denote |K| by the supremum of the distance between a point in K and 0.

Proposition 1.4.17. Let f be a continuous function on X×R 2 with |f | Lip finite. Assume that the projection of suppf on R v is contained in a compact set K. For all δ > 0, t > |K| + δ and x ∈ X, we have

E C f (x, t) = X G 0 -σ(h,y) f (hy, σ(h, y), u)dudµ(h)dν(y) + O K (δ + O δ /t)|f | Lip , (1.4.19)
where O K does not depend on δ, f, t, x, and the integral

0 -σ(h,y) = 0 if σ(h, y) < 0.
Remark 1.4.18. We decompose f into real and imaginary parts, then decompose these two parts into positive and negative parts. Each part satisfies the hypotheses of Proposition 1.4.17, with the support and the Lipschitz norm bounded by the original one. Thus, it is sufficient to prove this proposition for f positive.

The following lemma connects the operator E c with E.

Lemma 1.4.19. Under the assumptions of Proposition 1.4.17, let

f o (x, v, u) = 1 -v≤u<0 f (x, v, u)
.

Then E C f (x, t) = Ef o (x, t).
Before proving this proposition, we describe some regularity and independence properties. They are corollaries of analogous properties for the renewal process. The idea is to decompose the integral according to the last letter. The following lemma means that the residue process with cutoff has exponential decay with respect to the last jump.

Lemma 1.4.20. For t, s in R and x in X, we have

E C (1 v≥s )(x, t) = E(1 -v≤u<0,v≥s )(x, t) = O exp (s).
(1.4.20)

Proof. By Lemma 1.4.9 and finiteness of the exponential moment, we have

n≥0 µ ⊗ µ * n {(h, g) ∈ G 2 |σ(g, x) -t ∈ [-σ(h, gx), 0], σ(h, gx) ≥ s} ≤ n≥0 µ ⊗ µ * n {(h, g) ∈ G 2 |σ(g, x) -t ∈ [-κ(h), 0], κ(h) ≥ s} = κ(h)>s R(1 [-κ(h),0] )(x, t)dµ(h) κ(h)>s max{1, κ(h)}dµ(h) = O exp (s).
The proof is complete.

Lemma 1.4.21. There exists C > 0 such that for all t ∈ R and x ∈ X, we have

E C (1)(x, t) = E(1 -v≤u<0 )(x, t) ≤ C. (1.4.21)
This is a special case of Lemma 1.4.20. The following lemma quantifies the independence of the scalar part and the angle part. Abbreviate 1 d(y,x )≤e -s ,-v≤u<0 (y, v, u) to 1 d(y,x )≤e -s ,-v≤u<0 , and others are similar.

Lemma 1.4.22. For t > 5s > 0 and x, x in X, we have

E C (1 d(y,x )≤e -s )(x, t) = E(1 d(y,x )≤e -s ,-v≤u<0 )(x, t) = O exp (s).
(1.4.22)

Proof. Since

1 -v≤u<0 ≤ 1 d(y,x )≤e -s ,-v≤u<0,v≥s + 1 d(y,x )≤e -s ,0≤u+v<s ,
we have

E(1 d(y,x )≤e -s ,-v≤u<0 )(x, t) ≤ E(1 -v≤u<0,v≥s )(x, t) + E(1 d(y,x )≤e -s ,0≤u+v<s )(x, t).
By definition, we have

E(1 d(y,x )≤e -s ,0≤u+v<s )(x, t) = n≥0 1 d(hgx,x )≤e -s ,σ(h,gx)+σ(g,x)-t∈[0,s] dµ * n (g)dµ(h) = n≥1 1 d(gx,x )≤e -s ,σ(g,x)-t∈[0,s] dµ * n (g) = R(1 B(x ,e -s ),[0,s] )(x, t).
By Lemma 1.4.20 and Proposition 1.4.10, the result follows.

Lemma 1.4.23. For s > 0, t > max{10s, 10} and x, x o , x ∈ X, we have

n≥0 µ ⊗ µ * n {(h, g) ∈ G × G|σ(hg, x) ≥ t, σ(g, x) < t, d((hg) -1 x o , x ) ≤ e -s } = O exp (s).
By the same argument as in the proof of Lemma 1.4.22, we only need to replace Proposition 1.4.10 by Proposition 1.4.14. The difference between this lemma and Lemma 1.4.22 is the angle part (hg) -1 x.

Using ψ δ to regularize these functions, we write

f δ (x, v, u) = f o (x, v, u-u 1 )ψ δ (u 1 )du 1 = ψ δ * f o (x, v, u).
Lemma 1.4.24. Under the same hypotheses as in Proposition 1.4.17, we have

E(f δ )(x, t) = X G 0 -σ(h,y) f (hy, σ(h, y), u)dudµ(g)dν(y) + O K (δ + O δ t )|f | Lip .
Proof. We want to verify the conditions in Proposition 1.4.15 and then use this proposition. The integrable condition is valid because

| Ru f δ | = | Ru f o (x, v, u)du| = | 0 -v f (x, v, u)du| ≤ |K||f | ∞ .
For the Fourier transform, we have

F u f δ = F u (ψ δ * f o ) = ψδ F u f o .
We need to estimate the Lipschitz norm of F u f o . This function equals

f o (x, v, u)e iξu du = 0 -v f (x, v, u)e iξu du. Taking (x, v) = (x , v ), we have | 0 -v f (x, v, u)e iξu du - 0 -v f (x , v , u)e iξu du| ≤| 0 -v (f (x, v, u) -f (x , v , u))e iξu du| + |v -v||f | ∞ |K||f | Lip (d(x, x ) + |v -v |).

Then we have

Lemma 1.4.25 (Change of norm). Under the same hypotheses as in Proposition 1.4.17, we have

|F u f δ | L ∞ Lip ≤ |K||f | Lip , |∂ ξ Ff δ | L ∞ Lip ≤ |K| 2 |f | Lip .
Proof. Noting that in the integration |u| ≤ |v|, we get the second inequality by the same computation.

Therefore by Proposition 1.4.15, we have

E(f δ )(x, t) = 1 σ µ ∞ -t G X f δ (hy, σ(h, y), u)dν(y)dµ(h)du + O δ t |f | Lip (|K| + |K| 2 ) . Then ∞ -t f δ (x, v, u)du = ∞ -t 0 -v f (x, v, u 1 )ψ δ (u -u 1 )du 1 du = 0 -v f (x, v, u 1 ) ∞ -t ψ δ (u -u 1 )dudu 1 = 0 -v f (x, v, u 1 )du 1 - 0 -v f (x, v, u 1 ) -t-u 1 -∞ ψ δ (u)dudu 1 . Since t -δ ≥ |K|, we have -t -u 1 ≤ -t + v ≤ -δ. By -δ -∞ ψ δ ≤ C ψ δ, this implies that ∞ -t f δ (x, v, u)du = 0 -v f δ (x, v, u)du(1 + O(δ)). Using Lemma 1.4.21, we have | X G 0 -σ(h,y) f (hy, σ(h, y), u)dudµ(g)dν(y)| ≤ |f | ∞ E C (1) = O(|f | ∞ ). Therefore ∞ -t G X f δ (hy, σ(h, y), u)dν(y)dµ(h)du = X G 0 -σ(h,y) f (hy, σ(h, y), u)dudµ(g)dν(y) + O(δ|f | ∞ ).
The proof is complete.

Next lemma gives the difference between a function and its regularization.

Lemma 1.4.26. Let ϕ 0 (u) = 1 [b 1 ,b 2 ] (u)ϕ(u), where b 2 > b 1 and |ϕ | L ∞ < ∞, |ϕ| L ∞ ≤ 1.
Then we have

|ψ δ * ϕ 0 (u) -ϕ 0 (u)| ≤      (|ϕ | ∞ + 2)δ u ∈ [b 1 + δ, b 2 -δ], 2 u ∈ [b 1 -δ, b 1 + δ] ∪ [b 2 -δ, b 2 + δ], ψ δ * 1 [b 1 ,b 2 ] (u) u ∈ [b 1 -δ, b 2 + δ] c .
(1.4.23)

Proof. We will prove this inequality in each interval.

• When u is in [b 1 + δ, b 2 -δ], we have |(ψ δ * ϕ 0 -ϕ 0 )(u)| = ψ δ (t)(ϕ 0 (u -t) -ϕ 0 (u))dt ≤ δ -δ ψ δ (t)|ϕ 0 (u -t) -ϕ 0 (u)|dt + 2δ. When |t| ≤ δ, we have u -t ∈ [b 1 , b 2 ]. Since |ϕ 0 (u)| ≤ |ϕ | ∞ for u ∈ [b 1 , b 2 ], this implies that δ -δ ψ δ (t)|ϕ 0 (u -t) -ϕ 0 (u)|dt ≤ δ -δ ψ δ (t)|t||ϕ | ∞ dt ≤ δ|ϕ| ∞ . • When u ∈ [b 1 -δ, b 1 +δ]∪[b 2 -δ, b 2 +δ], we use the trivial bound |ψ δ * ϕ 0 (u)-ϕ 0 (u)| ≤ 2. • When u ∈ (-∞, b 1 -δ]∪[b 2 +δ, ∞], we have ϕ 0 (u) = 0, then |ψ δ * ϕ 0 | ≤ |ψ δ * 1 [b 1 ,b 2 ] |.
Thus collecting all together, we get the inequality.

Proof of Proposition 1.4.17. To simplifier the notation, we normalize f in such a way that |f | ∞ = 1. By Lemma 1.4.24, we only need to give an estimate of

E(|f δ -f o |)(x, t). Since f o (x, v, u) = 1 -v≤u<0 (u)f (x, v, u) with (x, v) fixed, Lemma 1.4.26 implies that |f δ -f o |(u) ≤      (|∂ u f | ∞ + 2)δ u ∈ [-v + δ, -δ], 2 u ∈ [-v -δ, -v + δ] ∪ [-δ, δ], ψ δ * 1 [-v,0] (u) u ∈ [-v -δ, δ] c .
By definition of |K|, the first term is less than

(|∂ u f | ∞ + 2)δ1 [-|K|+δ,-δ] . The third term equals 1 [-∞,-v-δ]∪[δ,∞] ψ δ * 1 [-v,0] (u) = 1 [-∞,-v-δ]∪[δ,∞] (u) 0 -v ψ δ (u -u 1 )du 1 = 1 [-∞,-v-δ]∪[δ,∞] (u) u+v u ψ δ (u 1 )du 1 .
By definition and the above arguments, we have

E(|f δ -f o |)(x, t) = n≥0 |f δ -f o |(hgx, σ(h, gx), σ(g, x) -t)dµ * n (g)dµ(h) ≤ n≥0 (|∂ u f | ∞ + 2)δ1 [-|K|,-δ] (σ(g, x) -t)+ + 21 [-σ(h,gx)-δ,-σ(h,gx)+δ]∪[-δ,δ] (σ(g, x) -t) + 1 [-∞,-σ(h,gx)-δ]∪[δ,∞] (σ(g, x) -t) σ(hg,x)-t σ(g,x)-t ψ δ (u 1 )du 1 dµ * n (g)dµ(h).
By Lemma 1.4.9, the first term is controlled by

(|∂ u f | ∞ + 2)δ|K|. The second term is less than R(1 [-δ,δ] )(x, t). Due to Proposition 1.4.8, it is controlled by 6δ(1/σ µ +C δ (1+2δ)/t).
For the third term, we need to change the order of integration. Since σ(g, x) -t > δ or σ(g, x) -t < -σ(h, gx) -δ, we have u 1 ≥ σ(g, x) -t > δ or u 1 ≤ σ(hg, x) -t = σ(h, gx) + σ(g, x) -t ≤ -δ. We integrate first with respect to u 1 , then the third term is less than

[-∞,-δ]∪[δ,∞] ψ δ (u 1 ) n≥0 µ ⊗ µ * n {(h, g)|σ(hg, x) ≥ u 1 + t, σ(g, x) ≤ u 1 + t}du 1 .
By Lemma 1.4.21, the above quantity is less than

C [-∞,-δ]∪[δ,∞] ψ δ (u 1 )du 1 δ.
Therefore, we have

E(|f δ -f |)(x, t) = O K (δ + C δ /t)|f | Lip .
The proof is complete.

Remark 1.4.27 (Minus case). The lemmas in this part concern plus and minus. The another version we need is for

E - C (f )(x, t) = E(1 0<u≤-v f )(x, t
), the proofs are exactly the same.

Proposition* 1.4.17. Under the assumptions of Proposition 1.4.17, we have

E - C (f )(x, t) = X G -σ(h,y) 0 f (hy, σ(h, y), u)dudµ(h)dν(y) + O K (δ + O δ /t)|f | Lip .

Residue process for the Cartan Projection

We consider the residue process for the cutoff of a function f on X 2 × R 2 , where the cocycle is replaced by the Cartan projection. We will give a limit not only with gx, but also with g -1 x .

As in the previous subsection, we can define a similar Lipschitz norm on the space of Lipschitz functions on X 2 × R 2 , using the same name |f | Lip . Define the operator from bounded Borel functions on X 2 × R 2 to functions on X 2 × R by

E P f (x , x, t) = n≥0 κ(g)<t≤κ(hg) f ((hg) -1 x , hgx, κ(hg) -κ(g), κ(g) -t)dµ(h)dµ * n (g). Proposition 1.4.28. Let f be a continuous function on X 2 × R 2 with |f | Lip finite.
Assume that the projection of suppf on R v is contained in a compact set K. For all δ > 0, t > max{2(|K| + δ), 20} and x , x in X, we have

E P f (x , x, t) = X 2 G 0 -σ(h,y) f (y , hy, σ(h, y), u)dudµ(h)dν(y)dν(y ) + O K (δ + O δ /t)|f | Lip , (1.4.24)
where O K does not depend on δ, f, t, x, x , the integral

0 -σ(h,y) = 0 if σ(h, y) < 0.
Proof. We introduce local notations here: for an element g in G and a continuous function

f on X 2 × R 2 , define gf (x , x, v, u) = f (g -1 x , x, v, u). Let f x (x, v, u) = f (x , x, v, u),
which emphasizes that the first coordinate is fixed. Let l = [ 5 t/σ µ ], where 5 < 1/10. We use the decomposition h = g n+1 , g = (g n , . . . , g l+1 ), j = (g l , . . . , g 1 ).

Recall that

N + t = n≥0 {(g n+1 , g n , . . . , g 1 )|κ(g n+1 • • • g 1 ) ≥ t > κ(g n • • • g 1 )}. Let N + t (n) = N + t ∩ G ×(n+1) = {(g n+1 , . . . , g 1 )|κ(gj) ≤ t, κ(hgj) > t}. Let T n (x, t) = {(g n+1 , . . . , g 1 ) ∈ G ×(n+1) |σ(hg, jx) > t -κ(j), σ(g, jx) ≤ t -κ(j)},
and let

G ,l = {(g l , . . . , g 1 )||κ(j) -lσ µ | ≤ l , d(x M g l •••g 1 , x
) ≥ e -l }, as well as

T n, = {(g n+1 , . . . , g 1 ) ∈ T n |(g l , . . . , g 1 ) ∈ G ,l }.
Step 1: Due to Corollary 1.2.9 and Corollary 1.2.11, the sum of the integrals N + t (n) for n ranging from t/(σ µ + ) -1 to t/(σ µ -) is exponentially small in t. In other words, we have

| [t/(σµ-)] n=[t/(σµ+ )] N + t (n) f ((hgj) -1 x , hgjx, κ(hgj) -κ(gj), κ(gj) -t)dµ ⊗(n+1) -E P f (x , x, t)| = O exp (t)|f | ∞ (1.4.25)
The following lemma replaces the Cartan projection with the cocycle.

Lemma 1.4.29. Under the same assumption as in Proposition 1.4.28, we have

| [t/(σµ-)] n=[t/(σµ+ )] N + t (n) f ((hgj) -1 x , hgjx, κ(hgj) -κ(gj), κ(gj) -t)dµ ⊗(n+1) (hgj) - [t/(σµ-)] n=[t/(σµ+ )] Tn, jf ((hg) -1 x , hgjx, σ(h, gjx), σ(g, jx) -(t -κ(j)))dµ ⊗(n+1) (hgj)| = O(δ + O δ /t)|f | Lip . (1.4.26)
This lemma will be proved later. We will decompose T n, (x, t) to apply the residue process for the cocycle. The space T n, (x, t) can be seen as a fibered space over G ,l . When the first l elements are fixed, the elements (h, g) such that hgj = (g n+1 , . . . , g 1 ) ∈ T n, (x, t), are the admitted elements in the residue process with cutoff, whose start point is jx and time is t -κ(j).

Since (n -l)(σ µ + ) ≤ t -κ(j) and (n -l)(σ µ -) ≥ t -κ(j),
we can apply Principle 1 to this residue process. Integrating over G ,l implies that

| [t/(σµ-)] n=[t/(σµ+ )] Tn, jf ((hg) -1 x , hgjx, σ(h, gjx), σ(g, jx) -(t -κ(j)))dµ ⊗(n+1) (hgj) - G ,l E I jf (x , jx, t -κ(j))dµ ⊗l (j)| = O exp (t)|f | ∞ .
(1.4.27) where

E I f (x , x, t) = n≥0 σ(g,x)<t≤σ(hg,x) f ((hg) -1 x , hgx, σ(h, gx), σ(g, x) -t)dµ(h)dµ * n (g).
The following inequality, whose proof relies on Lemma 1.4.23, will give a major term.

Lemma 1.4.30. Under the same assumption as in Proposition 1.4.28, for all j ∈ G ,l , we have

|E C f j -1 x (jx, t -κ(j)) -E I jf (x , jx, t -κ(j))| ≤ |f | Lip O exp (l), (1.4.28)
This lemma will be proved later. Integrating (1.4.28) over G ,l , we obtain

| G ,l E C f j -1 x (jx, t -κ(j)) -E I jf (x , jx, t -κ(j))dµ ⊗l (j)| ≤ |f | Lip O exp (t). (1.4.29)
By (1.4.25)(1.4.26)(1.4.27), it suffices to compute the major term

G ,l E C f j -1 x (jx, t -κ(j))dµ ⊗ (j).
Step 2: Recall that N 0 , P (0) are the two operators defined by N 0 ϕ = ϕdν, P (0)ϕ(x) = ϕ(gx)dν(g), where ϕ is a function in H γ (X). We have another property of transfer operators [BQ16, Lemma 11.18]: The spectral radius of P = P (0) restricted to ker N 0 is less than 1, which means that there exist ρ < 1, C > 0 such that for every function ϕ in H γ (X), we have

|P n ϕ -ϕdν| ∞ ≤ Cρ n |ϕ| γ . Thus by µ ⊗l G ,l = O exp (l), we have | G ,l ϕ(j -1 x)dµ ⊗l -ϕdν| = | G ×l ϕ(j -1 x)dµ ⊗l (j)-ϕdν|+O exp (l)|ϕ| ∞ = O exp (l)|ϕ| Lip . (1.4.30) By the definition of | • | Lip on X × R 2 , the function f j -1 x (x, v, u) has a finite | • | Lip value. Together with t -κ(j) ≥ t/2 ≥ |K| + δ, Proposition 1.4.17 implies that G ,l E C f j -1 x (jx, t -κ(j))dµ ⊗ (j) = G ,l X G 0 -σ(h,y) f j -1 x (y, σ(h, y), u)dudµ(h)dν(y)dµ ⊗l (j) + O K (δ + O δ /t)|f j -1 x | Lip = X G 0 -σ(h,y) G ,l f (j -1 x , y, σ(h, y), u)dµ ⊗l (j)dudµ(h)dν(y) + O K (δ + O δ /t)|f | Lip ).
(1.4.31)

With (x, v, u) fixed, f (x , x, v, u
) is a Lipschitz function on x , so it is a Hölder function.

Together with Lemma 1.4.21 and inequality (1.4.30), we have

G ,l E C f j -1 x (jx, t -κ(j))dµ ⊗ (j) = X G 0 -σ(h,y) X f (u, σ(h, y), y, y )dν(y )dudµ(h)dν(y) + (O exp (l) + O K (δ + O δ /t))|f | Lip .
(1.4.32)

The result follows.

It remains to prove Lemma 1.4.29 and Lemma 1.4.30.

Proof of Lemma 1.4.29. There exist S n+1,l,x ⊂ G ×(n+1) and S n,l,x ⊂ G ×n which satisfy the conditions in Lemma 1.2.14. Let S n (x) = S n+1,l,x ∩ (G × S n,l,x ). Then

µ ⊗(n+1) S n (x) c = O exp (l), (1.4.33)
and for (g n+1 , . . . , g 1 ) in S n (x), we have

|κ(hgj) -σ(hg, jx) -κ(j)| ≤ e -l |κ(gj) -σ(g, jx) -κ(j)| ≤ e -l .
In

N + t (n) ∩ S n (x) ∩ T n (x, t)
, we can replace the Cartan projection by the cocycle with exponentially small error. Fortunately, the difference of this set with N + t (n) and T n (x, t) has exponentially small measure. By definition, we have

N + t (n) ∩ S n (x) ⊂ {σ(hg, jx) > t -e -l -κ(j), σ(g, jx) ≤ t + e -l -κ(j)},
and

N + t (n) ⊃ {σ(hg, jx) > t + e -l -κ(j), σ(g, jx) ≤ t -e -l -κ(j)} ∩ S n (x).
Therefore

(N + t (n) ∩ S n (x) -T n (x, t)) ⊂{σ(hg, jx) ∈ [-e -l , 0] + t -κ(j)} ∪ {σ(g, jx) ∈ [0, e -l ] + t -κ(j)}, and
(T n (x, t) ∩ S n (x) -N + t (n)) ⊂{σ(hg, jx) ∈ [0, e -l ] + t -κ(j)} ∪ {σ(g, jx) ∈ [-e -l , 0] + t -κ(j)}.
Hence, these imply that

µ ⊗(n+1) (N + t (n) -N + t (n) ∩ S n (x) ∩ T n (x, t)) ≤ µ ⊗(n+1) S n (x) c + µ ⊗(n+1) (N + t (n) ∩ S n (x) -T n (x, t)) ≤ O exp (l) + µ ⊗(n+1) {σ(hg, jx) ∈ [-e -l , 0] + t -κ(j)} ∪ {σ(g, jx) ∈ [0, e -l ] + t -κ(j)}. and µ ⊗(n+1) (T n (x, t) -N + t (n) ∩ S n (x) ∩ T n (x, t)) ≤ µ ⊗(n+1) S n (x) c + µ ⊗(n+1) (T n (x, t) ∩ S n (x) -N + t (n)) ≤ O exp (l) + µ ⊗(n+1) {σ(hg, jx) ∈ [0, e -l ] + t -κ(j)} ∪ {σ(g, jx) ∈ [-e -l , 0] + t -κ(j)}.
Moreover, for (g n+1 , . . . , g 1 ) in the set

N + t (n) ∩ S n (x) ∩ T n (x, t), the definition of S n (x) implies that |f ((hgj) -1 x , hgjx, κ(hgj) -κ(gj), κ(gj) -t) -jf ((hg) -1 x , hgjx, σ(h, gjx), σ(g, jx) -(t -κ(j)))| ≤ e -γl |f | Lip .
(1.4.34)

Thus, for n ∈ [t/(σ µ + ) -1, t/(σ µ -)], we have | N + t (n) f ((hgj) -1 x , hgjx, κ(hgj) -κ(gj), κ(gj) -t)dµ ⊗(n+1) - Tn(x,t) jf ((hg) -1 x , hgjx, σ(h, gjx), σ(g, jx) -(t -κ(j)))dµ ⊗(n+1) | ≤ µ ⊗ (N + t (n) -N + t (n) ∩ S n (x) ∩ T n (x, t)) ∪ (T n (x, t) -N + t (n) ∩ S n (x) ∩ T n (x, t)) + µ ×(n+1) N + t (n) ∩ S n (x) ∩ T n (x, t)O exp (l)|f | Lip ≤ (O exp (l) + µ ⊗(n+1) {|σ(hg, jx) -t + κ(j)|, |σ(g, jx) -t + κ(j)| ≤ e -l })|f | ∞ + O exp (l)|f | Lip .
Sum up over all n ∈ [t/(σ µ + ) -1, t/(σ µ -)]. Then the above inequality becomes

| [t/(σµ-)] n=[t/(σµ+ )] N + t (n) f ((hgj) -1 x , hgjx, κ(hgj) -κ(gj), κ(gj) -t)dµ ⊗(n+1) (hgj)| - [t/(σµ-)] n=[t/(σµ+ )] Tn jf ((hg) -1 x , hgjx, σ(h, gjx), σ(g, jx) -(t -κ(j)))dµ ⊗(n+1) (hgj) ≤ tO exp (l)|f | Lip + |f | ∞ G ×l 2R(1 [-e -l ,e -l ] )(jx, t -κ(j))dµ ⊗l (j).
(1.4.35)

By (1.2.9), (1.2.12), we have µ ⊗l G ,l ≥ 1 -O exp (l). Thus combined with Lemma 1.4.21, we get

n≥l µ ⊗(n+1) (T n (x, t) -T n, (x, t)) = G c ,l E1(jx, t -κ(j))dµ ⊗l (j) = O exp (l).
This enables us to replace the integration domain T n by T n, with exponentially small error. It is sufficient to control the right hand side of (1.4.35).

The last term can be bounded by the similar argument as in (1.4.12), with Proposition 1.4.10 replaced by inequality (1.4.6). It follows that

G ×l 2R(1 [-e -l ,e -l ] )(jx, t -κ(j))dµ ⊗l (j) = O exp (l) + δO(1 + O δ /t).
(1.4.36)

The proof is complete.

Proof of Lemma 1.4.30. We want to replace (hgj) -1 x with (j) -1 x in the first coordinate in order to find the residue process with cutoff. The idea is always similar. We have a good approximation in a large set, whose complement has exponentially small measure. Let

Σ l = n≥0 {(h, g) ∈ G × G ×n |σ(g, jx) < t -κ(j) ≤ σ(hg, jx), d((hg) -1 x , x M j ) ≤ e -l }.
Since t -κ(j) ≥ t -(σ µ + )l ≥ 10 l and t -κ(j) ≥ t/2 > 10, we can use Lemma 1.4.23 with s = l and jx, x , x M j to obtain

µ ⊗ μΣ l = O exp (l). (1.4.37)
The definition of G ,l implies that d(x M j , x ) ≥ e -l and κ(j) ≥ (σ µ -)l. It follows from (1.2.4) that x M j = x m j -1 . Together with (1.2.1),(1.2.5), for (h, g) outside of the set Σ l , we have

d((hgj) -1 x , j -1 x ) = d(j -1 (hg) -1 x , j -1 x ) ≤ exp(-2κ(j -1 ) -log d(x m j -1 , x ) -log d(x m j -1 , (hg) -1 x )) ≤ exp(-2(σ µ -)l + 2 l). Therefore |f (j -1 x , x, v, u) -f ((hgj) -1 x , x, v, u)| ≤ |f | Lip d(j -1 x , (hgj) -1 x ) = |f | Lip O exp (l).
(1.4.38) In the bad part Σ l , we use inequality (1.4.37) to control. Outside of Σ l , we apply inequality (1.4.38). Thus we have

| n≥0 σ(hg,jx)>t-κ(j)≥σ(g,jx) f (j -1 x , hgjx, σ(h, gjx), σ(g, jx) + κ(j) -t) -f ((hgj) -1 x , hgjx, σ(h, gjx), σ(g, jx) + κ(j) -t)dµ(h)dµ * n (g)| ≤ |f | Lip (O exp (l) + O exp (l)E C 1(jx, t -κ(j))).
Then by Lemma 1.4.21, the proof is complete.

Remark 1.4.31 (Minus case). Let

E - P f (x , x, t) = n≥0 κ(g)≥t>κ(hg) f ((hg) -1 x , hgx, κ(hg) -κ(g), κ(g) -t)dµ(h)dµ * n (g).
Then by the same proof, we have Proposition* 1.4.28. Under the assumptions of Proposition 1.4.28, we have

E - P f (x , x, t) = X 2 G -σ(h,y) 0 f (y , hy, σ(h, y), u)dudµ(h)dν(y)dν(y ) + O K (δ + O δ /t)|f | Lip .

Main Approximation

In this section, we want to complete the proof in Section 1.3. It remains to prove Proposition 1.3.6 and the following Lemma 1.5.2 and Corollary 1.5.5.

Recall the definitions in Section 1.3: Let µ be a Borel probability measure on SL 2 (R) with a finite exponential moment, and assume that the subgroup Γ µ is Zariski dense. Let Σ = n∈N G ×n be the symbol space of all finite sequences with elements in G. Let μ be the measure on Σ defined by

μ = +∞ n=0 µ ⊗n , where µ ⊗0 = δ ∅ .
Let the integer ω(g) be the length of an element g in Σ. Let T be the shift map on Σ, defined by T g = T (g 1 , g 2 , . . . , g ω ) = (g 1 , g 2 , . . . , g ω-1 ), when ω(g) ≥ 2, and T g = ∅, when ω(g) = 1, 0. Let L be the left shift map on Σ, defined by Lg = L(g 1 , g 2 , . . . , g ω ) = (g 2 , . . . , g ω-1 , g ω ), when ω(g) ≥ 2, and Lg = ∅, when ω(g) = 1, 0.

The sets M + t , N + t are defined by

M + t = {g ∈ Σ| κ(T g) < t ≤ κ(g)}, N + t = ι(M + t ) = {g ∈ Σ|κ(Lg) < t ≤ κ(g)},
where ι(M ) equals {g -1 |g ∈ M } for any subset M of Σ.

Let μ be the pushforward of µ by the inverse action. It also satisfies the assumptions of Theorem 1.1.1. By definition μ(M + t ) = μ(N + t ). For x, y in X, write s 1 = 3 s and

M + t (x, y) = {g ∈ M + t ||κ(g) -κ(T g)| < s 1 , d(x m g , g -1 x) < e -t , d(g -1 x, x), d(g -1 x, y) > 2e -s 1 }.
We need some regularity properties of N + t . These lemmas are of the same type as the ones with the cocycle, using the Cartan projection instead. The correspondences are: Lemma 1.5.1 with Lemma 1.4.20, Lemma 1.5.2 with Lemma 1.4.21, Lemma 1.5.3 with Lemma 1.4.22. In fact, for all the regularity properties, there are similar versions for the Cartan projection. The subadditivity is sufficient. We follow the same procedure as in the proof for the cocycle.

Lemma 1.5.1. For s in R, we have

μ{g ∈ N + t ||κ(g) -κ(Lg)| > s} = O exp (s).
(1.5.1)

Proof. Subadditivity of Cartan projection implies κ(g ω ) ≥ |κ(g ω • • • g 1 )-κ(g ω-1 • • • g 1 )| = |κ(g) -κ(Lg)| > s and κ(Lg) ≥ κ(g) -κ(g ω ). Then μ{g ∈ N + t ||κ(g) -κ(Lg)| > s} = n≥0 µ ⊗ µ * n {(h, g) ∈ G × G|κ(g) < t ≤ κ(hg), |κ(hg) -κ(g)| > s} ≤ n≥0 µ ⊗ µ * n {(h, g) ∈ G × G|t -κ(h) ≤ κ(g) < t, κ(h) > s} = κ(h)>s R p (1 [-κ(h),0] )(t)dµ(h).
By Lemma 1.4.12 and finite exponential moment, we have

μ{g ∈ N + t ||κ(g) -κ(Lg)| > s} κ(h)>s max{1, κ(h) 2 }dµ(h) = O exp (s).
The proof is complete.

A special case is when s = 0. Applying the above lemma with μ, we have

Lemma 1.5.2. The measure μ(M + t ) = μ(N + t ) is uniformly bounded with t.
The following lemma quantifies the independence of the scalar part and the angle part of residue process for the Cartan projection.

Lemma 1.5.3. For s > 0, t > 10s and x, x o ∈ X, we have

μ{g ∈ N + t |d(x M g , gx) ≥ e -t } = O exp (t), (1.5.2) μ{g ∈ N + t |d(gx o , x) ≤ e -s } = O exp (s). (1.5.3)
The proof of the second inequality follows the same procedure as in the proof of Lemma 1.4.22, replacing Lemma 1.4.20 and Proposition 1.4.10 with Lemma 1.5.1 and Lemma 1.4.11. The first inequality is standard, using Principle 1 and Principle 2. When n ∈ [ t σµ+ -1, t σµ-], use Corollary 1.2.13, and when n is outside of this interval, use Corollary 1.2.9 and Corollary 1.2.11.

Joining Lemma 1.5.1 and Lemma 1.5.3, we have the following corollary Corollary 1.5.4. Let s > 0, t > 10s and let x, y be in X. Let

N + t (x, y) = {g ∈ N + t ||κ(g) -κ(Lg)| < s, d(x M g , gx) < e -t , d(gx, x), d(gx, y) > 2e -s }.
(1.5.4)

Then we have μ(N + t ) -μ(N + t (x, y)) = O exp (s).
(1.5.5)

Corollary 1.5.5. For s > 0, t > 10s and x, y in X, we have

μ(M + t ) -μ(M + t (x, y)) = O exp (s).
Proof. By definition, we have

μ(M + t ) -μ(M + t (x, y)) = μ(N + t ) -μ(N + t (x, y)).
Applying the above corollary with μ, we have completed the proof.

We start to proof Proposition 1.3.6. The central tool here is Lemma 1.2.7, which enables us to replace the cocycle with the sum of the scalar part and the angle part.

Proof of Proposition 1.3.6. We first replace the distance with the cocycle. By hypothesis, we have d(x m g , x) ≥ d(g -1 x, x) -d(x m g , g -1 x) ≥ 2e -s 1 -e -t ≥ e -s 1 . Using the same argument, we have d(x m g , y), d(x m g , x) ≥ e -s 1 . Then (1.2.1) and (1.2.5) imply

d(gx, gy) = d(x, y) exp(-σ(g, x) -σ(g, y)) ≤ exp(-2κ(g)) d(x m g , x)d(x m g , y)
≤ e -2(t-s 1 ) .

Applying the Newton-Leibniz formula (1.2.17) to φ at gx, gy, we have φ(gx) -φ(gy) = sign(gx, gy)

gx gy φ (θ)dθ.
Since κ(g) > t > s 1 , we have d(x m g , x) ≥ e -s 1 ≥ e -κ(g) . Then (1.2.18) implies that φ(gx) -φ(gy)) = sign(x, y, x m g ) gx gy φ (θ)dθ.

We need the arc length distance d a (•, •) on R/πZ. Since d(gx, gy) ≤ e -2(t-s 1 ) , for θ in the small arc gx gy, we have d a (θ, gx) ≤ e -2(t-s 1 ) . Therefore So we can replace the arc length distance with the sine distance. Again by hypothesis, we have d(x m g , g -1 x) ≤ e -t < d(g -1 x, x), d(g -1 x, y). When changing x m g to g -1 x, the relative place with respect to x, y does not change, therefore we get sign(x, y, x m g ) = sign(x, y, g -1 x).

Inequality (1.2.1), together with the above two inequalities, implies

|φ(gx) -φ(gy) -sign(x, y, g -1 x)φ (gx)d(x, y) exp(-σ(g, x) -σ(g, y))| ≤ |φ | ∞ 2e -4(t-s 1 )
.

(1.5.7) We may now replace the cocycle with the Cartan projection and the angle part. Since e -2κ(g)

+ d(x m g , g -1 x) d(g -1 x, x) ≤ 2e -t+s 1 < 1/2, Lemma 1.2.7 implies that |σ(g, x) -κ(g) -log d(g -1 x, x)| ≤ 2 e -2κ(g) + d(x m g , g -1 x) d(g -1 x, x) ≤ 4e -t+s 1 , |σ(g, y) -κ(g) -log d(g -1 x, y)| ≤ 2 e -2κ(g) + d(x m g , g -1 x) d(g -1 x, y) ≤ 4e -t+s 1 .
We have an inequality for z 1 , z 2 in C,

|e z 1 -e z 2 | ≤ max{e z 1 , e z 2 }|z 1 -z 2 |. Since σ(g, x) ≥ κ(g) + log d(x m g , x) ≥ t -s 1 and κ(g) + log d(g -1 x, x) ≥ t -s 1 , we have | exp(-σ(g, x)) -exp(-κ(g))/d(g -1 x, x)| ≤ e -t+s 1 4e -t+s 1 . Therefore by inequality |a 1 a 2 -b 1 b 2 | ≤ |(a 1 -b 1 )a 2 | + |(a 2 -b 2 )b 1 |, we have |e -σ(g,x)-σ(g,y) -e -2κ(g) /(d(g -1 x, x)d(g -1 x, y))| ≤ 8e -3(t-s 1 ) .
Then by the hypothesis |ξ| = e 2t+s and (1.5.7), we have e iξ(φ(gx)-φ(gy) -e iξφ (gx)sign(x,y,g -1 x)d(x,y) exp(-2κ(g))/(d(g -1 x,x)d(g -1 x,y))

≤|ξ||φ(gx) -φ(gy) -φ (gx)sign(x, y, g -1 x)d(x, y) exp(-2κ(g))/(d(g -1 x, x)d(g -1 x, y))| ≤|ξ||φ | ∞ 2e -4(t-s 1 ) + |ξ||φ d(x, y)||e -σ(g,x)-σ(g,y) -e -2κ(g) /(d(g -1 x, x)d(g -1 x, y))| ≤|ξ|(|φ | ∞ 2e -4(t-s 1 ) + 8|φ | ∞ e -3(t-s 1 ) ) ≤ 8(|φ | ∞ + |φ | ∞ )e -t+s+3s 1 .
Finally, for |Λ 0 -Λ|, it suffices to add the difference

|r(gx)r(gy) -r(gx) 2 | ≤ |r| ∞ |r | ∞ e -2(t-s 1 ) . Then |Λ 0 -Λ| ≤ |r| ∞ |r | ∞ e -2(t-s 1 ) + |r| 2 ∞ (|φ | ∞ + |φ | ∞ )e -t+s+3s 1 = O exp (s),
where O exp (s) does not depend on t, but depends on r, φ. The proof is complete.

Remark 1.5.6 (Minus case). The proof works the same for M - t .

Chapter 2

Fourier dimension and spectral gaps for random walks on SL m+1 (R)

List

Introduction

The purpose of this manuscript is to study the Fourier decay of stationary measures on projective spaces and some spectral properties of random walks on G = SL m+1 (R), the special linear group of degree m + 1. Let µ be a Borel probability measure on G. Let Γ µ be the subsemigroup of G generated by the support of µ. If Γ µ is Zariski dense in G, then we call µ a Zariski dense measure. We say that µ has a finite exponential moment if there exists > 0 such that

G g dµ(g) < ∞.
If we have a group action of G on a compact manifold X, then a Borel probability measure

ν on X is called µ-stationary if ν = µ * ν, which means ν = G g * νdµ(g)
and where g * ν is the pushforward measure, that is g * ν(E) = ν(g -1 E) for any Borel subset E of X. For a metric space X, let C γ (X) be the space of γ-Hölder functions on

X. For f in C γ (X) let c γ (f ) = sup x =x |f (x)-f (x )| d(x,x ) γ and |f | γ = |f | ∞ + c γ (f ).
Let (ρ, V ) be a finite dimensional irreducible linear representation of G with a norm. (For example V = R m+1 ) Let PV be the real projective space defined by (V \{0})/R * , the set of all the directions in V . Then we have the group action of G on PV , given by gRv = Rρ(g)v for g in G and Rv in PV . A result of Furstenberg says that when µ is Zariski dense, there exists a unique µ-stationary measure ν V on PV .

Let v 0 be a unit vector in V . Let v ⊥ 0 be the linear subspace of V , which is orthogonal to v 0 . Let U be the open subset of PV , which is the complement of the hyperplane Pv ⊥ 0 . We take an affine local chart (ψ, U ) of PV , given by

ψ : PV ⊃ U → v ⊥ 0 , Rv → v -v 0 , v v 0 v 0 , v ,
which is well defined on U . The inverse of ψ is simply given by ψ

-1 : v ⊥ 0 → U ⊂ PV, u → R(u + v 0 ).
Theorem 2.1.1. Let µ be a Zariski dense Borel probability measure on SL m+1 (R) with a finite exponential moment. Let V be an irreducible representation of SL m+1 (R). Let ν V be the µ-stationary measure on PV . Let r be a C 1 continuous function whose support is in U and r ∞ ≤ 1. Then there exists > 0 such that for every ς ∈ v ⊥ 0 with the norm ς sufficiently large, we have

v ⊥ 0 e i ς,u r(u)dν V (u) ≤ ς -.
Remark. For simplicity, we use the same notation ν V for the measure on PV and the measure on v ⊥ 0 . More precisely, the integral actually means PV e i ς,ψ(x) r(ψ(x))dν V (x). The constant only depends on µ and V , and inequality holds for ς sufficiently large only depending on µ, V , the support of r and c γ (r).

We state a stronger version for m = 1, SL 2 (R), which is a quantitative version of the main result in Chapter 1 Theorem 2.1.2. Let µ be a Zariski dense Borel probability measure on SL 2 (R) with a finite exponential moment. Let X = P(R 2 ) and let ν be the µ-stationary measure on X.

For every γ > 0, there exist 0 >, 1 > 0 depending on µ such that the following holds. For any f

∈ C 2 (X), r ∈ C γ (X) such that |ϕ | ≥ |ξ| -0 on the support of r, r ∞ ≤ 1 and ϕ C 2 + c γ (r) ≤ |ξ| 0 , then e iξϕ(x) r(x)dν(x) ≤ |ξ| -1 for all |ξ| large enough.
Remark 2.1.3. As a consequence of the case of SL 2 (R), the Fourier coefficients of the stationary measure ν on the circle converge to zero with a power decay. This is also a generalization of the same theorem for the Patterson-Sullivan measures as in [START_REF] Bourgain | Fourier dimension and spectral gaps for hyperbolic surfaces[END_REF]. This stronger version is not valid if we replace R 2 by higher dimensional representation V of SL 2 (R). Because the support of the stationary measure ν V is in a one dimensional subvariety of PV . We can always find a ϕ which is constant on the subvariety and satisfies similar assumptions in Theorem 2.1.2. Then we have no Fourier decay for this function ϕ.

Another result is an exponential remainder term in the renewal theorem. Define the renewal operator R as follows. For a positive bounded Borel function f on PV × R, a point x = Rv in PV and a real number t, we set

Rf (x, t) = +∞ n=0 G f (gx, log gv v -t)dµ * n (g).
Here log gv v is an analogue of the sum of i.i.d. real random variables. Because of the positivity of f , this sum is well defined. In [START_REF] Kesten | Renewal theory for functionals of a Markov chain with general state space[END_REF], Kesten proved a renewal theorem for Markov chains, which is valid in our case. The assumptions of [START_REF] Kesten | Renewal theory for functionals of a Markov chain with general state space[END_REF] were verified in [START_REF] Guivarc | Spectral gap properties for linear random walks and Pareto's asymptotics for affine stochastic recursions[END_REF]. Using spectral gap, more precisely by Proposition 2.4.22, we can give a version with exponential remainder term. Let σ V,µ be the Lyapunov constant defined by

σ V,µ = G PV log gv v dν(x)dµ(g), where x = Pv.
Theorem 2.1.4 (Renewal theorem). Let µ be a Zariski dense Borel probability measure on SL m+1 (R) with finite exponential moment. Let V be an irreducible representation of SL m+1 (R). There exists > 0 such that for f ∈ C ∞ c (R) and t ∈ R, we have

Rf (x, t) = 1 σ V,µ ∞ -t f (u)dLeb(u) + O f (e -|t| ),
where O f depends on the support and some Sobolev norm of f . It is a standard Fourier analysis argument which follows from the spectral gap. We give a proof in Section 2.4.5 for completeness. Now, we will introduce our results on spectral gaps. On PV , we fix a Riemannian distance and we define the transfer operator.

Definition. For z ∈ C with | z| small enough, let P z be the operator on the space of continuous functions, which is given by

P z f (x) = G e z log gv v f (gx)dµ(g), where x = Rv ∈ PV.
We keep the assumption that µ is a Zariski dense Borel probability measure on SL m+1 (R) with a finite exponential moment. The use of this transfer operator on the products of random matrices has been introduced by Guivarc'h. Due to the contracting action of G on X, when | z| is small enough, the operator P z preserves the Banach space C γ (PV ) for γ > 0 small enough. For z in a small ball centred at 0, the spectral radius of P z on C γ (PV ) is less than 1 except at 0. Due the non-arithmeticity of Γ µ , on the imaginary line, the operator P z also has spectral radius less than 1 except at 0. These were used to give limit theorems for products of random matrices by Le Page and Guivarc'h (Please see [START_REF] Le | Théorèmes limites pour les produits de matrices aléatoires[END_REF] and [START_REF] Benoist | Random Walks on Reductive Groups[END_REF]).

Theorem 2.1.5 (Spectral gap). Let µ be a Zariski dense Borel probability measure on SL m+1 (R) with finite exponential moment. Let V be an irreducible representation of SL m+1 (R). For every γ > 0 small enough, there exists δ > 0 such that for all |b| > 1 and |a| small enough the spectral radius of P a+ib acting on C γ (PV ) satisfies

ρ(P a+ib ) < 1 -δ.
Even in the case of SL 2 (R), the result is new and only known in some special case. When µ is supported on a finite number of elements in SL 2 (R) and these elements generate a Schottky semigroup, this result is due to Naud [START_REF] Naud | Expanding maps on Cantor sets and analytic continuation of zeta functions[END_REF]. When µ is absolutely continuous with respect to the Haar measure on SL 2 (R), this result can be obtained directly using oscillation integral.

This result should be compared with similar results for random walks on R. Let µ be a Borel probability measure on R with finite support. Then

lim inf |b|→∞ |1 -μ(ib)| = 0,
which is totally different from our case and where μ(z) is the Laplace transform of the measure µ, given by

μ(z) = R e zx dµ(x).
The proof is direct. Let {x 1 , . . . , x l } be the support of µ. Then μ(ib) = 1≤j≤l µ(x j )e ibx j , and we only need to find b such that all the terms are uniformly near 1. Using the fact that lim inf b→∞ d R l (b(x 1 , . . . , x l ), 2πZ l ) = 0, we have the claim.

We can also compare with the counting problem in the setting of hyperbolic surfaces. The spectral gap is used to obtain an exponential remainder term in the counting problem as in [START_REF] Lax | The Asymptotic Distribution of Lattice Points in Euclidean and Non-Euclidean Spaces[END_REF], [START_REF] Naud | Expanding maps on Cantor sets and analytic continuation of zeta functions[END_REF].

An analogous result is valid if we replace the projective space PV by the flag variety P. Let P be the full flag variety of SL m+1 (R) and let a be a Cartan subspace of Lie algebra sl m+1 R. For g ∈ G and η ∈ P, let σ(g, η) be the Iwasawa cocycle, which takes values in a. We fix a Riemannian distance on P. We can similarly define the space of γ-Hölder functions C γ (P). Let , θ be in a * . For a continuous function f on P and | | small enough, the transfer operator P +iθ on the flag variety is defined by

P +iθ f (η) = G e ( +iθ)σ(g,η) f (gη)dµ(g).
Theorem 2.1.6 (Spectral gap). Let µ be a Zariski dense Borel probability measure on SL m+1 (R) with finite exponential moment. For every γ > 0 small enough, there exists δ > 0 such that for all θ, in a * with |θ| > 1 and | | small enough the spectral radius of P +iθ acting on C γ (P) satisfies

ρ(P +iθ ) < 1 -δ.

Fourier decay

The key ingredient of the proof of the above results is the following Fourier decay property of the µ-stationary measure on the flag variety P. In order to state the Fourier decay on the flag variety, we need to introduce a special condition. Let r be a continuous function on P and let C > 0. For a C 2 function ϕ on P, we say ϕ is (C, r) good if it satisfies some assumptions on the Lipschitz norm and derivative, which will be defined later (Definition 2.4.1). When G = SL 2 (R), the (C, r) goodness is exactly the assumption of ϕ in Theorem 2.1.2, which is natural for having a Fourier decay. Recall that for a γ-Hölder function f , we have defined

c γ (f ) = sup x =x |f (x)-f (x )| d(x,x ) γ .
Theorem 2.1.7 (Fourier decay). Let µ be Zariski dense Borel probability measure on SL m+1 (R) with finite exponential moment. Let ν be the µ-stationary measure on the flag variety P.

For every γ > 0, there exist 0 > 0, 1 > 0 depending on µ such that the following holds. For any real function ϕ ∈ C 2 (P), r ∈ C γ (P) and ξ > 0 such that ϕ is (ξ 0 , r) good, r ∞ ≤ 1 and c γ (r) ≤ ξ 0 , then e iξϕ(η) r(η)dν(η) ≤ ξ -1 for all ξ large enough.

(2.1.1)

Remark 2.1.8. The decay rate only depends on the constants in the large deviation principles and the regularity of stationary measures. This should be compared with [START_REF] Bourgain | Fourier dimension and spectral gaps for hyperbolic surfaces[END_REF], where the spectral gap and the decay rate only depend on the dimension of the Patterson-Sullivan measure.

Now we explain the (C, r) good condition. In higher dimension, we observe that under the action of G there are some directions contracting slower than other directions. Roughly speaking, we will only consider these principal directions in the flag variety P and generalize the condition of SL 2 (R) to higher dimension. The exact definition is a little technique and all the notation will be explained in Section 2.4.1.

A key ingredient in the proof of Theorem 2.1.7 comes from the discretized sumproduct estimate, Proposition 2.3.17, which is a generalized version of a result of Bourgain in [START_REF] Bourgain | The discretized sum-product and projection theorems[END_REF]. The key input to use the machine of the discretized sum-product estimate is a non concentration hypothesis. A analogue hypothesis for measures on R is as follows.

Definition (non concentration). Let µ be a Borel probability measure on R. We say that µ satisfies non concentration hypothesis if there exist , κ, C > 0 such that for every n ∈ N and ρ = e -n ,

sup a∈R µ * n {x ∈ R| |x -a| ≤ ρ} ≤ Cρ κ .
But this hypothesis is never satisfied when the measure µ supports on a finite set, for example {x 1 , • • • , x l } ⊂ R. Because the convolution µ * n is supported on at most n l points, there exists a point y such that µ * n {y} n -l . The decay rate of sup a∈R µ * n (B(a, ρ)) is at most polynomial in n, which does not satisfy the definition of non concentration.

In Section 2.3, we will introduce a similar non concentration hypothesis and we will verify the hypothesis for our measure µ. The main ingredients are the large deviation principle, a Hölder regularity for stationary measures and highest weight representations. The strategy can be roughly summarized by saying that once the non concentration of the Iwasawa cocycle is verified, by the discretized sum-product estimate, we will have a Fourier decay and a speed in the equidistribution of the Iwasawa cocycle (the renewal theorem). This is in the similar spirit of the work of Bourgain and Gamburd on the spectral gap for compact Lie groups [START_REF] Bourgain | On the spectral gap for finitely-generated subgroups of su (2)[END_REF].

The action of the group G on the flag variety P is not conformal if the rank m is greater than 1, which is quite different from the theory of Kleinian groups. In Section 2.2, we will study the action of G on the tangent bundle of P and we will find directions of slowest contraction speed.

We will make use of some classical notation: for two real functions A and B, we write A = O(B), A B or B A if there exists a constant C > 0 such that |A| ≤ CB, where C only depends on the ambient group G and the measure µ. We write

A B if A B and B A. We write A = O (B), A B or B A if the constant C depends on an extra parameter > 0.
We always use 0 < δ < 1 to denote an error term and 0 < β < 1 to denote the magnitude. The quantity β -1 is supposed to be greater than δ -O(1) . If δ O(1) A ≤ B ≤ δ -O(1) A, then we say that A and B are of the same size.

Random walks on Lie groups

In this manuscript, we only consider G = SL m+1 (R).

Semisimple Lie groups and representations

We will introduce the vocabulary of semisimple algebraic connected real Lie groups. Please see [START_REF] Helgason | Differential geometry, Lie groups, and symmetric spaces[END_REF], [START_REF] Borel | Linear algebraic groups[END_REF] and [START_REF] Benoist | Random Walks on Reductive Groups[END_REF] for more details.

Semisimple Lie groups

Let G be a semisimple algebraic connected real Lie group. Let g be its Lie algebra. Since all the maximal compact subgroups are conjugate, we fix a maximal compact subgroup K of G. Let k be its Lie algebra. For X, Y in g, the Killing form is defined as

K(X, Y ) = tr(adXadY ).
The Killing form is non degenerate on g and negative definite on k. Let s be the orthogonal complement of k in g. Then the Killing form is positive definite on s. Let τ be the Cartan involution which fixes k and equals -id on s.

We say an element X in g is hyperbolic, if adX is diagonalizable over R. Let a be a maximal abelian subalgebra of s whose elements are hyperbolic. Such subalgebras are called Cartan subspaces, and they are conjugate under K. The dimension of a is called the real rank of G. The real rank of the group G will always be denoted by m. Endowed with the Killing form, the Cartan subalgebra a and its dual a * become Euclidean spaces. Let A be the algebraic subgroup of G with the Lie algebra a. We write exp for the exponential map from a to A.

Root systems and the Weyl group

Let R be the root system of g with respect to a, which is a finite subset of a * . Fix a choice of positive roots R + . Let Π be the collection of primitive simple roots of R + . Let a + be the Weyl chamber defined by {X ∈ a|α(X) ≥ 0, ∀α ∈ Π}. Let a ++ be the interior of Weyl chamber defined by {X ∈ a|α(X) > 0, ∀α ∈ Π}. Using the root system, we have a decomposition of g into eigenspaces of a,

g = z ⊕ α∈R g α ,
where z is the centralizer of a and g α is the eigenspace given by

g α = {X ∈ g| [Y, X] = α(Y )X for all Y ∈ a}.
The real Lie group is called split if a = z, which is equivalent to saying that g α are of dimension 1. The groups SL m+1 (R) are split groups.

Recall that for every root α in R, there is an orthogonal symmetry s α which preserves R and s α (α) = -α. For α ∈ R, let H α be the unique element in a such that s α (α ) = α -α (H α )α for α ∈ a * . The set {H α | α ∈ R} is called the set of duals roots in a. Since the Cartan involution τ equals -id on a, this implies τ g α = g -α for α ∈ R. Using the Killing form, we can prove that [g α , g -α ] = RH α . (See [Ser66, Cha. 4, Theorem 2] for more details) Hence, there is a unique choose (up to sign)

X α ∈ g α , Y α ∈ g -α such that [X α , Y α ] = H α and τ (X α ) = -Y α . Let K α = X α -Y α . Due to τ K α = K α , the element K α is in k.
Let W be the Weyl group of R. Then the group W acts simply transitively on the set of Weyl chambers. Let w 0 be the unique element in W which sends the Weyl chamber a + to the Weyl chamber -a + . Let ι = -w 0 be the opposition involution. The Weyl group also acts on a * by the dual action. Let N G (A) be the normalizer of A in G. An element in N G (A)/A induces an automorphism on the tangent space a. This gives an isomorphism from N G (A)/A to the Weyl group W . Hence w 0 can be realized as an element in G/A and its action on a is given by conjugation.

The Iwasawa cocycle

Let n = ⊕ α∈R + g α and n -= ⊕ α∈R + g -α . They are nilpotent Lie algebras. Let N be the connected algebraic subgroup of G with Lie algebra n. The group N is normalized by A. Let P = AN be a minimal parabolic subgroup. The flag variety P is defined to be the set of conjugations of P under the action of G. Since the normalizer of P in G is itself, we have an isomorphism G/P → P.

We write η o for the subgroup P seen as a point in P. The action of K on P is transitive. Hence P is a compact manifold. Let M = P ∩ K. The fact that the group G is split implies that M is discrete.

Recall that N is the nilpotent subgroup with Lie algebra n. We have an Iwasawa decomposition of G given by

G = KA e N,
where A e = exp(a) is the analytical connected component of A. This is a bijection between G and K × A e × N . Then we can define the Iwasawa cocycle σ from G × P to a. Let η be in P and g be in G. By the transitivity of K, there exists k ∈ K such that η = kη o . By the Iwasawa decomposition, there exists a unique element σ(g, η) in a such that

gk ∈ K exp(σ(g, η))N.
We can verify that this is well defined and σ is an additive cocycle, that is for g, h in G and η in P σ(gh, η) = σ(g, hη) + σ(h, η).

The Cartan decomposition

The Cartan decomposition says that G = KA + K, where A + is the image of the Weyl chamber a + under the exponential map. For g in G, by Cartan decomposition, we can write g = k g a g g with k g , g in K and a g in A + . The element a g is unique and there is a unique element κ(g) in a + such that a g = exp(κ(g)). We call κ(g) the Cartan projection of g. Then κ(g -1 ) = ικ(g), where ι is the opposition involution. Since A is contained in P , we can define ζ o = w 0 η o , where the element w 0 in the Weyl group is seen as an element in G/A. (As an element in P, ζ o is the opposite parabolic group with respect to P and

A) Let η M g = k g η o and ζ m g = -1 g ζ o .
When κ(g) is in a ++ , they are uniquely defined, independently of the choice of k g and g .

Representations and highest weight

Let (ρ, V ) be a linear finite dimensional algebraic representation of G. In this manuscript, we only consider finite dimensional representations. The set of restricted weights Σ(ρ) of the representation is the set of elements ω in a * such that the eigenspace

V ω = {v ∈ V |∀X ∈ a, dρ(X)v = ω(X)v}
is nonzero, where dρ is the tangent map of ρ from g to End(V ). We define a partial order on the restricted weights: For ω 1 , ω 2 in Σ(ρ),

ω 1 ≥ ω 2 ⇔ ω 1 -ω 2 is a sum of positive roots.
If ω is in Σ(ρ), then we say that ω is a weight of V and a vector v in V ω is said to have weight ω. We call ρ proximal if there exists χ in Σ(ρ) which is greater than the other restricted weights and such that V χ is of dimension 1. We should pay attention that a proximal representation is not supposed to be irreducible. The advantage of the splitness of G is that all the irreducible representations are proximal, which will be extensively used later on.

Let {χ α } α∈Π be the set of fundamental weights, which is the dual basis of the dual roots (H α ) α∈Π , a basis of a * . For an element χ in a, there exists a finite dimensional representation with highest weight χ if and only if it is a dominant weight, which means that χ is a sum of multiple of fundamental weights.

Suppose that (ρ, V ) is an irreducible representation. Let χ ∈ a * be the highest weight of (ρ, V ). We write V χ,η = ρ(g)V χ for η = gη o , which is well defined because the parabolic subgroup P fixes the subspace V χ . This gives a map from P to PV by

P → PV, η → V χ,η .
(2.2.1) Lemma 2.2.1. Let G be SL m+1 (R). There exists a family of representations (ρ α , V α ) α∈Π such that the highest weight of ρ α is χ α . Furthermore, the other weights of ρ α are of the form

χ α -α - β∈Π n β β, where n β ∈ N ≥0 .
The product of the maps given by (2.2.1)

P -→ α∈Π PV α , η → (V χα,η ) α∈Π ,
is an embedding of P to the product of projective spaces.

Please see [START_REF] Tits | Représentations linéaires irréductibles d'un groupe réductif sur un corps quelconque[END_REF]. The set of restrict weight is invariant under the Weyl group. Due to χ α (H β ) = 0 when β = α, we have s β (χ α -β) = χ α + β. Hence χ α -β is not a weight of V α except β = α. This explains the structure of weights of ρ α .

Definition 2.2.2 (Super proximal representation). Let (V, χ) be an irreducible representation of G. We call V super proximal if the exterior square ∧ 2 V is also proximal. This is equivalent to say that there is only one simple root α such that χ -α is a weight of V , and V χ-α is of dimension 1.

Lemma 2.2.3. Fundamental representations are super proximal.

Proof. Let α be a simple root. By Lemma 2.2.1, we only need to prove that V χα-α is of dimension 1. Let v be a nonzero vector with highest weight χ α . By [Ser66, Chapter 7, Proposition 2], the representation V is generated by vectors

Y β 1 • • • Y β k v,
where β 1 , . . . , β k are positive roots. Hence a vector of weight χ α -α can only be obtained by Y α v. The dimension of V χα-α is no greater than 1. Due to χ α -α = s α (χ α ), the element χ α -α is a weight of V α . The proof is complete.

Remark 2.2.4. We can prove that an irreducible representation is super proximal if and only if its highest weight is a multiple of fundamental weight, by using Freudenthal's multiplicity formula.

Representations and good norms

Let • be an euclidean norm on V . For g in GL(V ), let g be its application norm. We call • a good norm if ρ(A) is symmetric and ρ(K) preserves the norm. By [START_REF] Helgason | Differential geometry, Lie groups, and symmetric spaces[END_REF], [BQ16, Lemma 6.33], good norms exist on every representation of G. The application (2.2.1) enables us to get information on P by the representations.

Lemma 2.2.5. Let G be a connected algebraic semisimple real Lie group. Let (ρ, V ) be an irreducible linear representation of G with good norm. Let χ be the highest weight of V . For η in P and a non zero vector v ∈ V χ,η , we have

ρ(g)v v = exp(χσ(g, η)), (2.2.2) ρ(g) = exp(χκ(g)).
(2.2.3)

Please see [BQ16, Lemma 6.33] for the proof.

Algebraic characters

Let X(A) be the set of algebraic characters of A. For any character χ of A, there exists a unique weight χ ω in a * such that for any X in a, χ(exp(X)) = e χ ω (X) .

When G = SL m+1 (R), for every weight , which is in Γ(G) := ⊕ α∈Π Zχ α (the lattice generated by {χ α } α∈Π ), we can find an algebraic character χ in X(A) such that χ ω = . In fact, this is an isomorphism between X(A) and Γ(G). By the definition of eigenspace V ω , we have Lemma 2.2.6. Let (ρ, V ) be an irreducible representation of G. Let χ be an algebraic character of A. For a in A and v ∈ V χ ω , we have

ρ(a)v = χ(a)v.
Algebraic characters will be used to determiner the sign in Section 2.2.5. We will use the same symbol χ to denote a weight in Γ(G) and χ to denote its corresponding algebraic character in X(A), that is (χ ) ω = χ.

Examples

For the group SL m+1 (R), the maximal torus A can be taken as the diagonal subgroup and the Lie algebra a is the set of diagonal matrices with trace 0. For X in a, we write X = diag(x 1 , . . . , x m+1 ) with x i ∈ R and 1≤i≤m+1 x i = 0. The restriction of Killing form on a is a multiple of the standard euclidean norm on R m+1 . Let λ i in a * be the linear map given by λ i (X) = x i . The root system R is given by R = {λ i -λ j |i = j, and i, j ∈ {1, . . . , m + 1}}.

A choice of positive roots is λ i -λ j with i < j. The set of simple roots is Π = {λ i -λ i+1 |i = 1, . . . , m}. Let α i = λ i -λ i+1 . The Weyl chamber is

a + = {X ∈ a|x 1 ≥ x 2 ≥ • • • ≥ x m+1 }.
The fundamental weights are χ α i = λ 1 + • • • + λ i for i = 1, . . . , m. The fundamental representations are V α i = ∧ i R m+1 for i = 1, . . . , m. The maximal compact subgroup K is SO(m + 1) and the parabolic group P is the upper triangular subgroup and N is the subgroup of P with all the diagonal entries equal to 1. The flag variety P is the set of all flags

W 1 ⊂ W 2 ⊂ • • • ⊂ W m ,
where W i is a subspace of R m+1 of dimension i.

Let i,j be the square matrix of dimension m + 1 with the only nonzero entry at the i-th row and j-th column, which equals 1. The element H α i is i,i -i+1,i+1 . The element X α i , Y α i are given by i,i+1 , i+1,i . The Cartan involution τ is the additive inverse of the transpose, that is τ (X) = -t X for X in a.

The Weyl group W is isomorphic to the symmetric group S m+1 . The action on a is simply given by the permutation of coordinates and the element w 0 sends X = diag(x 1 , . . . , x m+1 ) to w 0 X = diag(x m+1 , . . . , x 1 ).

Linear actions on vector spaces

Let V be a vector space with euclidean norm. Then we have an induced norm on its dual space V * , exterior powers ∧ j V and tensor products ⊗ j V .

For x = Rv, x = Rv in PV , we define the distance between x, x by

d(x, x ) = v ∧ v v v . (2.2.4)
This distance has the advantage that it behaves well under the action of GL(V ). See for example Lemma 2.2.8. For y = Rf in PV * , let y ⊥ = P(ker f ) ⊂ PV be a hyperplane in PV . For x = Rv in PV , we define the distance of x to y ⊥ by

δ(x, y) = |f (v)| f v ,
which is explained by δ(x, y) = d(x, y ⊥ ) = min x ∈y ⊥ d(x, x ). Let K V be the compact group which preserves the norm. Let A + V be the set of diagonal elements such that

{a = diag(a 1 , • • • , a d )|a 1 ≥ a 2 ≥ • • • ≥ a d }, under the basis {e 1 , • • • , e d }. Let A ++ V be the interior of A + V .
For g in GL(V ), by the Cartan decomposition we can choose g = k g a g g , where a g ∈ A + V and k g , g ∈ K V .

(2.2.5) Let x M g = Rk g e 1 and y m g = R t ge * 1 be the density points of g on PV and t g on PV * , which is unique and independent of the choice of basis when a g is in A ++ V . For r > 0 and

g in GL(V ), let b M V,g (r) = {x ∈ PV |d(x, x M g ) ≤ r}, B m V,g (r) = {x ∈ PV |δ(x, y m g ) ≥ r}.
These two sets play important role when we want to get some ping-pong property. The elements in set B m V,g have distance at least r to the hyperplane determined by y m g . For g in GL(V ), let γ 1,2 (g) := ∧ 2 g g 2 be the gap of g.

Distance and norm

We start with general g in GL(V ), where V is a finite dimensional vector space with euclidean norm. We need some technical control of distance. These are quantitative versions of the same controls in [Qui02, Lemma 2.5, 4.3, 6.5].

For g in GL(V ) and x = Rv ∈ PV , we define an additive cocycle σ

V : GL(V )×PV → R by σ V (g, x) = log gv v . (2.2.6)
This is called cocycle, because for g, h in G, we have

σ V (gh, x) = σ V (g, hx) + σ V (h, x).
We fix the operator norm • on GL(V ).

Lemma 2.2.7. For any g in GL(V ) and x in PV , we have

δ(x, y m g ) ≤ gv g v ≤ 1.
(2.2.7)

Please see [BQ16, Lem 14.2] for the proof.

Lemma 2.2.8. Let δ > 0. For g in GL(V ), if β = γ 1,2 (g) ≤ δ 2 , then
• the action of g on B m V,g (δ) is βδ -2 -Lipschitz and

gB m V,g (δ) ⊂ b M V,g (βδ -1 ) ⊂ b M V,g (δ),
• the restriction of the real valued function σ

V (g, •) on B m V,g (δ) is 2δ -1 -Lipschitz. Proof. Due to [BQ16, Lem 14.2], d(gx, x M g )δ(x, y m g ) ≤ γ 1,2 (g) = β. Hence d(gx, x M g ) ≤ βδ(x, y m g ) -1 ≤ βδ -1 ,
which implies the inclusion. For x = Rv and x = Rv in B m V,g (δ), by (2.2.7), we have

d(gx, gx ) = gv ∧ gv v ∧ v v ∧ v v v v v gv gv ≤ γ 1,2 (g)d(x, x )δ -2 ,
which implies the Lipschitz property of g.

For the Lipschitz property of σ V (g, •), please see [START_REF] Benoist | Random Walks on Reductive Groups[END_REF]Lemma 17.11].

For two different points x = Rv and x = Rv in PV , we write

x ∧ x = R(v ∧ v ) ∈ P(∧ 2 V ).
Lemma 2.2.9. For any g in GL(V ) and two different points x = Rv, x = Rv in PV , we have

γ 1,2 (g)δ(x ∧ x , y m ∧ 2 g ) ≤ d(gx, gx ) d(x, x ) . (2.2.8)
Proof. By definition and (2.2.7), we have

d(gx, gx ) = gv ∧ gv v ∧ v v ∧ v v v v v gv gv ≥ γ 1,2 (g)δ(x ∧ x , y m ∧ 2 g )d(x, x ).
The proof is complete.

Actions on Flag varieties

Representations and Density points

Now, suppose that V is a representation of G with a good norm. Recall that V χ is the eigenspace of the highest weight. Let V * be the dual space of V . The representation of G on V * is the dual representation given by: for g ∈ G and f ∈ V * , let ρ * (g)f = t ρ(g -1 )f . This definition gives

ρ * (g)f, ρ(g)v = t ρ(g -1 )f, ρ(g)v = f, v ,
for f in V * and v in V . Then the highest weight of V * is ιχ. The following results explain the relation between different definitions by using combinatoric information on root systems and representations.

Lemma 2.2.10. We claim that for every irreducible representation V and weight χ,

V χ,ζo = V w 0 χ .
(2.2.9)

Proof. This can be verified as follows: For X in a and

v in V χ , dρ(X)ρ(w 0 )v = w 0 dρ(w 0 X)v = χ(w 0 X)w 0 v = (w 0 χ)(X)w 0 v.
The proof is complete.

Lemma 2.2.11. Let V be a proximal representation of G. Then we have

x M ρ(g) = ρ(k g )V χ and y m ρ(g) = t ρ( g )(V * ) -χ .
(2.2.10)

If V is irreducible, then we have x M ρ(g) = V χ,η M g and y m ρ(g) = V * ιχ,ζ m g .
Proof. Let {e 1 , . . . , e d } be an orthonormal basis of V composed of eigenvectors of ρ(A) such that e 1 ∈ V χ . Then ρ(A) is diagonal. For g = exp(X) ∈ A + , since χ is the highest weight, we have

a 1 = exp(χ(X)) ≥ a 2 , . . . , a d .
By the definition of a good norm, ρ(K) preserves the norm. Hence for g in G, the formula ρ(g) = ρ(k g )ρ(a g )ρ( g ) is a decomposition which satisfies (2.2.5) in the previous paragraph with some permutation of {e 2 , . . . , e d }. But these permutations do not change the density points. Hence we have

x M ρ(g) = Rρ(k g )e 1 = ρ(k g )V χ . If V is irreducible we have x M ρ(g) = V χ,η M g .
In the dual space, we can verify that e * 1 has weight -χ, which is the lowest weight in weights of V * . By the same argument as in PV , we have

y m ρ(g) = R t ρ( g )e * 1 = t ρ( g )(V * ) -χ .
We also have a map from P to PV * . Hence by (2.2.9) with representation V * and weight ιχ, we know V * ιχ,ζo = (V * ) w 0 ιχ = (V * ) -χ . For ζ = gζ o in P, by definition,

V * ιχ,ζ = gV * ιχ,ζo = g(V * ) -χ . (2.2.11) Since V is irreducible, by (2.2.11) we have y m ρ(g) = t ρ( g )(V * ) -χ = ρ * ( -1 g )(V * ) -χ = V * ιχ,ζ m g .

Distance on Flag varieties

For α in Π, we abbreviate

V χα,η , V * ιχα,ζ to V α,η , V * α,ζ . For g in G, by Lemma 2.2.11, we find x M ρα(g) = V α,η M g and y m ρα(g) = V * α,ζ m g . For η, η in P, let d α (η, η ) = d(V α,η , V α,η )
be its distance between their images in PV α . We define a distance on the flag variety. It is the maximal distance induced by projections,

d(η, η ) = max α∈Π d(V α,η , V α,η ).
(2.2.12)

We have another embedding of the flag variety In particular, because the images of

P → α∈Π P(V * α ).
η o , ζ o in PV α , PV * α are V χα , (V * ) -χα , we know δ(V α,ηo , V * α,ζo ) = δ(V χα , (V * ) -χα ) = 1, and then δ(η o , ζ o ) = 1. (2.2.13) We write b M Vα,g (r) = {x ∈ PV α |d(x, x M ρα(g) ) ≤ r}, B m Vα,g (r) = {x ∈ PV α |δ(x, y m ρα(g) ) ≥ r}. They are subsets of PV α . Write b M g (r) = {η ∈ P|∀α ∈ Π, V α,η ∈ b M Vα,g (r)} = {η ∈ P|d(η, η M g ) ≤ r}, B m g (r) = {η ∈ P|∀α ∈ Π, V α,η ∈ B m Vα,g (r)} = {η ∈ P|δ(η, ζ m g ) ≥ r}.
They are subsets of P.

Distance and norms

We need a multidimensional version of the lemmas in Section 2.2.2. Recall that G = SL m+1 (R). They are about the similar quantities on flag varieties. The idea is to use all the fundamental representations ρ α . For an element X in a, we have

sup α∈Π |χ α (X)| ≤ X sup α∈Π |χ α (X)|. ( 2
σ(g, η) -κ(g) | log δ(η, ζ m g )|.
For g in G and α ∈ Π, by Lemma 2.2.5,

γ 1,2 (ρ α (g)) = ∧ 2 ρ α (g) ρ α (g) 2 = e (2χα-α-2χα)κ(g) = e -ακ(g) . Let γ(g) = sup α∈Π e -ακ(g) .
(2.2.15)

We call it the gap of g.

Lemma 2.2.13. Let δ > 0. For g in G, if β = γ(g) = sup α∈Π exp(-ακ(g)) ≤ δ 2 , then

• the action of g on B m g (δ) is βδ -2 -Lipschitz and

gB m g (δ) ⊂ b M g (βδ -1 ) ⊂ b M g (δ),
• the restriction of the a-valued function σ(g,

•) on B m g (δ) is O(δ -1 )-Lipschitz.
These properties tell us that the action of an element g on a large set of the flag variety P behaves like uniformly contracting map.

We also need to compare the distance on the projective space and the flag variety. Recall the map from P to PV defined in (2.2.1).

Lemma 2.2.14. Let (ρ, V ) be an irreducible representation of G with highest weight χ. There exists a constant C > 0 depending on the chosen norm such that for η, η in P,

d(V χ,η , V χ,η ) ≤ Cd(η, η ).
(2.2.16)

The intuition is that a differentiable map between two compact Riemannian manifolds is Lipschitz. For more details, please see Corollary 2.5.6 in Appendix 2.5.2.

Actions on the tangent bundle of the Flag variety

In this section, we will study the action of G on the tangent bundle of P. Recall that P G/P is the flag variety and P = AN is a parabolic subgroup.

We first study the tangent bundle of the homogeneous space

P 0 = G/A e N.
Recall that A e is the analytical connected component of A, given by exp(a). Note that the left action of K on P 0 is simply transitive (due to the Iwasawa decomposition). Let z o be the base point A e N in P 0 . We can identify the left K-invariant vector fields as

T zo P 0 = T zo (G/A e N ) g/p.
Hence the tangent bundle of P 0 has an isomorphism

T P 0 P 0 × g/p,
that is because we can identify the tangent space at z o and z = kz o by the left action of k. We denote by (z, Y ) a point of T P 0 where z is in P 0 and Y is in g/p. We use elements in n -= ⊕ α∈R + g -α as representative elements in g/p.

Then we describe the left action of G on T P 0 . Take Y in g -α and z = kz o in P 0 . For g in G, by the Iwasawa decomposition we have a unique k in K and a unique σ(g, k) in a such that gk = k p ∈ k exp(σ(g, k))N , where p ∈ A e N . Here σ(g, k) is understood as σ(g, kη o ). Due to

gk exp(tY )z o = k p exp(tY )z o = k exp(tAd p Y )z o ,
by taking derivative at t = 0, the left action of g on the tangent vector (z, Y ) satisfies

L g (z, Y ) = (z , Ad p Y ),
where z = k o and Ad is the adjoint action of P on g/p.

Now we restrict our attention to simple roots. Let α be a simple root. Due to Y ∈ g -α , we have Ad N Y ⊂ Y + a + n, which implies that the unipotent part N acts trivially on (g -α + p)/p. By p ∈ exp(σ(g, k))N , we have

Ad p Y = exp(-ασ(g, k))Y on (g -α + p)/p.
(2.2.17)

This means that the line bundle P 0 × g -α is stable under the left action of G, and we call it the α-bundle.

The flag variety P is a quotient of P 0 by the right action of group M , due to A = M A e . We use π to denote the quotient map. The right action of M also induces an action on the tangent bundle. For (z, Y ) in T P 0 and m in M , by k exp(tY

)mz o = km exp(tAd m -1 Y )z o , we have R m (kz o , Y ) = (kmz o , Ad m -1 Y ).
(2.2.18)

Descending to the quotient implies the tangent bundle of P satisfies

T P P 0 × M g/p,
which is the quotient space of P 0 × g/p by the equivalence relation generated by the action of M , (2.2.18). Due to M < A, its adjoint action preserves the line g -α in g/p. Hence the α-bundle on P 0 descends to a line bundle on P, and we call it P α , a subbundle of the tangent bundle. The integral curves of α-bundle on P 0 are closed, and we call them α-circles on P 0 . At a point z = kz o in P 0 , it is given by

γ α : R → P 0 , t → k exp(tK α )z o . (2.2.19)
This can be verified directly, because the tangent vector of the curve at time t is (γ α (t), K α ) = (γ α (t), Y α ), due to the definition of g/p, which belongs to the α-bundle.

The one parameter subgroup {exp(tK α ) : t ∈ R} is a compact subgroup of G, which is isomorphic to SO(2). We call it O α . Under the right action of M , the α-circles on P 0 descends to the α-circles on P.

Lemma 2.2.15. Under the map (2.2.1), the image of the α-circle containing η = kη 0 in PV α is the projective line generated by ρ α (k)V χα and ρ α (k)V χα-α . Let α be another simple root. The image of an α-circle in PV α is a point.

Proof. Since α-bundle is left K-invariant, the set of α-circles are also left K-invariant.

It is sufficient to consider the α-circle containing η 0 . Let (ρ, V ) be an irreducible representation of highest weight χ. By (2.2.19) and (2.2.1), the image of α-circle is given by ρ(O α )V χ . Consider the Lie algebra s α generated by H α , X α , Y α , which is isomorphic ot sl 2 . For v in V χ , we have dρ(H α )v = χ(H α )v. Due to the classification of the irreducible representation of sl 2 , the irreducible representation V 1 of s α generating by V χ is of dimension χ(H α ) + 1.

When χ = χ α , we have χ α (H α ) = 0, which implies V 1 is a trivial representation and ρ(O α ) acts trivially on V 1 . Hence the image of the α-circle is a point.

When

χ = χ α , the same argument implies V 1 is of dimension 2. Another eigenspace of V 1 is V χα-α . The group ρ(O α ) acts as SO(2) on V 1 , which implies the result.
Remark 2.2.16. If we introduce the partial flag variety P Π-{α} , then α-circle is simply the fiber of the quotient map P → P Π-{α} . This point of view also implies Lemma 2.2.15.

Generally, the α-bundle on P is non trivial in the sense of line bundle.

Example 2.2.17. Let G be SL 3 (R). Recall that

a = {X = diag(x 1 , x 2 , x 3 )| x 1 + x 2 + x 3 = 0, x 1 , x 2 , x 2 ∈ R},
and α 1 , α 2 are two simple roots given by α 1 = λ 1 -λ 2 and α 2 = λ 2 -λ 3 . The group M is {e, diag(1, -1, -1), diag(-1, 1, -1), diag(-1, -1, 1)} (Z/2Z) 2 . We have

Ad diag(1,-1,-1) Y α 1 = Ad diag(1,-1,-1)   0 0 0 1 0 0 0 0 0   = -Y α 1 .
In this case the action of M is nontrivial and it is not a normal subgroup of K = SO(3). The α-bundle on P restricted to an α-circle is roughly a Möbius band.

In this case, α 1 -circles are given by {W 1 ⊂ W 2 }, where W 2 is a fixed two dimensional subspace of R 3 and W 1 varies in one dimensional subspaces of W 2 . On the contrary, α 2circles are given by {W 1 ⊂ W 2 } with W 1 fixed and W 2 varying in two planes which contain W 1 . From this description, we can easily see the G invariance of the set of α circles.

It is better to work on P 0 , where the α-bundle is trivial. One difficulty is that in the covering space P 0 , we need to capture the missing information of group M . More precisely, for h in G and z, z in P 0 if hπ(z), hπ(z ) are close, we do not know whether hz, hz are close or not. This will be answered at the end of Section 2.2.5.

Remark 2.2.18. In an abstract language as in [BQ14, Lemma 4.8], we have a principal bundle M → P 0 → P, where the action of M on P 0 is a right action. We also have a left action of a semigroup Γ in G on P 0 and P (Γ will be taken as Γ µ in our case). Suppose that we have a Γ-minimal set Λ Γ in P. The lifting of Λ Γ to P 0 has different possibilities. Let η be a point in Λ Γ and z = kz o be a lifting in P 0 . Let M z = {m ∈ M |Γkm = Γk}. Then we have a nice equivalence {Γ -minimal orbit in P 0 } ←→ M z \M.

In particular, if Γ is a semigroup of matrices of positive entries, then M z = {e} and Γ has the maximal number of minimal orbits in P 0 .

The sign group

Recall the notation for Lie groups and Lie algebras. Let N -be the subgroup with Lie algebra n -. We have a Bruhat decomposition of the Lie group G, where the main part is given by

N -× M × A e × N → G.
The image U is a Zariski open subset of G and the map is injective. For elements in U , we can define a map m to the group M , mapping an element g to the part of M in the Bruhat decomposition.

In order to study the M part, we will use fundamental representations defined in Lemma 2.2.1. This is in the same spirit as the treatment of the sign group M in [START_REF] Benoist | Convexes divisibles iii[END_REF]. Let v α be a non zero eigenvector with highest weight χ α in V α . Let sg be the sign function on R.

Lemma 2.2.19. For g in U , we have

sg v α , ρ α (g)v α = χ α (m(g)),
where χ α is the corresponding algebraic character on A of the fundamental weight χ α .

Proof. Since v α is N -invariant and the transpose of

N -is N , v α , ρ α (N -M A e N )v α ) = ρ α ( t (N -))v α , ρ α (M A e N )v α = v α , ρ α (M A e )v α .
The action of A e does not change the sign, hence by Lemma 2.2.6 we have

sg v α , ρ α (g)v α = sg v α , ρ α (m(g))v α = χ α (m(g)).
The proof is complete.

In the case G = SL m+1 (R), the algebraic character χ α i is given by

χ α i (a) = a 1 • • • a i for a = diag(a 1 , • • • , a m+1 ) ∈ A. Hence we have Lemma 2.2.20. The function Π α∈Π χ α : M → R m given by Π α∈Π χ α (m) = (χ α (m)) α∈Π for m ∈ M, is injective. Definition 2.2.21. We define the sign function from G × G to M ∪ {0} by m(g, g ) = m( t gg ) if t gg ∈ U, 0 if not,
where g, g in G.

This definition exploits the relation between g and g . More precisely, for u, v in V α we have v, ρ α ( t gg )u = ρ α gv, ρ α g u , which explains the definition. Due to t N = N -, the sign function m factors through G/A e N × G/A e N = P 0 × P 0 .

We now explain the sign function for the case m = 1, that is SL 2 (R). We only need to consider the representation of SL 2 (R) on R 2 . Let v 0 = 1 0 be a vector with highest weight in R 2 . Then v 0 , t gg v 0 = gv 0 , g v 0 , which is the inner product of the first column of g and g . The sign function is used to determine whether these two vectors gv 0 , g v 0 have an acute angle or not. By the Bruhat decomposition, we have the following two lemmas.

Lemma 2.2.22. For g, g in G and m in M , we have

m(g, g m) = m(gm, g ) = m(g, g )m. v 1 v ⊥ 1 v 3 v 2 θ 1 θ 2 Figure 2.1: Angle Lemma 2.2.23. Take a Cartan decomposition of g, that is g = k g a g g ∈ KA + K. Then for h in G, m(k g , gh) = m( -1 g , h)
The key observation here is that the sign function is locally constant. Recall that ζ o is point in P and its image in PV * α is the linear functional on V α which vanishes on the hyperplane perpendicular to

V χα . Recall that δ(η, ζ) = min α∈Π δ(V α,η , V * α,ζ ) and d(η, η ) = max α∈Π d(V α,η , V α,η ). Lemma 2.2.24. For k 1 , k 2 , k 3 in K, if δ(k 2 η o , k 1 ζ o ) > d(k 2 η o , k 3 η o ), then m(k 1 , k 2 ) = m(k 1 , k 3 )m(k 2 , k 3 ). Proof. By definition, we have δ(k 2 η o , k 1 ζ o ) = δ( t k 1 k 2 η o , ζ o ) and m(k 1 , k 2 ) = m(id, t k 1 k 2 ).
Hence, we can suppose that k 1 = e, the identity element in K. Lemma 2.2.19 and Lemma 2.2.20 imply that it is sufficient to prove that if δ(k 2 η o , ζ o ) > d(k 2 η 0 , k 3 η 0 ) and m(k 2 , k 3 ) = e, then for every simple root α, we have

sg v α , ρ α (k 2 )v α = sg v α , ρ α (k 3 )v α . Fix a simple root α in Π. Abbreviate v α , ρ α (k 2 )v α , ρ α (k 3 )v α to v 1 , v 2 , v 3 .
Let θ 1 be the angle between the vector v 2 and the hyperplane v ⊥ 1 and let θ 2 be the angle between v 2 and v 3 . Due to m(k 2 , k 3 ) = e, this implies

0 < v 1 , t k 2 k 3 v 1 = k 2 v 1 , k 3 v 1 = v 2 , v 3 , the angle θ 2 is acute. The image of ζ 0 in PV * α is given by R v 1 , • . The hypothesis δ(k 2 η o , ζ o ) > d(k 2 η 0 , k 3 η 0 ) implies that sin θ 1 = v 1 , v 2 > v 2 ∧ v 3 = sin θ 2 .
Hence θ 2 < θ 1 and v 2 , v 3 are in the same side of the hyperplane v ⊥ 1 , which implies

sg v 1 , v 2 = sg v 1 , v 3 . Please see figure 2.1.
We state a consequence of Lemma 2.2.24 which will be used in Section 2.4.2 to get independence of certain measures λ j .

Lemma 2.2.25. Let δ < 1/2, let g, h be in G and k, k in K. If h, k, k satisfy

d(kη o , k η o ) < δ, kη o , k η o ∈ B m h (δ), η M h ∈ B m g (3δ) and γ(h) < δ 2 , then m(k g , ghk) = m( -1 g , hk )m(k, k ). Proof.
ζ o ) > δ > d(kη o , k η o ) implies m( -1 h , k ) = m( -1 h , k) = e. By Lemma 2.2.23, we conclude that e = m(k h , hk) = m( -1 h , k) = m( -1 h , k ) = m(k h , hk ).
Here we need a distance d 0 on P 0 , which is defined in Appendix 2.5.2. Let z = kz o and z = k z o . By Lemma 2.5.5,

d 0 (hz, hz ) ≤ d 0 (hz, z h ) + d 0 (z h , hz ) ≤ d(hkη 0 , η M h ) + d(η M h , hk η 0 ). (2.2.22)
Hence by (2.2.22), we have d 0 (hz, hz ) ≤ 2δ < 1, which implies m(hk, hk ) = e due to Lemma 2.5.5.

The proof of Lemma 2.2.26 also says that if z, z are close and away from the subvariety defined by h, the gap of h is large, then hz, hz are also close.

Derivative

Let ϕ be a C 1 function on P 0 . We will give some property of the directional derivative of ϕ. We write ∂ α ϕ for the directional derivative ∂ Yα ϕ, where α is a simple root. It turns out later that these directions are the major directions when we consider the action of G on P 0 . Definition 2.2.27 (Arc length). Let z 1 , z 2 be two points in the same α-circle in P 0 . If m(z 1 , z 2 ) = e, we define the arc length distance between z 1 , z 2 by

d A (z 1 , z 2 ) := arcsin d(πz 1 , πz 2 ).
Remark 2.2.28. This is a restriction of left K-invariant distance, which can be induced by the K-invariant Riemann metric d 2 in the appendix.

Lemma 2.2.29 (The Newton-Leibniz formula). Let z 1 , z 2 be two points in the same αcircle on P 0 such that m(z 1 , z 2 ) = e. Let u = d A (z 1 , z 2 ) and let γ : [0, u] → P 0 be the curve in the α-circle connecting z 1 , z 2 with unit speed (in the sense of arc length). Then for g in G ϕ(gz 1 ) -ϕ(gz 2 ) = ± u 0 ∂ α ϕ gγ(s) e -ασ(g,γ(s)) ds,

(2.2.23)

where the sign depends on the direction of γ.

Remark 2.2.30. The α-circle already has an orientation given by Y α . The sign is negative if the curve γ is negatively oriented.

Proof. Without loss of generality, suppose that γ is positively oriented. Recall that K α = Y α -X α for α ∈ Π. The images of K α and Y α coincide in g/p. Then k 2 = k 1 exp(uK α ) and γ(s) = k 1 exp(sK α )z o for s ∈ [0, u]. By the Newton-Leibniz formula and (2.2.17 The proof is complete.

Since a root α lies in Γ(G), the lattice generated by fundamental weights, there is a corresponding algebraic character α of A. For m in M and α in Π, by Lemma 2.2.6 with the adjoint representation of G on g, due to

Y α ∈ g -α , we have Ad m Y α = (-α) (m)Y α = α (m) -1 Y α = α (m)Y α .
The last equality is due to α (m) ∈ {±1}. Thanks to (2.2.18), we have Lemma 2.2.31. Let m be in M and let ϕ be a C 1 function on P 0 which is right Minvariant. We have for z = kz o in P 0

∂ α ϕ kmzo = α (m)∂ α ϕ z .
We say a function ϕ on P 0 is the lift of a function on PV α , if there exists a function ϕ 1 on PV α such that for z = kz o ∈ P 0 ϕ(z) = ϕ 1 (V α,kη 0 ). By Lemma 2.2.15, we have Lemma 2.2.32. If ϕ is a C 1 function on P 0 , which is the lift of a C 1 function on PV α , then ∂ α ϕ = 0 for α = α, α ∈ Π.

Changing Flags

This part is trivial for SL 2 (R), where the flag variety P(R 2 ) is a single α-orbit. In this section, we suppose that the rank m is no less than two.

On the flag variety, we have many directions in the tangent space. Roughly speaking, the action of g is contracting and the contraction speed on Y α is given by e -ακ(g) , α ∈ R + . Due to κ(g) being in the Weyl chamber a + , the slowest directions are given by simple roots. Other directions are negligible. The main result Lemma 2.2.40 is a quantitative version of this intuition.

We have already seen that if two points η, η are in the same α-circle, then we have a nice formula for the difference of the value of a real function ϕ at gη and gη , where g ∈ G. We want to do this for η, η in general position. For this purpose, we need to change the point according to g. This is a key new observation in higher rank.

If we are on the euclidean space E n and we are only allowed to move along the directions of coordinate vectors. For any two points x, x , we can walk from x to x with at most n moves. But this is not true for the flag variety P. Suppose that we are only allowed to move along α circles with α ∈ Π. Then for two general points η, η in P, it takes more than m = #Π moves to walk from one point to the other point. We try to move in each α circle at most one time and to make the resulting points as close as possible.

Recall that V is a finite dimensional vector space with euclidean norm. Let l = R(v 1 ∧ v 2 ) be a point in P(∧ 2 V ), which is also a line in PV . Lemma 2.2.33. Let x = Rw 1 be a point in PV and l = R(v 1 ∧ v 2 ) be a line in PV . Then we have

d(l, x) := min x ∈l d(x , x) = v 1 ∧ v 2 ∧ w 1 v 1 ∧ v 2 w 1 .
Proof. The geometric meaning of v 1 ∧ v 2 ∧ w 1 is the volume of the parallelepiped generated by three vectors v 1 , v 2 , w 1 . This volume can also be calculated as the product of the area of the parallelogram generated by v 1 and v 2 , that is v 1 ∧v 2 , and the distance of w 1 to the plane generated by v 1 and v 2 , that is d(w 1 , Span(v 1 , v 2 )). Hence, we have the formula

v 1 ∧ v 2 ∧ w 1 = v 1 ∧ v 2 d(w 1 , Span(v 1 , v 2 )).
(2.2.24)

The distance d(w 1 , Span(v 1 , v 2 )) equals w 1 d(l, x), because the geometric sense of d(l, x) is the sine of the angle between the vector w 1 and the plane Span(v 1 , v 2 ). Together with (2.2.24), we have the result .

Lemma 2.2.34. Let x be a point in PV and l be a line in PV . If g ∈ GL(V ) satisfies that δ(x, y m g ), δ(l, y m ∧ 2 g ) > δ, then

d(gl, gx) ≤ δ -2 γ 1,3 (g)d(l, x),
where γ 1,3 (g) = ∧ 3 g ∧ 2 g g .

Compared with Lemma 2.2.9, with more degree of freedom the contracting speed is significantly greater.

Proof. By definition and l

= R(v 1 ∧ v 2 ), x = Rw 1 , we have d(gl, gx) = ∧ 2 g(v 1 ∧ v 2 ) ∧ gw 1 ∧ 2 g(v 1 ∧ v 2 ) gw 1 ≤ ∧ 3 g v 1 ∧ v 2 ∧ w 1 ∧ 2 g(v 1 ∧ v 2 ) gw 1 ,
Then by Lemma 2.2.7, we have

d(gl, gx) ≤ ∧ 3 g v 1 ∧ v 2 ∧ w 1 δ 2 ∧ 2 g v 1 ∧ v 2 g w 1 = ∧ 3 g δ 2 ∧ 2 g g d(l, x).
The proof is complete.

Lemma 2.2.34 can also be understood that there exists a point x = Rv ∈ l such that v ∧ w 1 is orthogonal to the vector of highest weight in ∧ 2 V . Then the distance between gx and gx will be roughly γ 1,3 (g).

We will start to change the flags. Recall that for α ∈ Π and η, η in P, the function d α (η, η ) is the distance between the images of η and η in PV α . If one wants to change a flag in the α-circle in P, there are some constraints from the structure of flags. We introduce the following definition which explains the constraint.

Definition 2.2.35. Let η = {W 1 ⊂ W 2 ⊂ • • • ⊂ W m+1 = R m+1 }
be a flag in P. Recall that W r are r-dimensional subspaces of R m+1 . Let i r be the natural embedding of the Grassmannian to projective spaces, that is G r (R m+1 ) → P(∧ r R m+1 ). We write l r,η = l αr,η

:= i r (W r+1 ⊃ W r ⊃ W r-1 )
for a line in P(∧ r R m+1 ), which is the image of all the r dimensional subspace W r of R m+1 such that W r-1 ⊂ W r ⊂ W r+1 . Take W 0 = {0}. We also write l r,η when the line l r,η is seen as a point in P(∧ 2 (∧ r R m+1 )).

Recall that V αr = ∧ r V α 1 = ∧ r R m+1 and e 1 , . . . , e m+1 is the standard basis of R m+1 .

α 1 -orbit

gη 0 gη 0 gη 1 gη 1 α 2 -orbit Figure 2.2: Changing Flag for SL 3 (R)
The point is that the contraction speed β implies that the term δ -2 βe -ακ(g) is of smaller magnitude than e -ακ(g) . The objective is to walk from gη to gη only through α circles and to preserve information of distance. Since we can neglect error term, it is simpler to walk from gη to gη l 1 through some α circles and to walk from gη to gη l 2 through the other α circles, which means the corresponding simple roots are different from the first walk. After this operation, the distance between gη l 1 and gη l 2 is negligible, due to (2.2.26). The distance of the move in the α circle is approximately the distance between the images of gη and gη in PV α , due to (2.2.25).

Proof of Lemma 2.2.40. If we have already found (η 0 , . . . , η j ) and j < l 1 , we want to find η j+1 . Let α = α r ∈ Π 1 be a root that does not appear in the chain. Hence by Lemma 2.2.39, V αr,η j = V αr,η 0 = V αr,η .

(2.2.27)

Due to Lemma 2.2.38, the neighbour simple roots α r-1 and α r+1 are not in Π 1 , since α r-1 + α r and α r+1 + α r are roots. By Lemma 2.2.39,

V α r-1 ,η j = V α r-1 ,η and V α r+1 ,η j = V α r+1 ,η .
We are in the situation of Lemma 2.2.34 with

V = V αr = ∧ r R m+1 , x = V
+ O(δ -2 βe -αrκ(g) ),
which is (2.2.25). Please see Figure 2.2, where an element in the flag variety is represented by a projective line with a point. We need to verify the distance between gη l 1 and gη l 2 . Without loss of generality, suppose that α ∈ Π 1 . Then by Lemma 2.2.39, the construction and (2.2.28),

d α (gη l 1 , gη l 2 ) = d α (gη l 1 , gη ) = d α (gη j+1 , gη ) ≤ δ -2 βe -ακ(g) ,
where j is the unique number such that α(η j , η j+1 ) = α.

Remark 2.2.41. In the case of SL 3 (R), we know that ∧ 2 V α 1 and ∧ 2 V α 2 are isomorphic to V α 2 and V α 1 , respectively. The condition in Lemma 2.2.40 is equivalent to η, η in B m g (δ).

In the case of SL m+1 (R), the representations V r = ∧ r R m+1 are fundamental representation. Since SL m+1 (R) is split, ∧ 2 V r is again proximal, but may not be irreducible. In Lemma 2.2.58, we will proceed to give a control on y m ∧ 2 (∧ r g) . The condition of Lemma 2.2.40 is not really important, what we need is that the condition is true with a loss of exponentially small measure when we consider the random walks on SL m+1 (R).

Lemma 2.2.42. With the same assumption and construction in Lemma 2.2.40, if we also have η, η ∈ B m g (δ), then gη j , gη l are in b M g (βδ -O(1) ) for 1 ≤ j ≤ l 1 and 1 ≤ l ≤ l 2 .

Proof. By hypothesis, Lemma 2.2.13 implies that gη, gη ∈ b M g (βδ -1 ). By (2.2.25), d(gη j , gη j+1 ) ≤ 2βδ -1 + O(δ -2 βe -ακ(g) ) ≤ βδ -O(1) .

Hence by induction, we have gη j ∈ b M g (βδ -O(1) ) for all j. Similarly the results hold for gη l .

Random walks and Large deviation principles

The study of random walks on projective spaces and flag varieties are connected by representation theory.

Let X be P or PV , where V is an irreducible representation of G. There is a natural group action of G on X. Let µ be a Borel probability measure on G. Then a Borel probability measure ν on X is called µ-stationary if

ν = µ * ν := G g * νdµ(g),
where g * ν is the pushforward measure of ν under the action of g on X.

Proposition 2.2.48 is a multidimensional version of Proposition 2.2.47. Proposition 2.2.49 (Hölder regularity). If (ρ, V ) is an irreducible representation of G, then there exist constants C > 0, c > 0 such that for every y in PV * and r > 0 we have

ν V ({x ∈ PV | δ(x, y) ≤ r}) ≤ Cr c .
(2.2.33)

The proximality of the representation is also needed in Proposition 2.2.49. This result is due to Guivarc'h [START_REF] Guivarc | Produits de matrices aléatoires et applications aux propriétés géométriques des sous-groupes du groupe linéaire[END_REF]. See [BQ16, Thm 14.1] for more details. As a corollary of Proposition 2.2.49, we have the following.

Corollary 2.2.50. If (ρ, V ) is an irreducible representation of G with highest weight χ, then there exist constants C > 0, c > 0 such that for every y in PV * and r > 0 we have

ν({η ∈ P| δ(V χ,η , y) ≤ r}) ≤ Cr c .
(2.2.34)

Proof. By Lemma 2.2.43, we have

ν({η ∈ P| δ(V χ,η , y) ≤ r}) = ν V ({x ∈ PV | δ(x, y) ≤ r}).
Hence Corollary 2.2.50 follows from Proposition 2.2.49.

All the results in this section mean that the quantities considered here are really flexible. We can always image that things happen as wished in a large probability, a very positive expectation. Bad things are near some algebraic subvariety and have exponential small measures. For later convenience, we introduce the following definition. Definition 2.2.51 (Good element). For n ∈ N, > 0 and η, ζ ∈ P, we say that an element h is (n, , η, ζ) good if

κ(h) -nσ µ ≤ n/C A and δ(η, ζ m h ), δ(η M h , ζ) > 2e -n/C A , ( 2

.2.35)

where C A is a constant greater than 2, which is only depend on the whole group and will be determined in Lemma 2.2.53.

Lemma 2.2.52. We have that h is (n, , η, ζ) good outside an exponentially small set, that is to say there exist C > 0, c > 0 such that

µ * n {h is not (n, , η, ζ) good.} ≤ Ce -c n .
Proof. This is due to the large deviation principle (2.2.29), (2.2.31) and (2.2.32).

Lemma 2.2.53. Let δ = e -n and β = max α∈Π e -ασµn . Suppose that is small enough such that β < δ 3 . If h is (n, , η, ζ m g ) good, then γ(h) ≤ βδ -1 ≤ δ 2 and σ(gh, η) -κ(g) -nσ µ ≤ n.

Proof. By hypothesis, γ(h) = max α∈Π e -ακ(h) = sup α∈Π e -αnσµ e α(nσµ-κ(h)) ≤ βδ -1 , if we take C A large enough such that for all simple roots α and X in a, we have |α(X)| ≤ C A X . By Lemma 2.2.13, we have hη

∈ b M h (γ(h)/δ) ⊂ b M h (δ) ⊂ B m g (δ). Hence by Lemma 2.2.12 σ(gh, η) -κ(g) -nσ µ = σ(g, hη) -κ(g) + σ(h, η) -nσ µ | log δ(hη, ζ m g )| + | log δ(η, ζ m h )| + κ(h) -nσ µ n/C A .
Hence if C A is large enough depending on the whole group, the inequality holds.

For later usage in Section 2.3, we will define another notation of goodness.

Definition 2.2.54. For n ∈ N, > 0 and ζ ∈ P, we say that an element h is (n, , ζ)

good if κ(h) -nσ µ ≤ n/C A and δ(η M h , ζ) > 2e -n/C A . (2.2.36)
Lemma 2.2.55. Let δ = e -n and β = max α∈Π e -ασµn . Let η d be a flag which is different from η o only in d-dimensional subspace, that is

η d = {Re 1 ⊂ • • • ⊂ Re 1 ⊕ • • • ⊕ Re d-1 ⊂ Re 1 ⊕ • • • ⊕ Re d-1 ⊕ Re d+1 ⊂ • • • }. (2.2.

37)

If h is (n, , ζ m g ) good, then for η = l -1 h η d , we have e χ j (σ(gh,η)-κ(g)-nσµ) ∈ [δ, δ -1 ] for j = d and e χ d (σ(gh,η)-κ(g)-nσµ) ≤ βδ -1 .

(

χ d σ(h, η d ) = log hv v = log exp(κ(h))v v = (χ d -α d )κ(h).
(2.2.39) By (2.2.2) and (2.2.3), we have χ d (σ(g, hη) -κ(g)) ≤ 0. Together with (2.2.39), e χ d (σ(gh,η)-κ(g)-nσµ) = e χ d (σ(g,hη)-κ(g)) e χ d (σ(h,η)-nσµ) ≤ e (χ d -α d )κ(h)-nχ d σµ = e -α d κ(h) .

By (2.2.36), the proof is complete.

This Lemma tells us that by changing the image of η in one projective space, the value of Iwasawa cocycle only changes in that space. There is some independence of the value of Iwasawa cocycle with respect to η.

Let V be a representation of G. Let G 2 (V ) := {2-planes in V } be the Grassmannian variety of V . Let q λ : ∧ 2 V → ∧ 2 V be the G-equivalent projection on the sum of all the irreducible subrepresentations of ∧ 2 V with highest weight equal to λ. Lemma 2.2.56. Let V be an irreducible representation of G with highest weight χ. For a simple root α, let q 2χ-α be the G-equivalent projection from ∧ 2 V to ∧ 2 V . There exists c > 0 such that for all v, v in

V α∈Π q 2χ-α (v ∧ v ) ≥ c v ∧ v .
Proof. By Lemma 2.2.57, we know that α∈Π q 2χ-α (v∧v ) v∧v : G 2 (V ) → R ≥0 is a positive continuous function. Since G 2 (V ) is a compact space, on which positive continuous function has a lower bound, the result follows.

The following lemma is similar to [BQ12, Lemma 3.3].

Lemma 2.2.57. With the same assumption as in Lemma 2.2.56, then α∈Π q 2χ-α does not contain any pure wedge.

Proof. Let W be the intersection of all the kernels, that is W = α∈Π q 2χ-α . The two sets G 2 (V ) and PW are closed subset of P(∧ 2 V ) and G invariant. Therefore their intersection is again a G invariant closed subvariety which is complete. Let B be the Borel subgroup of G, which is solvable. By [Bor90, Thm.10.4], the action of a solvable algebraic connected group on a complete variety has fixed points. We claim that the fixed points of B on G 2 (V ) are the lines with the highest weight. Then the result follows by the fact that these lines do not belong to W .

Suppose that there exit v, u in V such that v ∧ u is B invariant. We can decompose v, u as a sum v = λ v λ and u = λ u λ . Since we can replace v, u by bv, bu for b in B, we can suppose that the component of highest weight v χ is non zero. Since the dimension of V χ is 1, we can suppose that u χ = 0. Let ρ = χ be a highest weight such that u ρ is nonzero. The B invariance of R(v ∧ u) also implies that the action of X α , for α simple roots, fixes the line. Hence X α (v ∧u) = X α v ∧u+v ∧X α u ∈ Rv ∧u. The weight χ+ρ+α is higher than all the weights appear in v ∧ u, hence v χ ∧ X α u ρ = 0 for all simple roots α. This implies that ρ = χ -α for some simple root α. Therefore v ∧ u contains v χ ∧ u χ-α . Since v ∧ u is also A invariant, all the components in the weight decomposition have the same weight. Hence v ∧ u = v χ ∧ u χ-α which is a vector of highest weight in ∧ 2 V .

We want to prove a large deviation principle for a special reducible representation. This lemma will be used in Lemma 2.4.10 to control y m ∧ 2 g in Lemma 2.2.9 and Lemma 2.2.40.

Lemma 2.2.58. Let V be a super proximal representation of G (Definition 2.2.2). For > 0 there exist C, c > 0 such that the following holds. For x = Rv, x = Rv ∈ PV with x = x , we have

µ * n {g ∈ G|δ(x ∧ x , y m ∧ 2 ρ(g) ) < e -n } ≤ Ce -c n .
Due to Definition 2.2.2, there is only one simple root α such that q 2χ-α (∧ 2 V ) is non zero. Write ∧ 2 V = W ⊕ W , where W is the irreducible representation generated by the vector corresponding to the highest weight in ∧ 2 V , and W is the G-invariant complementary subspace. Then q 2χ-α (∧ 2 V ) = W , and we write P r W = q 2χ-α .

Proof of Lemma 2.2.58. By (2.2.10), we see that a non zero vector in y m ∧ 2 g vanishes on W and y m ∧ 2 g can be seen as an element in PW * . We only need to consider the projection of v ∧ v onto W and use large deviation principle (2.2.30). By Lemma 2.2.56,

δ(x ∧ x , y m ∧ 2 g ) = |f (v ∧ v )| v ∧ v = |f (P r W (v ∧ v ))| P r W (v ∧ v ) P r W (v ∧ v ) v ∧ v ≥ cδ(P r W (x ∧ x ), y m ∧ 2 g ),
where f is a unit vector in y m ∧ 2 g . The proof is complete.

Non concentration condition

We want to verify the main input for the sum-product estimate, the non concentration condition. If we want to get the non concentration directly, then this becomes an effective local limit estimate, which is difficult due to the lack of spectral gap. Hence, we transfer it to the Hölder regularity of stationary measure.

For the first time read, the reader can neglect g in the left of h. The main idea of the proof is already there. Adding g is a technical step, which is needed in its application. (We only need an additional condition on η M h to control κ(gh).)

Projective, Weak and Strong non concentration

Recall that m is the real rank of G and χ 1 , • • • , χ m are the fundamental weights, where we change the subscript from α ∈ Π to i ∈ {1, • • • , m}. Recall that α 1 , • • • , α m are the simple roots of a * , which are linear combinations of fundamental weights χ i with integral coefficients.

In order to distinguish different objects, we will use capital letter X to denote functions or random variables and use small letter x to denote vectors or indeterminates.

Let L be the d × d square matrix which changes the basis

(χ 1 , • • • , χ m ) of a * to the basis (-α 1 , • • • , -α m ), that is L ij = -α i (H j ).
Then L is an integer matrix. Hence, we can define E d , a rational map from (R * ) m to (R * ) d , which is given by y = E d (x) for x ∈ (R * ) m where

y i = Π 1≤j≤m x L ij j .
Fix an element g in G. Let X g (n, h, η) = (e χ 1 (σ(gh,η)-κ(g)-nσµ) , . . . , e χm(σ(gh,η)-κ(g)-nσµ) ), Y n g (h, η) = (e -α 1 (σ(gh,η)-κ(g)-nσµ) , . . . , e -αm(σ(gh,η)-κ(g)-nσµ) )

for η in P and h in G. By definition, E d X g (n, h, η) is the vector which is composed of the first d components of Y n g (h, η), that is

p d Y n g (h, η) = E d X g (n, h, η), (2.3.1)
where p d : R m → R d is the map which takes a vector x of R m to the vector of R d composed of the first d components of x. In the following argument g is fixed or g equals identity. Hence we will abbreviate X g , Y n g , Y n e to X, Y n , Y n 0 . We define an affine determinant

A d on (R d ) d+1 . For d + 1 vectors y 1 , • • • , y d+1 in R d , let A d be the determinant of the (d + 1) × (d + 1) matrix y 1 • • • y d+1 1 • • • 1 , which
is the volume of the d + 1-dimensional parallelogram generated by vectors (y i , 1) for i = 1, . . . , d + 1. Let e i be the vector in R d with only i-th coordinate nonzero and equal to 1. By identifying e 1 ∧ • • • ∧ e d with number 1, we can also define A d by

A d (y 1 , • • • , y d+1 ) = 1≤i≤d+1 (-1) i+d+1 y 1 ∧ • • • ∧ y i ∧ • • • ∧ y d+1 . For d + 1 vectors x 1 , • • • , x d+1 in R m , let B d be a rational function defined by B d (x 1 , • • • , x d+1 ) = A d (E d x 1 , • • • , E d x d+1 ).
We introduce the notation

h d+1 = (h 1 , . . . , h d+1 ),
which is an element in G ×(d+1) . Let

A n d (h d+1 , η) := B d (X(n, h 1 , η), . . . , X(n, h d+1 , η)).
Definition 2.3.1. We say that µ satisfies the projective non concentration (PNC) on dimension d, if for every > 0 there exist c, C > 0 such that for all n in N, η in P and

g in G sup a∈R,v∈S d-1 µ * n {h ∈ G|| v, Y n (h, η) -a| ≤ e -n } ≤ Ce -c n ,
where v is regarded as a vector in

R d × {0} m-d ⊂ R m .
More geometrically, this is equivalent to say that the measure of Y n (h, η) close to an affine hyperplane is exponentially small. Definition 2.3.2. We say that µ satisfies the weak non concentration (WNC) on dimension d, if for every > 0 there exist c, C > 0 such that for all n in N, η in P and g in

G (µ * n ) ⊗(d+2) {(h d+1 , ) ∈ G ×(d+2) ||A n d (h d+1 , η)| ≤ e -n } ≤ Ce -c n .
Definition 2.3.3. We say that µ satisfies the strong non concentration (SNC) on dimension d, if for every > 0 there exist c, C > 0 such that for all n in N, η in P and g in G

(µ * n ) ⊗(d+1) {h d+1 ∈ G ×(d+1) ||A n d (h d+1 , η)| ≤ e -n } ≤ Ce -c n .
We will proceed by induction. When d = 0, we make the convention that A d 0 = 1 and it is trivial that SNC holds. Then

• SNC on dimension d ⇒ WNC on dimension d (By definition)

• PNC on dimension d ⇔ SNC on dimension d (Lemma 2.3.7)

• WNC on dimension d ⇒ PNC on dimension d (Lemma 2.3.9)

• SNC on dimension d -1 ⇒ WNC on dimension d (Lemma 2.3.10).
In the above implications, the constants C, c will change. We can conclude Proposition 2.3.4. Let µ be a Zariski dense Borel probability measure on G with exponential moment. Then µ satisfies PNC on dimension m.

Away from affine hyperplanes

We need a lemma of linear algebra, which relates different non concentrations. This lemma is already known from [EMO05, Lemma 7.5]. Recall that for two subsets A, B of a metric space (X, d), the distance between A and B is defined as

d(A, B) = inf x∈A,y∈B d(x, y) Lemma 2.3.5. Let C > 0, c > 0. Let u 1 , • • • , u d+1 be vectors in R d with length less than C.

Consider the following conditions:

i. There exists an affine hyperplane l such that for i = 1, . . . , d + 1,

d(u i , l) ≤ c. ii. We have 1≤i≤d+1 (-1) i u 1 ∧ • • • ∧ u i ∧ • • • ∧ u d+1 < c,
where u i means this term is not in the wedge product.

iii. There exists i in {1, . . . , d} such that d(u i , Span aff (u d+1 , u 1 , . . . , u i-1 )) < c, where Span aff is the affine subspace generated by the elements in the bracket.

Then i(c) ⇒ ii(2 d+1 C d-1 c), ii(c) ⇒ iii(c 1/d ) and iii(c) ⇒ i(c).
Proof. We first transfer the affine problem to a linear problem. Let v i = u i -u d+1 for i = 1, . . . , d. Then v i are vectors with length less than 2C. The above three conditions are equivalent to (with change of constants in i) i'. There exists a linear subspace l of codimension 1 such that for i = 1, . . . , d

d(v i , l) ≤ c. ii'. We have v 1 ∧ • • • ∧ v d < c.
iii'. There exists i such that

d(v i , Span(v 1 , . . . , v i-1 )) < c,
where Span is the linear subspace generated by the elements in the bracket.

iii (c) ⇒ i (c): Let the hyperplane l be Span(v 1 , • • • , vi , • • • , v d ). Then i (c) follows from iii (c). i (c) ⇒ ii (2 d C d-1 c):
Due to i , the volume of the parallelogram generated by {v i } 1≤i≤d is less than (2C) d-1 2c, which is ii .

ii (c) ⇒ iii (c 1/d ): Due to the same argument as in Lemma 2.2.33, we have a formula of volume,

v 1 ∧ • • • ∧ v d = Π 1≤i≤d d(v i , Span(v 1 , . . . , v i-1 )),
from which the result follows.

As a corollary, we have the following lemma, which is general and deals with random variables.

Corollary 2.3.6. Let X 1 , . . . , X d+1 be i.i.d. random vectors in R d bounded by C > 0. Let l be an affine hyperplane in R d . Then for any c > 0, we have d+1) , we have

P{d(X 1 , l) < c} d+1 ≤ P{ (-1) i X 1 ∧ • • • ∧ Xi ∧ • • • ∧ X d+1 < 2 d+1 C d-1 c}, (2.3.2) and P{ (-1) i X 1 ∧ • • • ∧ Xi ∧ • • • ∧ X d+1 < c} ≤ 1≤i≤d P{d(X i , Span aff (X d+1 , X 1 , • • • , X i-1 )) < c 1/d }. (2.3.3) Lemma 2.3.7. PNC on dimension d is equivalent to SNC on dimension d. Proof. Let X i = E d X(n, h i , η) for i = 1, • • • , d + 1,
, • • • , ρ d+1 ) ∈ O(d) ×(
(µ * n ) ⊗(d+1) {h d+1 ∈ G ×(d+1) ||A d (ρ 1 E d X(n, h 1 , η), . . . , ρ d+1 E d X(n, h d+1 , η)| ≤ e -n } ≤ Ce -c n .
By Lemma 2.3.7, SNC implies PNC. We adopt the notation in the proof of Lemma 2.3.7. By (2.3.3) and the fact that O(d) preserves the distance,

P{ (-1) i ρ 1 X 1 ∧ • • • ρ i X i • • • ∧ ρ d+1 X d+1 < c} ≤ 1≤i≤d P{d(ρ i X i , l i ) < c 1/d } = 1≤i≤d P{d(X i , ρ -1 i l i ) < c 1/d },
where

l i = Span aff (ρ d+1 X d+1 , ρ 1 X 1 , • • • , ρ i-1 X i-1
). Therefore SNC implies the stronger form of SNC.

Lemma 2.3.9. WNC on dimension d implies PNC on dimension d.

WNC is weaker than SNC, because WNC is not uniform on position η. Let f (η) be (µ * n ) ⊗(d 2 ) {...η} in SNC (Definition 2.3.3). Then WNC only asks that f ( η)dµ * n ( ) is small, whereas SNC asks that f (η) is small for every η. The cocycle property is the key point to obtain an estimate uniform on position from an estimate not uniform on position.

Proof of Lemma 2.3.9. Let δ = e -n . We first prove the result for 2n. Recall that h is a random variable which takes values in G with the distribution µ * 2n . Let h = 1 such that 1 and have distribution µ * n . Then the cocycle property implies

Y n (h, η) = Y n ( 1 , η) = Y n ( 1 , η)Y n 0 ( , η). Fubini's theorem implies E := sup a,v µ * 2n {h| v, Y 2n (h, η) ∈ B(a, δ)} ≤ G sup a,v µ * n { 1 | v, Y n ( 1 , η)Y n 0 ( , η) ∈ B(a, δ)}dµ * n ( ).
The cocycle property is crucial here. Fix and fix a, v. We can write

v, Y n ( 1 , η)Y n 0 ( , η) = R v , Y n ( 1 , η) , where R = v • Y n 0 ( , η) ≥ min 1≤j≤d |Y n 0 ( , η) j |.
Here v is a vector of norm 1, defined by v = v • Y n 0 ( , η)/R, depending on v, l and η. By Lemma 2.2.52 and Lemma 2.2.53, for outside an exponentially small set independent of a, v, we have R ≥ δ 1/2 . Therefore

E ≤ G sup a,v µ * n { 1 | v, Y n ( 1 , η) ∈ B(a, δ 1/2 )}dµ * n ( ) + O (δ c ), (2.3.4)
where c > 0 comes from the large deviation principle (Lemma 2.2.52). By Hölder's inequality,

G sup a,v µ * n { 1 | v, Y n ( 1 , η) ∈ B(a, δ 1/2 )}dµ * n ( ) ≤ (sup a,v µ * n { 1 | v, Y n ( 1 , η) ∈ B(a, δ 1/2 )}) d+1 dµ * n ( ) 1/(d+1)
.

(2.3.5)

By the same argument as in Lemma 2.3.7

sup a,v µ * n { 1 | v, Y n ( 1 , η) ∈ B(a, δ 1/2 )} d+1 ≤ µ * (d+1)n {(h d+1 )||A n d (h d+1 , η)| ≤ 2δ 1/4 } + O (δ c ).
Therefore, by (2.3.4) and (2.3.5), we have

E d+1 ≤ µ * (d+2)n {(h d+1 , )||A n d (h d+1 , η)| ≤ 2δ 1/4 } + O (δ c ).
The proof for 2n ends by Definition 2.3.2. It remains to prove the same result for 2n+1. Let h = such that has distribution µ * (n+1) and 1 has distribution µ * n . Following the same argument, we have

E d+1 ≤ µ * (d+1)n+(n+1) {(h d+1 , )||A n d (h d+1 , η)| ≤ 2δ 1/4 } + O (δ c ).
Since only changes the position η, the uniformity of WNC implies that

µ * (d+1)n+(n+1) {(h d+1 , )||A n d (h d+1 , η)| ≤ 2δ 1/4 } = l 3 ∈G µ * (d+2)n {(h d+1 , l 2 )||A n d (h d+1 , l 2 (l 3 η))| ≤ 2δ 1/4 }dµ(l 3 ) δ c .
The proof is complete.

Hölder regularity

In this section, we will prove Lemma 2.3.10. SNC on dimension d -1 implies WNC on dimension d.

Using other representations, we can get more information on the Iwasawa cocycle. This idea has already been used in [START_REF] Aoun | Transience of algebraic varieties in linear groups-applications to generic zariski density[END_REF] for problem concerning transience of algebraic subvariety of split real Lie groups. It is also used in the work of Bourgain-Gamburd on the spectral gap of dense subgroups in SU(n), for establishing transience of subgroups.

The key tool is the following estimate. See [BQ16, Proposition 14.3] or [START_REF] Guivarc | Produits de matrices aléatoires et applications aux propriétés géométriques des sous-groupes du groupe linéaire[END_REF] for example.

Lemma 2.3.11. Let V be an irreducible representation of G. Let µ be a Zariski dense Borel probability measure on G with exponential moment. For every > 0 there exist c, C > 0 such that for v in V and f in V * we have

µ * n { ∈ G| |f ( v)| ≤ f e -n } ≤ Ce -c n .
The intuition is that if a function f is not small at some point, then it is robustly large for almost all points.

In this part, we write V j = V χ j for the fundamental representation and we write V j,η for the image of η ∈ P in PV j for j = 1, . . . , m. Let v j be a nonzero vector in V j,η . For in G, we abbreviate ρ j ( )v j to v j . Since v j lives in V j , we use the same symbol • for norms on different V j , which makes no confusion. For a vector x in R m , we denote by x i the i-th coordinate. We use upper script to denote different vectors.

Before proving Lemma 2.3.10, we introduce some linear algebras. We want to construct a linear form. Recall that E d is a rational map, A d is the affine determinant, B d is the composition of A d and E d and

A n d (h d+1 , η) := B d (X(n, h 1 , η), . . . , X(n, h d+1 , η)),
where

X(n, h, η) = (e χ j (σ(gh,η)-κ(g)-nσµ) ) 1≤j≤m = ghv j e χ j (κ(g)+nσµ) v j 1≤j≤m , (2.3.6)
and the second equality is due to (2.2.3) and (2.2.2). Let X i (n, η) := X(n, h i , η).

(2.3.7)

In order to use Lemma 2.3.11, we need to linearise some function related to A n d (h n+1 , η) with h n+1 fixed. We will multiply B d by its denominator, and all the Galois conjugate to get a polynomial on X i j 2 , which can be realized as a linear functional. The function B d can be seen as a rational function on

(x) := (x 1 , • • • , x d+1 ) = (x i j ) 1≤i≤d+1,1≤j≤m .
By definition, B d has a special form. Each term in B d can be expressed as a quotient of two monomials. Let D d be the lowest common denominator of B d such that D d B d is a polynomial on (x). In other words, suppose that

B d = n∈Z m(d+1) b n 1≤j≤m,1≤i≤d+1 (x i j ) n ij ,
where n is a multi index and b n is the coefficient. Let

q ij = sup n∈Z m(d+1) {-n ij , 0} for 1 ≤ j ≤ m, 1 ≤ i ≤ d + 1. Then D d = Π 1≤j≤m,1≤i≤d+1 (x i j ) q ij .
Definition 2.3.12. Let F be a polynomial on

(x 1 , • • • , x k ) where x 1 , • • • x k are vectors in R n . Then we call F a multi homogeneous polynomial of degree q = (q 1 , • • • , q n ) ∈ N n if for ξ in (R * ) n we have F (ξx 1 , • • • , ξx k ) = ξ q F (x 1 , • • • , x k ),
where ξ q = Π 1≤j≤n ξ q j j .

Let Γ be the finite group (Z/2Z) d(d+1) which acts on R d(d+1) . Let (y

) := (y 1 , • • • , y d+1 ) = (y i j ) 1≤i≤d+1,1≤j≤d ∈ (R d ) d+1
. For ρ ∈ Γ, we write ρ(y) for the action on the coefficient y i j , which is of dimension d(d + 1). Due to the definition of Γ, the product

Π ρ∈Γ A d ρ(y 1 , . . . , y d+1 ) is invariant under the action Γ, hence it is a polynomial on (y i j ) 2 . Let F d (x 1 , . . . , x d+1 ) = ρ∈Γ D d A d ρ(E d x 1 , . . . , E d x d+1 ), (2.3.8) then Lemma 2.3.13. F d is a multi homogeneous polynomial on ((x 1 ) 2 , • • • , (x d+1 ) 2 ) with degree q = (q 1 , • • • , q m ) ∈ N m .
Proof. We only need to verify that F d is a multi homogeneous polynomial. The fact that the determinant is a multilinear function implies that for λ and

y i in R d A d (λy 1 , • • • , λy d+1 ) = det(λ)A d (y 1 , • • • , y d+1 ), (2.3.9)
where

det(λ) = λ 1 • • • λ d .
The functions E d and D d are group morphisms due to definition. Hence we have

E d (ξx) = E d (ξ)E d (x) and D d (ξx 1 , • • • , ξx d+1 ) = D d (ξ, • • • , ξ)D d (x 1 , • • • , x d+1 ).
(2.3.10) Therefore by (2.3.8), (2.3.9) and (2.3.10), for ξ and x i in R m ,

F d (ξx 1 , • • • , ξx d+1 ) = ρ∈Γ D d A d ρ(E d (ξx 1 ), • • • , E d (ξx d+1 )) = ρ∈Γ D d A d ρ(E d (ξ)E d (x 1 ), • • • , E d (ξ)E d (x d+1 )) = ρ∈Γ D d A d ρ(E d (x 1 ), • • • , E d (x d+1 )) det(E d (ξ))D d (ξ, • • • , ξ) = ξ q F d (x 1 , • • • , x d+1 ),
where q is a vector in N m such that ξ q = (det

(E d (ξ))D d (ξ, • • • , ξ)) |Γ| .
For h d+1 ∈ G ×(d+1) and η in P, we write

F (h d+1 , η) = F d (X(n, h 1 , η), . . . , X(n, h d+1 , η)).
Since F/A n d is a polynomial on X i j , for h d+1 which is η good, we have

F/A n d = D d Π ρ∈Γ,ρ =e D d A n d ρ ≤ δ -O(
M := µ * (d+2)n {(h d+1 , )||A n d (h d+1 , η)| ≤ e -n } ≤ µ * (d+2)n {h d+1 is η good, ∈ G||A n d (h d+1 , η)| ≤ e -n } + O 2 (δ c ) ≤ µ * (d+2)n {h d+1 is η good, ∈ G||F (h d+1 , η)| ≤ e -n δ -O(1) } + O 2 (δ c ) ≤ µ * (d+2)n {(h d+1 , )||F (h d+1 , η)| ≤ e -n δ -O(1) } + O 2 (δ c ) (2.3.12)
Step 2: Lemma 2.3.13 implies that F is a multi homogeneous polynomial on (x i j ) 2 of degree q = (q 1 , . . . , q d+1 ). Lemma 2.3.14 implies that

F (h d+1 , η) = F 1 (⊗ j ((v j ) 2 ) ⊗q j )/Π v j 2q j ,
where F 1 is a linear functional on V 0 = j (Sym 2 V j ) ⊗q j . To be more precise, F 1 will be restricted to a linear form on W , the unique irreducible representation of V 0 with maximal weight. (This is specific for real split Lie groups)

It remains to show that for most h d+1 in G ×(d+1) , the norm of F 1 is robustly large. It is sufficient to find one η such that |F (h d+1 , η)| is large. We will prove that |D d A d ρ| is large for each ρ in Γ, which implies that |F (h d+1 , η)| is large.

Using the d + 1-th column expansion of the matrix

y 1 • • • y d+1 1 • • • 1 , we have A d (y 1 , • • • , y d+1 ) = -A d-1 (r d y 1 , • • • , r d y d )y d+1 d + other terms, = 1≤j≤d (-1) j+d+1 A d-1 (r j y 1 , • • • , r j y d )y d+1 j + det(y 1 , • • • , y d ),
(2.3.13) where r j : R d → R d-1 is the map forgetting the j-th coordinate. Replacing

y i by E d x i , due to r d E d x i = E d-1 x i , we obtain A d (E d x 1 , • • • , E d x d+1 ) = -A d-1 (E d-1 x 1 , • • • , E d-1 x d )(E d x d+1 ) d + other terms.
(2.3.14) Using SNC on dimension d-1, we are able to give a lower bound of

A d-1 (E d-1 X 1 , • • • , E d-1 X d )
with a loss of exponentially small probability of h d+1 . But the problem is in other similar terms. Due to y d+1 j = Π 1≤i≤m (x d+1 i ) -α j (H i ) and the structure of root system, the degree of

x d+1 d in y d+1 j = (E d x d+1 ) j is -α d (H d ) = -2 and -α j (H d ) ≥ 0 for j < d.
(2.3.15)

Hence, we will make X d+1 d ≤ β, which makes the first term in (2.3.13) greater than δ O(1) β -2 , and the other terms are less than δ -O(1) . Now, here is the precise proof. Take h d+1 good, that means h d+1 is (n, 2 , ζ m g ) good (Definition 2.2.54). We take

η = -1 h η d (2.3.16)
as in Lemma 2.2.55. By Lemma 2.2.55

X d+1 j ∈ [δ, δ -1 ] for j = d and X d+1 d ≤ βδ -1 .
(2.3.17)

Let Γ d-1 = (Z/2Z) (d-1)d
, seen as a subgroup of Γ, which acts on R (d-1)d . Then we demand that h d satisfies

|A n d-1 ρ(h d , η)| ≥ δ for all ρ ∈ Γ d-1 and h d is η good. (2.3.18) Recall that h d is η good means that h i is (n, 2 , η, ζ m g ) good for 1 ≤ i ≤ d. By Lemma 2.2.53 and (2.3.6), X i j (η) ∈ [δ, δ -1 ], for 1 ≤ i ≤ d, 1 ≤ j ≤ m. (2.3.19)
Recall that W is the unique irreducible subrepresentation of V 0 with the highest weight.

Lemma 2.3.16. We claim that if h d+1 is good ((n, 2 , ζ m g ) good), η is taken as in (2.3.16) and the assumption (2.3.18) is satisfied for h d , then the operator norm satisfies

F 1 | W ≥ δ O(1) .
Proof ot Lemma 2.3.16. As we have already explained, it is sufficient to prove that for ρ in Γ, we have

|D d A n d ρ(h d , η)| ≥ δ O(1)
. The proof is similar for ρ in Γ, we will only prove the case ρ = e.

By (2.3.13) and (2.3.14)

D d A d (E d x 1 , • • • , E d x d+1 ) = -A d-1 (E d-1 x 1 , • • • , E d-1 x d )D d (E d x d+1 ) d + 1≤j<d (-1) j+d+1 A d-1 (r j E d x 1 , • • • , r j E d x d )D d (E d x d+1 ) j + D d det(E d x 1 , • • • , E d x d ) (2.3.20)
where r j : R d → R d-1 is the map forgetting the j-th coordinate. Since x d+1 d only appears in E d x d+1 , by (2.3.15), we know that the degree of

x d+1 d in D d equals α d (H d ) = 2, which implies that D d ≤ δ -O(1) β 2 .
Hence by (2.3.17)-(2.3.19) and the property (2.3.15) that the degree of

X d+1 d in (E d X d+1 ) d is -2, the degree in (E d X d+1 ) j is non negative for j < d, we have D d (E d X d+1 ) d ≥ δ O(1) , |A d-1 (E d-1 X 1 , • • • , E d-1 X d )| ≥ δ O(1) , D d (E d X d+1 ) j ≤ δ -O(1) β 2 , |A d-1 (r j E d X 1 , • • • , r j E d X d )| ≤ δ -O(1) for 1 ≤ j < d and D d det(E d X 1 , • • • , E d X d ) ≤ δ -O(1) β 2 .
(2.3.21) By (2.3.20) and (2.3.21), we have

|D d A n d | ≥ δ O(1) -δ -O(1) β 2 ≥ δ O(1) .
The proof is complete.

Step 3. We return to the proof of Lemma 2.3.10. We write v for the vector ⊗ j ( (v j ) 2 ) ⊗q j in V 0 . Then Rlv is exactly the image of η in PW . Using the Fubini theorem and (2.3.12), we have

M ≤ dµ * n (h d+1 ) dµ * (d-1)n (h d )µ * n |F 1 ( v)| F 1 | W ≤ e -n δ -O(1) F 1 | W -1 + O 2 (δ c ). Using SNC on dimension d -1, for all ρ ∈ Γ d-1 , we have µ * (d-1)n {(h d )||A n d-1 ρ(h d , η)| ≤ δ} = O 2 (δ c ). (This is a stronger form of SNC on dimension d -1. Due to Γ d-1 ∈ O(d -1)
h d satisifes (2.3.18) dµ * (d-1)n (h d )µ * n |F 1 ( v)| F 1 | W ≤ e -n δ -O(1) F 1 | W -1 + O 2 (δ c ). (2.3.22)
Due to Lemma 2.3.16, when 2 is small enough with respect to , we have (δ = e -2 n and

F 1 | W δ -O(1) ) e -n δ -O(1) F 1 | W -1 ≤ e -n δ -O(1) ≤ e -n/2 .
Using Lemma 2.3.11 with V = W , due to v in W we conclude that under the condition of Lemma 2.3.16,

µ * n F 1 ( v) F 1 | W ≤ e -n δ -O(1) F 1 | W -1 ≤ e -c

Combinatoric tool

Proposition 2.3.17. Fix κ 1 > 0. Let C 0 > 0. Then there exist 3 and k ∈ N, > 0 depending only on κ 1 such that the following holds for τ large enough depending on

C 0 . Let λ 1 , . . . λ k be Borel measures on ([-τ 4 , -τ -4 ] ∪ [τ -4 , τ 4 ]) m ⊂ R m where 4 = min{ 3 , 3 κ 0 }/10k
, with total mass less than 1. Assume that for all ρ ∈ [τ -2 , τ -3 ] and j = 1, . . . , k

sup a∈R,v∈S m-1 (π v ) * λ j (B R (a, ρ)) = sup a,v λ j {x| v, x ∈ B R (a, ρ)} ≤ C 0 ρ κ 1 . (2.3.24)
Then for all ς ∈ R m , ς ∈ [τ 3/4 , τ 5/4 ] we have

exp(i ς, x 1 • • • x k )dλ 1 (x 1 ) • • • dλ k (x k ) ≤ τ -3 .
This is Proposition 3.4.4 in Chapter 3, based on a discretized sum-product estimate by He-de Saxcé. When n = 1, this is due to Bourgain in [START_REF] Bourgain | The discretized sum-product and projection theorems[END_REF]. The assumption (2.3.24) is called the projective non concentration in the introduction (Definition 2.1).

Application to our measure

From Proposition 2.3.4, we fix 2 < 1 10 min α∈Π {ασ µ } and we can find c 1 such that PNC holds. Let ( 2 /2, c ) be the constants in Lemma 2.2.53. Take

κ 0 = 1 10 min{c 1 , c }.
Using Proposition 2.3.17 with κ 1 = κ 0 , we get 3 , 4 . For g, h in G and η in P, recall that Y n (h, η) = (e -α(σ(gh,η)-κ(g)-nσµ) ) α∈Π ∈ R m . Let λ g,η be a pushforward measure on R m of µ * n restricted on a subset G n,g,η of G, which is defined by

λ g,η (E) = µ * n {h ∈ G n,g,η |Y n (h, η) ∈ E}, for any Borel subset E of R m , where G n,g,η = {h ∈ G|h is (n, , η, ζ m g ) good} (2.3.25)
and where µ ≥ > 0 will be determined later.

PNC is only at one scale, we need to verify all the scales needed in the sum-product estimate. The idea is to separate the random variable and try to use PNC in other scale, where we need the cocycle property to change scale.

Proposition 2.3.18 (Change scale). With small enough depending on 4 2 , there exists C 0 independent of n such that the measure λ g,η satisfies the conditions in Proposition 2.3.17 with constant τ = e 2 n for all n ∈ N.

Proof. We abbreviate λ g,η to λ. By taking small depending on 4 2 , Lemma 2.2.53 implies that the support of λ is contained in the cube [τ -4 , τ 4 ] m .

Then we verify (2.3.24

). Let ρ ∈ [τ -2 , τ -3 ]. Let n 1 = [ | log ρ| 2 2 ]. and n 2 = n -n 1 . Then n 1 lies in [ 3 n/2, n]. We separate h = h 1 h 2 such that h 1 , h 2 have distributions µ * n 1 , µ * (n-n 1 )
, respectively. We have

Y n (h, η) = Y n 1 (h 1 , h 2 η)Y n 2 0 (h 2 , η), (2.3.26)
We can not use the cocycle property directly to change the scale. The problem is in (2.3.26), where the term Y n 2 0 behaves bad if n 2 n 1 , that is to say that the probability of h 2 such that Y n 2 0 (h 2 , η) is smaller than ρ = e -2 2 n 1 is large. In order to overcome this difficulty, we use the support of Y n . We will prove that if Y n 2 0 is too small, then the support of Y n will force Y n 1 to become large, which can be controlled by the large deviation principle. Now we give the details of the proof. For (2.3.24), due to the fact that the support of λ is contained in [τ -4 , τ 4 ] m , we have

(π w ) * λ(B(a, ρ)) ≤ sup h 2 ,v µ * n 1 {h 1 | v, Y n 1 (h 1 , h 2 η) ∈ R -1 B(a, ρ), Y n (h 1 h 2 , η) ∈ [τ -4 , τ 4 ] m }, (2.3.27) where R = wY n 2 0 (h 2 , η) depends on h 2 . • If R ≥ ρ 1/2 , then ρR -1 ≤ ρ 1/2 = e -2 n 1 . It follows by PNC at scale n 1 that µ * n 1 {h 1 | v, Y n 1 (h 1 , h 2 η) ∈ B(a, e -2 n 1 )} 2 e -c 1 2 n 1 ≤ ρ κ 0 .
(2.3.28)

• If R ≤ ρ 1/2 . There exists one coordinate α such that |Y n 2 0 (h 2 , η) α | ≤ ρ 1/2 , which implies that Y n 1 (h 1 , h 2 η) α = Y n (h, η) α /Y n 2 0 (h 2 , η) α ≥ τ -4 ρ -1/2 . Due to 3 ≥ 4 4 and n 1 ≥ 3 n/2, we have 2 n 1 ≥ 2 4 2 n. Therefore τ -4 ρ -1/2 = τ -4 e 2 n 1 ≥ e 2 n 1 /2 . For such h 2 , we have µ * n 1 {h 1 |Y n 1 (h 1 h 2 , η) ∈ [τ -4 , τ 4 ] m } ≤ α∈Π µ * n 1 {h 1 |Y n 1 (h 1 , h 2 η) α ≥ e 2 n 1 /2 }. (2.3.29) It follows from Lemma 2.2.53 that µ * n 1 {h 1 |Y n 1 (h 1 , h 2 η) α ≥ e 2 n 1 /2 } ≤ µ * n 1 {h 1 | σ(gh 1 , h 2 η) -κ(g) -n 1 σ µ ≥ 2 n 1 /2} 2 e -c 2 n 1 ≤ ρ κ 0 .
(2.3.30)

By (2.3.27)-(2.3.30), for ρ ∈ [τ -2 , τ -3 ] we have (π w ) * λ(B(a, ρ)) 2 ρ κ 0 .
The proof is complete.

Proof of the main theorems

In this section, we will use the results of Section 2.2 and Section 2.3 to give the proofs of the main theorems. We will add many assumptions on the elements of G and P. The assumptions seem complicate. In fact, they are not really important. They are taken to make the result work outside a set of exponentially small measure. These assumptions says that the elements are away from certain closed subvarieties of G or P, which also explains that they are true almost everywhere.

(C, r) good function

For a C 1 function ϕ on the flag variety P. We first lift it to P 0 = G/A e N . Let ∂ α ϕ = ∂ Yα ϕ be the directional derivative on P 0 . By Lemma 2.2.31 the action of the group M only changes the sign of the directional derivative ∂ α ϕ, hence |∂ α ϕ| is actually a function on P. Recall that for η, η in P and simple root α, we have defined

d α (η, η ) = d(V α,η , V α,η ).
Definition 2.4.1. Let r be a continuous function on P. Let J be the open set in P, which is the 1/C-neighbourhood of the support of r. Let ϕ be a C 2 function on P. For a simple root α, let v α = sup η∈suppr |∂ α ϕ(η)|. We say that ϕ is (C, r) good if:

(G1) For η, η in J such that d(η, η ) ≤ 1/C, |ϕ(η) -ϕ(η )| ≤ C α∈Π d α (η, η )v α ,
(2.4.1) (G2) For every simple root α and for every η in the support of r, we have

|∂ α ϕ(η)| ≥ 1 C v α , (2.4.2) (G3) For z, z in π -1 (J) ⊂ P 0 , |∂ α ϕ(z) -∂ α ϕ(z )| ≤ Cd 0 (z, z )v α . (2.4.3) (G4) sup α∈Π v α ∈ [1/C, C].
(2.4.4)

In the above definition, only G3 assumption (2.4.3) really need P 0 , where the Lipschitz norm is defined with respect to the distance d 0 on P 0 . G1 assumption is new in higher dimension which means that we can bound the difference by its difference in each fundamental representation, and in a fundamental representation the directional derivative |∂ α ϕ| can bound the Lipschitz norm. G2 and G3 assumptions are natural generalizations of the case m = 1, SL 2 (R). G4 assumption is used to normalize the function.

The role of J is to simplify the verification of (C, r) goodness. With this definition, we only need to verify assumptions on a neighbourhood of the support of r.

From sum-product estimates to Fourier decay

In this subsection we will prove Theorem 2.1.7, an estimate of Fourier decay, by using the results established in Section 2.2 and Section 2.3.

Recall that we have fixed ( 2 , c 1 ) for Proposition 2.3.4 in Section 2.3.5, the constant ( 2 /2, c ) in Lemma 2.2.52 and

κ 0 = 1 10 min{c 1 , c }.
Take k, 3 , 4 from Proposition 2.3.17 with this κ 0 . Let be a positive number to be determined later (the only constant which is not fixed yet). The constant 0 in the hypothesis of Theorem 2.1.7 is defined as

0 = max α∈Π {(2k + 1)ασ µ + 2 } + (2.4.5)
which will be fixed once is fixed.

Here, we define and give relations of different constants. Let v be the vector in R m whose components are v α = sup η∈suppr |∂ α ϕ(η)|, for α ∈ Π. Then by G4 assumption (2.4.4), we have sup

α∈Π v α ∈ [ξ -0 , ξ 0 ]. (2.4.6)
Let n be the minimal integer such that

e 2 n ≥ ξ max α∈Π {v α e -(2k+1)ασµn }. (2.4.7)
The existence is guaranteed by the positivity of Lyapunov constant, that is ασ µ > 0 for α ∈ Π (Lemma 2.2.45). Let the regularity scale δ be given by

δ = e -n < 1/2,
where we take ξ large enough such that n is large enough. Let the contraction scale β given by

β α = e -ασµn , β = max α∈Π {β α }.
The point is that the contraction speed β decides the magnitude of a term and δ is only an error term, much larger than β.

Let the frequency τ be defined by τ = e 2 n . By (2.4.7), we have

τ ≥ ξ max α∈Π {v α β 2k+1 α } ≥ C 2 τ, (2.4.8)
where C 2 = e -2 min α∈Π {e -(2k+1)ασµ }. By (2.4.6), there exists α o in Π such that v αo ≥ ξ -0 . Then (2.4.8) and (2.4.5) imply that

ξ ≤ τ v -1 αo β -2k-1 αo ≤ ξ 0 τ β -(2k+1) α 0 ≤ ξ 0 e n 1-0 0 .
Hence the regularity scale satisfies

ξ 0 ≤ e n = δ -1 .
(2.4.9)

Notation: We state some notation which will be used throughout Section 2.4.2.

• Let g = (g 0 , . . . , g k ) be an element in G ×(k+1) .

• Let h = (h 1 , . . . , h k ) be an element in G ×k .

• We write g ↔ h = g 0 h 1 • • • h k g k ∈ G
for the product of g, h.

• We write

T g ↔ h = g 0 h 1 • • • g k-1 h k ∈ G.
• For l ∈ N, let µ l,n be the product measure on G ×l given by µ

l,n = µ * n ⊗ • • • ⊗ µ * n l times .
• Recall that for g, h in G and η in P, we define

Y n g (h, η) α = exp(-α(σ(gh, η) - κ(g) -nσ µ )) and Y n g (h, η) = (Y n g (h, η) α ) α∈Π ∈ R m . • For z in P 0 , let Ỹ n g (h, z) α = α (m( -1 g , hz))Y n g (h, η) α ,
where α is the corresponding algebraic character of the simple root α and we make a choice of g and η = π(z).

• For g in G, z in P 0 and η = π(z), let λg,z be the pushforward measure on R m of µ * n restricted to a subset G n,g,η under the map Ỹ n g (•, z). In other words, for a Borel set E,

λg,z (E) = µ * n {h ∈ G n,g,η | Ỹ n g (h, z) ∈ E}. Recall that the set G n,g,η is defined by G n,g,η = {h ∈ G|h is (n, , η, ζ m g ) good}.
• After fixing g, we will also fix a choice of k g j , g j for g j and let z g j = k g j z o , m j (h) = m( -1 g j-1 , hk g j ) and λ j = λg j-1 ,zg j , for j = 1, . . . , k.

Lemma 2.4.2. The measure λg,z satisfies the same property (2.3.24) as λ g,η with C 0 replaced by 2 m C 0 , where η = π(z).

Proof. Since the difference is only in the sign, we have

(π v ) * λg,z (B R (a, ρ)) ≤ f ∈(Z/2Z) m (π f v ) * λ g,η (B R (a, ρ)),
where we identify (Z/2Z) m with {-1, 1} m ⊂ R m . The result follows from this inequality.

First step: For η, η in P, let f (η, η ) = G e iξ(ϕ(gη)-ϕ(gη )) r(gη)r(gη )dµ * (2k+1)n (g).

(2.4.10)

Lemma 2.4.3. We have

P e iξϕ(η) r(η)dν(η) 2 ≤ P 2 f (η, η )dν(η)dν(η ).
(2.4.11)

Proof. By the definition of µ-stationary measure and the Cauchy-Schwarz inequality,

P e iξϕ(η) r(η)dν(η) 2 = P×G e iξϕ(gη) r(gη)dµ * (2k+1)n (g)dν(η) 2 ≤ G P e iξϕ(gη) r(gη)dν(η) 2 dµ * (2k+1)n (g) = P 2 G
e iξ(ϕ(gη)-ϕ(gη )) r(gη)r(gη )dµ * (2k+1)n (g)dν(η)dν(η ).

The proof is complete.

Recall that for η in P, we write V α,η for its image in PV α and d α (η, η ) = d(V α,η , V α,η ).

Definition 2.4.4 (Good Position). Let η, η be in P. We say that they are in good position if ∀α ∈ Π, d α (η, η ) ≥ δ.

We fix η, η in good position, which means that η, η are far in all PV α . We rewrite the formula.

Lemma 2.4.5. We have

P e iξϕ(η) r(η)dν(η) 2 ≤ η,η good f (η, η )dν(η)dν(η ) + O(δ c ).
(2.4.12)

Proof. By the regularity of stationary measure (2.2.34), we have

ν{η ∈ P|d α (η, η ) ≤ δ} = ν{η ∈ P|d(V α,η , V α,η ) ≤ δ} ≤ Cδ c . (2.4.13)
Therefore by (2.4.13) and Fubini's theorem,

ν ⊗ ν{(η, η ) ∈ P 2 | d α (η , η) < δ} = η∈P ν{η ∈ P|d α (η, η ) ≤ δ}dν(η) δ c .
Summing over simple roots α, we obtain the result by r ∞ ≤ 1.

Second step: The purpose of this part is to give a Ping-Pong Lemma in measure sense. We will eliminate sets with negligible measure such that the Ping-Pong condition is almost preserved by iteration on the complement.

We fix g j for j = 0, . . . , k -1 which satisfies κ(g j ) -nσ µ ≤ n/C A .

(2.4.14)

Recall that C A is a constant in Definition 2.2.51. We also demand that

h j+1 is (n, , η M g j+1 , ζ m g j ) good.
(2.4.15)

Recall that the Cartan subspace a is equipped with the norm induced by the Killing form, and with this norm a is isomorphic to the euclidean space R m . Lemma 2.4.6. Suppose that g, h satisfy the above conditions (2.4.14) and (2.4.15). Then the action of

T g ↔ h on b M Vα,g k (δ) is β 2k α δ -O(1)
Lipschitz and

e -ασ(g 0 h 1 ,x M g 1 ) • • • e -ασ(g k-1 h k ,x M g k ) ≤ β 2k α δ -O(1) , (2.4.16) for every α in Π. For t ∈ b M g k (δ), let t j = g j h j+1 • • • h k t for j = 0, . . . , k, where we let t k = t. Then t j ∈ b M g j (βδ -2 ) ⊂ b M g j (δ), (2.4.17) σ(g j h j+1 , t j+1 ) -σ(g j h j+1 , η M g j+1 ) βδ -O(1) .
(2.4.18)

Remark 2.4.7. The contraction constant β here is a little different from the gap γ(g j ), but γ(g j )/β is in the interval [δ O(1) , δ -O(1) ] by Lemma 2.2.53. Hence they are of the same largeness and we will not distinguish them.

The intuition here is that by controlling κ(g), η M g , ζ m g , all the other position or length will also be controlled, which is similar to hyperbolic dynamics.

Proof. For every α in Π, using Lemma 2.2.8 2k times, we obtain the Lipschitz property. By Lemma 2.2.53, we have (2.4.16) from (2.4.15) for all α in Π at the same time.

We use induction to prove the inclusion. For j = k, it is due to the hypothesis of Lemma 2.4.6.

Suppose that the property holds for j + 1. By definition, t j = g j h j+1 t j+1 . We abbreviate g j , h j+1 , t j+1 , η M g j+1 to g, h, η, η . The condition becomes

d(η, η ) ≤ δ, κ(g) -nσ µ ≤ n/C A and h is (n, , η , ζ m g ) good.
By Lemma 2.2.53, we have γ(h) ≤ βδ -1 . By Lemma 2.2.13, due to

η ∈ B(η , δ) ⊂ B m h (δ), we have hη ∈ b M h (β/δ 2 ) ⊂ B m g (δ). Therefore ghη ∈ b M g (β/δ 2 )
, which is the inclusion condition.

Then we will prove (2.4.18) and we keep the notation g, h,

By the same argument, due to Lemma 2.2.13 and η, η

∈ B(η , β/δ 2 ) ⊂ B m h (δ), we have hη, hη ∈ b M h (β/δ 2 ) ⊂ B m g (δ)
. Therefore by the Lipschitz property of Lemma 2.2.13

σ(gh, η) -σ(gh, η ) (d(η, η ) + d(hη, hη ))δ β/δ 3 .
The proof is complete.

Lemma 2.4.8. Suppose that g, h satisfy the conditions (2.4.14) and (2.4.15). Let s be in {z ∈ P 0 |d 0 (z, z g k ) ≤ δ}. Let s j = g j h j+1 • • • h k s for j = 0, . . . , k, where we let s k = s.

We have

m(s 0 , k g 0 ) = Π 1≤j≤k m( -1 g j-1 , h j k g j ) = Π 1≤j≤k m j (h j ). (2.4.19)
Proof. We let η = π(s), then η is in b M g k (δ). By (2.4.17) with j = 1 and (2.4.15) with j = 0, Lemma 2.2.25 implies

m(s 0 , k g 0 ) = m(k g 0 , g 0 h 1 s 1 ) = m( -1 g 0 , h 1 k g 1 )m(s 1 , k g 1 ).
Iterating this formula, we obtain the result.

Third step: Here we mimic the proof of [START_REF] Bourgain | Fourier dimension and spectral gaps for hyperbolic surfaces[END_REF], where they heavily use the properties of Schottky groups and symbolic dynamics. But in our case, the group is much more complicate from the point of view of dynamics. We use the large deviation principle to get a similar formula.

By very careful control of g l , with a loss of an exponentially small measure, we are able to rewrite the formula in a form to use the sum-product estimates. The key point is that by controlling the Cartan projection and the position of η M g and ζ m g of each g l , we are able to get good control of their product g ↔ h.

We should notice that the element g j will be fixed, and we will integrate first with respect to h j . This gives the independence of the cocycle σ(g j-1 h j , η M g j ), that is for different j they are independent, which is an important point to apply sum-product estimates.

We return to (2.4.12). We call g "good" with respect to η, η if

g satisfies (2.4.14), g k satisfies conditions in Lemma 2.2.40, η M g 0 ∈ suppr and δ(η, ζ m g k ), δ(η , ζ m g k ), δ(V α,η ∧ V α,η , y m ∧ 2 ραg k ) ≥ 4δ.
(2.4.20)

Lemma 2.4.9. If η and η are in good position and g is "good", then g k η, g k η are in b M g k (δ), and for α ∈ Π the d α distance between g k η and g k η is almost β α , that is

d α (g k η, g k η ) ∈ β α [δ O(1) , δ -O(1) ].
Proof. The inclusion is due to Lemma 2.2.13. Since g is good (2.4.20), by (2.2.8) we have the lower bound and by the Lipschitz property in Lemma 2.2.8 we have the upper bound.

For η, η in P, we can rewrite the formula of f (η, η ) as

f (η, η ) = e iξ(ϕ(g↔hη)-ϕ(g↔hη )) r(g ↔ hη)r(g ↔ hη )dµ k,n (h)dµ k+1,n (g). (2.4.21)
We call h is g-regular if h satisfies (2.4.15). Let

f g (η, η ) = g-regular
e iξ(ϕ(g↔hη)-ϕ(g↔hη )) dµ k,n (h).

Lemma 2.4.10. For η, η in P

|f (η, η )| ≤ g"good |f g (η, η )|dµ k+1,n (g) + O (δ c ), (2.4.22)
if is small enough with respect to γ, that is ≤ min α∈Π {ασ µ γ/(2 + 2γ)}.

Proof. Let fg (η, η ) = g-regular e iξ(ϕ(g↔hη)-ϕ(g↔hη )) r(g ↔ hη)r(g ↔ hη )dµ k,n (h).
We call g "semi-good" if g satisfies (2.4.20) except the assumption of η M g 0 ∈ suppr in (2.4.20). By large deviation principle (Proposition 2.2.46, Proposition 2.2.48, Lemma 2.2.58), we conclude that µ k+1,n {g not "semi-good" } ≤ O (δ c ).

(2.4.23)

Then by (2.4.21), Lemma 2.2.52 and (2.4.23),

|f (η, η )| ≤ g | fg (η, η )|dµ k+1,n (g) + O (δ c ) ≤ g"semi-good | fg (η, η )|dµ k+1,n (g) + O (δ c ).
(2.4.24) By Lemma 2.4.9, (2.4.17) with j = 0 and c γ (r)

≤ ξ 0 ≤ δ -1 , |r(η M g 0 ) 2 -r(g ↔ hη)r(g ↔ hη )| ≤ 2 r ∞ c γ (r)(βδ -2 ) γ ≤ 2β γ δ -1-2γ ≤ 2δ,
if is small enough with respect to γ. Hence

| fg (η, η )| ≤ g-regular e iξ(ϕ(g↔hη)-ϕ(g↔hη )) r(η M g 0 ) 2 dµ k,n (h) + O(δ c ) ≤ r(η M g 0 ) 2 |f g (η, η )| + O(δ c ).
(2.4.25)

If r(η M g 0 ) = 0, then that g is "semi-good" implies g is "good". Combined with (2.4.24) and (2.4.25), by r ∞ ≤ 1, we have

|f (η, η )| ≤ g"semi-good r(η M g 0 ) 2 |f g (η, η )| + O(δ c ) dµ k+1,n (g) + O (δ c ) ≤ g"good |f g (η, η )|dµ k+1,n (g) + O (δ c ).
The proof is complete.

Recall that β is the magnitude which is really small, δ is only an error term and τ is the frequency for applying the sum-product estimate, which lies between δ -1 and β -1 . Proposition 2.4.11. Let I τ = [τ 3/4 , τ 5/4 ]. The following formula is true for η, η in good position and g "good",

|f g (η, η )| ≤ sup ς ∈Iτ e i ς,x 1 •••x k dλ 1 (x 1 ) • • • λ k (x k ) + O(βδ -O(1) τ ),
(2.4.26)

when is small enough with respect to 2 .

Remark 2.4.12. This is the most complicate step, where the difficulty comes from higher rank. We need to use the technique of changing flags to find the direction of slowest contraction speed, where we can use Newton-Leibniz's formula. Since the action of the sign group M is non trivial on the slowest directions, we also carefully treat the sign.

Proof. The element η, η and g are already fixed. Since g k satisfies the conditions in Lemma 2.2.40, we obtain two chains (η = η o , η 1 , . . . , η l 1 ) and (η = η o , η 1 , . . . , η l 2 ) as in Lemma 2.2.40. Then we write

ϕ(g ↔ hη) -ϕ(g ↔ hη ) = 0≤j≤l 1 -1 (ϕ(g ↔ hη j ) -ϕ(g ↔ hη j+1 )) - 0≤j≤l 2 -1 (ϕ(g ↔ hη j ) -ϕ(g ↔ hη j+1 )) + ϕ(g ↔ hη l 1 ) -ϕ(g ↔ hη l 2 ) , (2.4.27)
The terms for different j and for η, η are similar. We fix j and we simplify α(η j , η j+1 ) to α. We compute the term ϕ(g ↔ hη j ) -ϕ(g ↔ hη j+1 ). In order to treat the sign, we will work on P 0 = G/A e N . Recall that π : P 0 → P is the projection and we use z = kz o to denote the element kA e N in P 0 .

By Lemma 2.2.42 and (2.4.20), we know that g k η j , g k η j+1 are in b M g k (δ), which satisfy the condition of Lemma 2.4.6. Let z 0 , z 1 be preimages of g k η j and g k η j+1 in P 0 such that m(z 0 , z 1 ) = e. Notice that z 0 , z 1 are in the same α-circle. By Lemma 2.2.40 (2.2.25) and Lemma 2.4.9

d(g k η j , g k η j+1 ) = d α (g k η, g k η ) + O(βe -ακ(g k ) δ O(1) ) ∈ β α [δ O(1) , δ -O(1) ].
Due to m(z 0 , z 1 ) = e, the arc length distance also satisfies

d A (z 0 , z 1 ) = arcsin d(g k η j , g k η j+1 ) ∈ β α [δ O(1) , δ -O(1) ].
(2.4.28)

Now, we lift ϕ to P 0 , becoming a right M -invariant function. By abuse of notation, we also use ϕ to denote the lifted function. Let γ be an arc connecting z 0 , z 1 with unit speed in the α-circle with length less than π/2. Without loss of generality, we suppose that γ is in the positive direction (If not, we add minus in the right hand side of (2.4.29)). By Newton-Leibniz's formula (2.2.23), we have

ϕ(T g ↔ hz 0 ) -ϕ(T g ↔ hz 1 ) = u 0 ∂ α ϕ(T g ↔ hγ(t))e -ασ(T g↔h,γ(t)) dt, (2.4.29)
where

u = d A (z 0 , z 1 ). Fix a time t in [0, u], let s j = g j h j+1 • • • h k γ(t). Then π(γ(t)) is in b M g k (δ), because g k η j and g k η j+1 are in b M g k (δ)
and by (2.4.28). By (2.4.17), the element π(s 0 ), the image of s 0 = T g ↔ hγ(t) in P, is in b M g 0 (βδ -O(1) ). Recall that we have made a choice of the Cartan decomposition of every g j for 0 ≤ j ≤ k. In particular, k g 0 is given in the decomposition of g 0 = k g 0 a g 0 g 0 ∈ KA + K. Let m 0 = m(s 0 , k g 0 ) and s 0 = s 0 m 0 , then m(s 0 , k g 0 ) = e. By Lemma 2.2.31,

∂ α ϕ s 0 = ∂ α ϕ s 0 m 0 = α (m 0 )∂ α ϕ s 0 .
(2.4.30) By Lemma 2.5.5 and πs 0 , πz g 0 = η M g 0 in b M g 0 (βδ -O(1) ), we have

d 0 (s 0 , z g 0 ) ≤ d(πs 0 , πz g 0 ) < βδ -O(1) . (2.4.31)
Due to g good (2.4.20), we have η M g 0 ∈ suppr. By G2 assumption (2.4.2), we have

|∂ α ϕ(z g 0 )| ≥ δv α . By (2.4.31), the point πs 0 is in J, the δ neighbourhood of suppr. By G3 assumption (2.4.3), |∂ α ϕ(s 0 ) -∂ α ϕ(z g 0 )| ≤ δ -1 v α d 0 (s 0 , z g 0 ), which implies ∂ α ϕ(s 0 )/∂ α ϕ(z g 0 ) ∈ [1 -βδ -O(1) , 1 + βδ -O(1) ].
By Lemma 2.4.6 (2.4.18), we have

(1-βδ -O(1) )e -O(β/δ) ≤ ∂ α ϕ(s 0 )e -ασ(g 0 h 1 ,s 1 ) • • • e -ασ(g k-1 h k ,s k ) ∂ α ϕ(z g 0 )e -ασ(g 0 h 1 ,x M g 1 ) • • • e -ασ(g k-1 h k ,x M g k ) ≤ (1+βδ -O(1) )e O(β/δ) . (2.4.32) By (2.4.16), B α := e -ασ(g 0 h 1 ,x M g 1 ) • • • e -ασ(g k-1 h k ,x M g k ) ≤ β 2k α δ -O(1)
. Together with (2.4.28)-(2.4.32)

|ϕ(g ↔ hη j ) -ϕ(g ↔ hη j+1 ) -d A (z 0 , z 1 )α (m 0 )∂ α ϕ(z g 0 )B α | ≤ ββ 2k+1 α δ -O(1) v α . (2.4.33)
We deal with the error term which comes from the process of changing flags. The Lipschitz property in Lemma 2.4.6 and Lemma 2.2.40 (2.2.26) imply that

d α (g ↔ hη l 1 , g ↔ hη l 2 ) ≤ β 2k α δ -O(1) d α (g k η l 1 , g k η l 2 ) ≤ β 2k+1 α βδ -O(1) ,
Due to (2.4.17) in Lemma 2.4.6 and Lemma 2.2.42, the two points g ↔ hη l 1 , g ↔ hη l 2 are in J, the δ neighbourhood of suppr. Due to G1 assumption (2.4.1)

|ϕ(g ↔ hη l 1 ) -ϕ(g ↔ hη l 2 )| ≤ δ -1 α v α d α (g ↔ hη l 1 , g ↔ hη l 2 ). Therefore |ϕ(g ↔ hη l 1 ) -ϕ(g ↔ hη l 2 )| ≤ δ -O(1) β α v α β 2k+1 α .
(2.4.34)

We collect information for different simple roots. Recall that for a fixed g in G and for h ∈ G, z ∈ P 0 , we have defined Ỹ n g (h, z) α = e -α(σ(gh,z)-κ(g)-nσµ) α(m( g , hk)). Let

ς α := ξd A (z 0 , z 1 )α (m 0 )∂ α ϕ(z g 0 )B α Π k l=1 Ỹ n g l-1 (h l , z g l ) α . Let ς = (ς α ) α∈Π ∈ R m .
|ξ(ϕ(g ↔ hx) -ϕ(g ↔ hx )) -ς, Π k l=1 Ỹ n g l-1 (h l , z g l ) | ≤ βδ -O(1) α β 2k+1 α v α ξ βδ -O(1) τ.
(2.4.35) We want to verify that ς ∈ I τ . By (2.4.19), we have

ς α = ξd A (z 0 , z 1 )∂ α ϕ(z g 0 )β k α e -ακ(g 0 )-•••-ακ(g k-1 ) .
By (2.4.14), (2.4.28), (2.4.20) and (2.4.2) we have

|ς α | ∈ ξv α β 2k+1 α [δ O(1) , δ -O(1)
]. Therefore by (2.4.8),

ς ∈ sup α ξv α β 2k+1 α [δ O(1) , δ -O(1) ] ∈ τ [δ O(1) , δ -O(1) ] ⊂ [τ 3/4 , τ 5/4 ] = I τ .
By definition, the distribution of Ỹ n g l-1 (h l , z g l ), where h l satisfies (2.4.15) with distribution µ * n , is the measure λ l . Finally, due to |e ix -e iy | ≤ |x -y| for x, y ∈ R, the inequality (2.4.35) implies (2.4.26).

Fourth step:

We are able to apply sum-product estimates.

Proof of Theorem 2.1.7. For l = 1, 2, . . . k, Proposition 2.3.18 and Lemma 2.4.2 tell us that with small enough depending on 4 , there exists C 0 such that the measures λ l satisfy the assumptions in Proposition 2.3.17 with τ .

Proposition 2.3.17 implies that for τ large enough,

exp(i ς, x 1 • • • x k )dλ 1 (x 1 ) . . . dλ k (x k ) ≤ τ -3 .
Then by (2.4.12), (2.4.22) and (2.4.26), we have

e iξϕ(η) r(η)dν(η) 2 ≤ O (δ c ) + O(βδ -O(1) τ ) + τ -3 .
Due to βδ -O(1) τ = max α∈Π e (-ασµ+O(1) + 2 )n , take small enough. The proof is complete.

Remark 2.4.13. Another difference with [START_REF] Bourgain | Fourier dimension and spectral gaps for hyperbolic surfaces[END_REF] is that we avoid using the renewal idea, which simplifies the proof of this part. The renewal idea is that instead of using µ * n , we use a renewal measure µ t , which is defined to be the distribution of g 1 • • • g n for the first time that its Cartan projection exceeds t, where g 1 , g 2 . . . are i.i.d. random variables with distribution µ. This is because we generalize the sum-product estimate to a form that the measure can have a support which depends on the frequency, and we use the large deviation principle to prove that our measure has a support not too large with respect to the frequency.

Examples of Fourier decay

In this section, we give a nice application of Theorem 2.1.7, that is Theorem 2.1.1. This application also serves as a "baby case" for Section 2.4.4.

Recall that v 0 is a unit vector in V and ς is a vector in v ⊥ 0 . We fix the direction, that is u 0 := ς/ ς , and we let ξ = ς . Then for x = Rv, we have ς, ψ(v) = ξ u 0 , ψ(v) = ξ u 0 , v / v 0 , v , and we take

ϕ(x) = u 0 , v v 0 , v .
Since we are only interested in the value on the support of ν V , which is contained in the image of P in PV . The functions ϕ, r can be lifted to functions on P. We use the same notation ϕ to denote the lifted functions. We first calculate the directional derivative of ϕ. Recall that the inner product on the exterior square ∧ 2 V is given by

v 1 ∧ v 2 , w 1 ∧ w 2 = v 1 , w 1 v 2 , w 2 -v 1 , w 2 v 2 , w 1 , (2.4.36) for v 1 , v 2 , w 1 , w 2 in V .
Recall that q 2χ-α is the projection of ∧ 2 V on the subrepresentations of highest weight 2χ -α. By the same proof as in Lemma 2.2.3, we see that the multiplicity of an irreducible representation of highest weight 2χ -α is at most 1 in ∧ 2 V . Hence the image of the projection q 2χ-α is an irreducible subrepresentation or zero. Let e 1 be a unit vector of highest weight in V .

Lemma 2.4.14. Let v 0 , u 0 be two unit vectors in V . Let ϕ be defined as above. Then for a simple root α and z = kz 0 ∈ P 0 ,

∂ α ϕ(z) = v 0 ∧ u 0 , v ∧ u v 0 , v 2 = q 2χ-α (v 0 ∧ u 0 ), v ∧ u v 0 , v 2 , ( 2.4.37)
where v = ke 1 and u = kY α e 1 .

Proof. By definition,

∂ α ϕ(z) = ∂ t u 0 , k exp(tY α )e 1 v 0 , k exp(tY α )e 1 t=0 = u 0 , kY α e 1 v 0 , ke 1 -u 0 , ke 1 v 0 , kY α e 1 v 0 , ke 1 2 .
By (2.4.36), we have the first equality. The vector e 1 ∧ Y α e 1 is a vector of weight 2χ -α, which is in the irreducible subrepresentation of ∧ 2 V with highest weight 2χ -α. The vector v ∧ u = k(e 1 ∧ Y α e 1 ) is also in this subrepresentation, hence

v 0 ∧ u 0 , v ∧ u = q 2χ-α (v 0 ∧ u 0 ), v ∧ u .
The proof is complete.

For a vector v in an euclidean space W , let v * be the linear linear functional on W given by v * (w) = v, w for w ∈ W.

Proof of Theorem 2.1.1 from Theorem 2.1.7. Let δ > 0 be a constant to be fixed later. Recall that ϕ, r have been lifted to functions on P. In order to use Theorem 2.1.7, we need to verify the (ξ 0 , r) goodness assumption for ϕ. Let C 0 > 0 be a constant such that

c γ (r) ≤ C 0 , | v 0 , v | ≥ v /C 0 for Rv ∈ V χ,η and η ∈ suppr, (2.4.38) max α∈Π q 2χ-α (v ∧ u) v ∧ u ≥ 1/C 0 for every couple v, u in V with v ∧ u = 0.
(2.4.39)

The existence of C 0 for (2.4.39) is due to Lemma 2.2.56. We want to verify that ϕ is (ξ 0 , r) good. Let l α = q 2χ-α (v 0 ∧ u 0 ) and θ α = l α . The main problem is to verify G2, because ∂ α ϕ may vanish. We need a cutoff. Let τ be a smooth function on

R such that τ | [0,∞) = 1, τ takes values in [0, 1], suppτ ⊂ [-1, ∞) and |τ | ≤ 2. Set τ δ (x) = τ (x/δ) for x ∈ R. Let r 1 = r • Π α∈Π τ α , where τ α = τ δ (δ(V 2χ-α,η , Rl * α ) -2δ).
If l α = 0, then we let τ α = 1. Let J be the ξ -0 neighbourhood of the support of r 1 , an open set of P. When ξ is large enough, we can suppose that for η ∈ J and v ∈ V χ,η we have

| v 0 , v | > v /(2C 0 ).
We claim that if δ = ξ -0 /2 and ξ is large enough such that δ -1 ≥ CC m+4 0 , where C is a constant only depending on the group G and the norm on V , which is defined in Lemma 2.5.7. Then ϕ, r 1 satisfy the assumptions of Theorem 2.1.7.

For η in the support of r 1 , due to (2.4.37) and δ(V 2χ-α,η , Rl * α ) > δ, we have

|∂ α ϕ(η)| = | l α , v ∧ u | v 0 , v 2 = θ α δ(V 2χ-α,η , Rl * α ) v 0 , v -2 ≥ δθ α . (2.4.40) Due to | v 0 , v | ≥ v /C 0 for η ∈ suppr and v ∈ V χ,η , v α = sup η∈suppr 1 |∂ α ϕ(η)| = sup η∈suppr 1 | l α , v ∧ u | v 0 , v 2 ≤ C 2 0 θ α .
(2.4.41)

Then for η in suppr 1 , by (2.4.40) and (2.4.41) we have

|∂ α ϕ(η)| ≥ C -2
0 δv α which implies G2 assumption (2.4.2). The inequality (2.4.40) also implies that

v α ≥ δθ α , (2.4.42)
that is v α and θ α are of the same magnitude. Hence by (2.4.39), we have

sup α∈Π v α ∈ [δ, C 2 0 ] sup α∈Π θ α ⊂ [δC -1 0 , C 2 0 ],
which is G4 assumption (2.4.4). Now, we verify G1 assumption (2.4.1). If χ is a fundamental weight χ α , then (2.4.39)

implies θ α = q 2χ-α (v 0 ∧ u 0 ) ≥ 1/C 0 . Hence, for η, η in J and unit vectors v ∈ V χ,η , v ∈ V χ,η , we have |ϕ(η) -ϕ(η )| = u 0 ∧ v 0 , v ∧ v v 0 , v v 0 , v ≤ 4C 2 0 u 0 ∧ v 0 v ∧ v ≤ 4C 3 0 θ α d(V α,η , V α,η ) ≤ 4δ -1 C 3 0 v α d(V α,η , V α,η ).
(2.4.43)

For general case, this step is more complicate. Please see Lemma 2.5.7. For G3 assumption (2.4.3), for z = kz 0 , z = k z 0 in π -1 (J)

⊂ P 0 and v = k e 1 , u = k Y α e 1 |∂ α ϕ(z) -∂ α ϕ(z )| ≤ C 4 0 (| l α , v ∧ u -v ∧ u | + | l α , v ∧ u v 0 , v -v |) C 4 0 θ α d 0 (z, z )
where the last inequality is due to Lemma 2.5.3.

We also need to calculate c γ (r 1 ). Lemma 2.2.14 implies c γ (τ α )

δ -γ . Hence c γ (r 1 ) δ -γ + c γ (r) ≤ δ -γ + C 0 .
The claim is true and Theorem 2.1.7 implies that

e iξϕ(η) r 1 (η)dν(η) ≤ ξ -1 .
Finally, by regularity of stationary measure, Corollary 2.2.50, the set where

r 1 = r has measure bounded by O(δ c ) = O(ξ -0 c/2 ), that is there exist C, c > 0 such that for all δ > 0 ν{η ∈ P| δ(V 2χ-α,η , Rl * α ) ≤ 2δ} ≤ Cδ c . The proof is complete.
Remark 2.4.15. In higher dimension, the differential dϕ at a point always vanishes in some direction of the tangent space. The cutoff in the proof can be understood as removing a neighbourhood of the zero locus of dϕ in the unit tangent bundle of PV . The language of flag variety makes the proof obscure, but this language is really powerful.

From Fourier decay to spectral gap

Derivative of the cocycle

This part is devoted to the derivative of the cocycle. The results of this part imply that for most g, h in G, the difference of the Iwasawa cocycle σ(g, •) -σ(h, •) satisfies the (C, r) good condition in Definition 2.4.1 (See Lemma 2.4.25). Since the α-bundle is trivial on P 0 , we will work on P 0 . We need to lift the Iwasawa cocycle σ to P 0 and we use the same notation σ.

Let V be an irreducible representation of G with a good norm. Recall that σ

V (g, x) = ρ(g)v v
for g in G and v in V . We will abbreviate ρg to g in the proof, because (ρ, V ) is the only representation to be studied in this part. Let α be a simple root. Let e 1 be a unit vector of highest weight in V and let e 2 = Y α e 1 .

Lemma 2.4.16. Let V be an irreducible representation of G with a good norm. For z = kz o in P 0 , we have

∂ α σ V (g, z) = ρgv, ρgu ρgv 2 , where v = ke 1 and u = ke 2 .
Proof. Without loss of generality, we suppose that z = z o . Since Y α is a left K invariant vector field on P 0 , we have

∂ Yα σ V (g, e) = ∂ t σ V (g, exp(tY α )z o )| t=0 = ∂ t log g exp(tY α )e 1 exp(tY α )e 1 t=0 = ge 1 , gY α e 1 ge 1 2 - e 1 , Y α e 1 e 1 2 .
Since the norm is good, eigenvectors of different weights are orthogonal, we have e 1 , Y α e 1 = 0. The result follows.

Form this lemma, we know that the derivative of the cocycle σ V in the direction Y α is nonzero only if χ -α is a weight of V . Lemma 2.2.32 only implies that ∂ α σ α = 0 for α = α , which works for fundamental representations σ V α = σ α . We fix the distance d 0 on P 0 , which is defined in Appendix 2.5.2.

Lemma 2.4.17. Let δ < 1/2. Let B m V,g (δ) be the preimage of B m V,g (δ) ⊂ PV in P 0 . For z = kz o ∈ B m V,g (δ), |∂ α σ V (g, z)| ≤ δ -O(1) .
(2.4.44)

We also have

Lip P 0 (∂ α σ V (g, •)| B m V,g (δ)
) ≤ δ -O(1) .

(2.4.45)

Proof. By Lemma 2.4.16, the hypothesis that Rke 1 ∈ B m V,g (δ) and (2.2.7)

|∂ α σ V (g, z)| = gke 1 , gke 2 gke 1 2 ≤ Y α g 2 e 1 2 g 2 δ 2 e 1 2 .
Since the operator norm of Y α is bounded, we have

|∂ α σ V (g, z)| ≤ δ -O(1) .
The estimate of Lipschitz norm is more complicate.

Let v = ke 1 , v = k e 1 , u = ke 2 , u = k e 2 . We have |∂ α σ V (g, z) -∂ α σ V (g, z )| = | gv, gu gv 2 -gv , gu gv 2 | gv 2 gv 2 .
By the same argument, due to v = ke 1 ∈ B m V,g (δ), we use (2.2.7) to give a lower bound of the denominator, that is

gv 2 gv 2 ≥ δ 4 g 4 v 2 v 2 = δ 4 g 4 e 1 4 .
Use the difference to give a upper bound of the numerator, that is

| gv, gu gv 2 -gv , gu gv 2 | g 3 e 1 3 ( gv -gv + gu -gu ) g 4 v 3 ( v -v + u -u ).
Therefore we have

|∂ α σ V (g, z) -∂ α σ V (g, z )| δ -O(1) ( ke 1 -k e 1 + ke 2 -k e 2 ).
Then by Lemma 2.5.3, the proof is complete.

Let V be a finite dimensional vector space with euclidean norm. Recall that ∧ 2 Sym 2 V is the exterior square of the symmetric square of V . It is a linear space generated by vectors of the form v 1 v 2 ∧ v 3 v 4 where v i are in V , for i = 1, 2, 3, 4. For g, h in GL(V ), let F g,h be the linear functional on ∧ 2 Sym 2 V , whose action on the vector v 1 v 2 ∧ w 1 w 2 is defined by

F g,h (v 1 v 2 ∧ w 1 w 2 ) = hv 1 , hv 2 gw 1 , gw 2 -gv 1 , gv 2 hw 1 , hw 2 .
This formula is well defined because v 1 , v 2 and w 1 , w 2 are symmetric, respectively. We also have

F g,h (v 1 v 2 ∧ w 1 w 2 ) = -F g,h (w 1 w 2 ∧ v 1 v 2 ). Since the vectors of form v 1 v 2 ∧ w 1 w 2 generate the space ∧ 2 Sym 2 V , the linear form F g,h is uniquely defined.
Suppose that V is a super proximal representation of G with highest weight χ (Definition 2.2.2). Let α be the unique simple root such that χ -α is a weight of V . The space ∧ 2 Sym 2 V may be reducible. The two highest weights of Sym 2 V are 2χ, 2χ -α, whose eigenspaces have dimension 1. Hence, the highest weight of ∧ 2 Sym 2 V is 4χ -α, and the eigenspace has dimension 1. Let W be the irreducible subrepresentation of ∧ 2 Sym 2 V with the highest weight χ 1 := 4χ -α. In the following lemma, we abbreviate ρ(g), ρ(h) to g, h.

Lemma 2.4.18. Let δ < 1/2. Let V be a super proximal representation of G and let α be the unique simple root such that χ -α is a weight of

V . Recall that V χ 1 ,η is the image of η ∈ P in PW . If g, h in G and z = kz o ∈ P 0 , η = π(z) satisfy (1) -1 h V χ , -1 h V χ-α ∈ B m V,g (δ), γ 1,2 (g) ≤ δ 3 , (2) δ(V χ 1 ,η , F g,h | W ) > δ and V χ,η ∈ B m V,g (δ) ∩ B m V,h (δ), then |∂ α (σ V (g, z) -σ V (h, z))| ≥ δ O(1) .
Remark 2.4.19. This is similar to the non local integrability property as defined in [START_REF] Dolgopyat | On decay of correlations in Anosov flows[END_REF] [Nau05] and [START_REF] Stoyanov | Spectra of Ruelle transfer operators for Axiom A flows[END_REF]. Although the above two conditions are complicate, we will see later that in the measure sense, most pairs g, h satisfy these conditions.

The key idea here is to use other representation to linearise polynomial functions on V . As long as the function is linear, we will have a good control of it. Another point is that the image of P stays in the same irreducible subrepresentation.

Proof of Lemma 2.4.18. By Lemma 2.4.16, let

L := ∂ α (σ V (g, z) -σ V (h, z)) = F g,h (v 2 ∧ vu) gv 2 hv 2 , (2.4.46)
where v = ke 1 and u = kY α e 1 as in Lemma 2.4.16.

Lemma 2.4.20. If g, h satisfy assumption (1), then the operator norm satisfy

F g,h | W ≥ δ O(1) g 2 h 2 .
Proof. Using the Cartan decomposition and good norm, we can suppose that h is diagonal and h = diag(a 1 , a 2 , • • • , a n ) with a 1 ≥ a 2 ≥ • • • ≥ a n . By Definition 2.2.2, we know that he 1 = a 1 e 1 and he 2 = a 2 e 2 . The assumption (1) becomes δ(Re 1 , y m g ), δ(Re 2 , y m g ) > δ, γ 1,2 (g) ≤ δ 3 .

(2.4.47)

In (2.4.46), let z = z o , then v = e 1 , u = e 2 , which make hv, hu = a 1 e 1 , a 2 e 2 = 0.

Therefore, due to

v 1 , v 2 ≥ v 1 v 2 -v 1 ∧ v 2 , for v 1 , v 2 in V , we have F g,h (e 2 1 ∧ e 1 e 2 ) = a 2 1 ge 1 , ge 2 ≥ a 2 1 ( ge 1 ge 2 -ge 1 ∧ ge 2 ).
Then (2.2.7) and (2.4.47) imply

F g,h (e 2 1 ∧ e 1 e 2 ) ≥ h 2 g 2 (δ 2 -γ 1,2 (g)).
The proof is complete.

By Definition 2.2.2, the representation ∧ 2 Sym 2 V is a proximal representation. Due to R(v 2 ∧ vu) = Rk(e 2 1 ∧ e 1 e 2 ) = kV χ 1 , the line R(v 2 ∧ vu) is contained in the K-orbit of the subspace of highest weight V χ 1 . Since V χ 1 is in W , we see that v 2 ∧ vu is also in W . By (2.4.46),

L = F g,h (v 2 ∧ vu) F g,h | W g 2 h 2 gv 2 hv 2 F g,h | W g 2 h 2 .
When η satisfies assumption (2), the result follows by applying (2.2.7) to gv 2 , hv 2 and by Lemma 2.4.20.

Proof of the spectral gap

Here we will prove the theorem of uniform spectral gap. The first part is classic, where we use some ideas of Dolgopyat [START_REF] Dolgopyat | On decay of correlations in Anosov flows[END_REF] to transform the problem to an effective estimate Proposition 2.4.24, see also [START_REF] Naud | Expanding maps on Cantor sets and analytic continuation of zeta functions[END_REF] and [START_REF] Stoyanov | Spectra of Ruelle transfer operators for Axiom A flows[END_REF]. The key observation is that this effective estimate (Proposition 2.4.24) can be obtained by the Fourier decay, regarding the difference of cocycle as a function on P. The intuition here is from Lemma 2.4.18. When g, h are in general position and η not too close to ζ m g , ζ m h , the difference ϕ(η) = σ(g, η) -σ(h, η) will be (C, r) good (Definition 2.4.1). But in order to accomplish this, we need some sophisticate cutoff, which makes the proof complicate.

Recall that the Iwasawa cocycle takes values in the Cartan subspace a. We can write θ in a * as a linear combination of fundamental weights, {χ α |α ∈ Π}, that is

θ = α∈Π θ α χ α . Set |θ| = max α∈Π |θ α |.
We want to treat the spectral gap on the flag variety P and the projective space PV at the same time, where V is an irreducible representation of G with good norm. Let X be P or PV . Let σ : G × X → E be the cocycle, which is given by the Iwasawa cocycle and E = a when X = P, and given by σ V (defined in (2.2.6)) and E = R when X = PV . Let E C = E ⊗ R C and E * C be the dual space of E C . For z ∈ E * C , write z = + iθ, where θ, are elements in E * . Recall that the transfer operator P z is defined as: For | | small enough and for f in C 0 (X), x in X P z f (x) = G e zσ(g,x) f (gx)dµ(g).

Recall that for

f in C γ (X) let c γ (f ) = sup x =x |f (x)-f (x )| d(x,x ) γ and |f | γ = |f | ∞ + c γ (f ).
Remark 2.4.21. Here we should be careful that the distances on PV and P are defined in (2.2.4) and (2.2.12). They are not the Riemannian distances defined in the introduction. But on a compact Riemannian manifold, different Riemannian distances are equivalent.

In particular, every Riemannian distance on P is equivalent to the K-invariant Riemannian distance on P. By Corollary 2.5.6, we know it is equivalent to the distances defined (2.2.12). The case of the projective space PV is similar. Hence, the norm | • | γ induced by different distances are equivalent.

We state our main result of this section Proposition 2.4.22. Let µ be a Zariski dense Borel probability measure on G with a finite exponential moment. For γ > 0 small enough, there exist ρ < 1, C > 0 such that for all θ and in E * with |θ| large enough, | | small enough and f in C γ (X), n in N we have

|P n +iθ f | γ ≤ C|θ| 2γ ρ n |f | γ .
Theorem 2.1.5 and Theorem 2.1.6 follow directly from Proposition 2.4.22. The assumption on µ will be needed throughout this section.

We start with standard a priori estimates. When z = 0, we will write P for P 0 .

Proposition 2.4.23. For every γ > 0 small enough, there exist C > 0 and 0 < ρ < 1 such that for all f in C γ (X), | | small enough and n ∈ N

|P n z f | ∞ ≤ C | |n |f | ∞ , (2.4.48) |P n f | ∞ ≤ X f dν + Cρ n |f | γ , (2.4.49) c γ (P n z f ) ≤ C(C | |n |θ| γ |f | ∞ + ρ n c γ (f )).
(2.4.50)

The inequality (2.4.48) is a consequence of exponential moment and the Hölder inequality. For (2.4.49), please see [BL85, V, Thm.2.5] and [BQ16, Prop 11.10, Lem.13.5] for more details. This inequality (2.4.49) is a consequence of the fact that the action of G on X is contracting. The third inequality (2.4.50) is called the Lasota-Yorke inequality. The proof is classic and we include a proof in the appendix for completeness.

We reduce Proposition 2.4.22 to Proposition 2.4.24. The reduction is standard, using Proposition 2.4.23. Please see [START_REF] Dolgopyat | On decay of correlations in Anosov flows[END_REF] and [START_REF] Naud | Spectral gaps and abelian covers of convex co-compact surfaces[END_REF] for more details. We also include a proof in the appendix for completeness. For f in C γ (X), we define another norm |f | γ,θ = |f | ∞ + c γ (f )/|θ| γ for θ = 0. Proposition 2.4.24. For every γ > 0 small enough, for |θ| large enough and | | small enough, there exist 2 , C 2 > 0 such that for f in C γ (X) and |f | γ,θ ≤ 1, we have

P [C 2 ln |θ|] +iθ f 2 dν ≤ e -2 ln |θ| .
(2.4.51) Now we will distinguish two cases. We claim that the case of PV is a corollary of the case of P up to a constant. Recall that the stationary measure on PV is written as ν V . Let f be a function in C γ (PV ) and |f | γ,θ ≤ 1. The estimate only depends on the value of f on the support of the stationary measure ν V . By Lemma 2.2.43, the stationary measure on PV is the pushforward measure of the stationary measure ν on P. Hence we can define the function f on P by

f (η) = f (V χ,η ),
where χ is the highest weight of V . Then by σ V (g, V χ,η ) = χσ(g, η) (see (2.2.2)),

P [C 2 ln |θ|] +iθ f 2 dν V = P [C 2 ln |θ|] ( +iθ)χ f 2 dν.
We will verify that f satisfies | f | γ,θ 1. By (2.2.16), for two distinct points η, η in P we have

| f (η) -f (η )| d(η, η ) γ = | f (η) -f (η )| d(V χ,η , V χ,η ) γ d(V χ,η , V χ,η ) γ d(η, η ) γ |f (V χ,η ) -f (V χ,η )| d(V χ,η , V χ,η ) γ = |f | γ .
Hence with some change of constant, we can deduce the case of PV from the case of P.

We only need to prove Proposition 2.4.24 for the case of P.

From Fourier decay to Proposition 2.4.24. We need to reduce (2.4.51) to Fourier decay (Theorem 2.1.7). Let n = [C 2 log |θ|] and δ = e -n (2.4.52)

(with C 2 ≥ max α∈Π {1/ασ µ } + 1 and > 0 to be determined later), and let G n, ,α be the subset of G × G, defined as the set of couples which satisfy Lemma 2.4.18 (1) with

V = V α . Let G n, = {g ∈ G| κ(g) -nσ µ ≤ n } 2 α∈Π G n, ,α ⊂ G × G. For |f | γ,θ ≤ 1, let A g,h := X e zσ(g,η)+zσ(h,η) f (gη) f (hη)dν(η).

Then

|P n z f | 2 dν = e zσ(g,η)+zσ(h,η) f (gη) f (hη)dν(η)dµ * n (g)dµ * n (h) = Gn,

A g,h dµ * n (g)dµ * n (h) + G c n,
A g,h dµ * n (g)dµ * n (h).

(2.4.53)

We first compute the term with (g, h) outside of G n, , where the behaviour is singular. By the Cauchy-Schwarz inequality,

G c n, A g,h dµ * n (g)dµ * n (h) 2 ≤ µ(G c n, ) |A g,h | 2 dµ * n (g)dµ * n (h). ( 2
A g,h |θ| -1 /2 + |θ| -c /C 2 .
Combined with (2.4.53) and (2.4.57), the proof is complete by setting 2 = min{ 1 2 , c 4C 2 }.

It remains to prove Lemma 2.4.25.

Proof of Lemma 2.4.25. We first verify that ϕ is (|θ| 0 , r) good. Since will be taken small enough, we can suppose |θ| -0 ≤ δ/4. Let J be the |θ| -0 neighbourhood of suppr.

Then for η ∈ J, we have δ α (η, ζ m g ) ≥ δ/2 for α in Π. The function ϕ is a sum of functions. Each function is the lift of a function on PV α for some simple root α. We write ϕ = α∈Π ϕ α where ϕ α (η) = |θ| -1 θ α (σ α (g, η) -σ α (h, η)). By Lemma 2.2.32, that is ∂ α ϕ α = 0 for α = α, in order to verify (|θ| 0 , r) good condition, it is enough to verify G1-G3 assumptions (2.4.1)-(2.4.3) for ϕ α and the G4 assumption (2.4.4) for ϕ. Since G1-G3 are linear, we can forget the coefficients |θ| -1 θ α in ϕ α . Now, we verify G1-G3 assumptions and we fix a simple root α and consider ϕ = ϕ α = σ α (g, •) -σ α (h, •). Recall that v α = sup η∈suppr |∂ α ϕ(η)|. Since J satisfies the hypothesis of Lemma 2.4.17 with V = V α , we have

v α , Lip P 0 (∂ α ϕ| π -1 J ) < δ -O(1) .
(2.4.65) Since (g, h) ∈ G n, satisfies Lemma 2.4.18(1) and the support of r satisfies Lemma 2.4.18(2), for η in the support of r, by Lemma 2.4.18,

|∂ α ϕ(η)| > δ O(1) ≥ δ O(1) v α
which is G2 assumption (2.4.2). This also implies

v α > δ O(1) , (2.4.66)
G4 assumption (2.4.4). By (2.4.65), we have G3 assumption (2.4.3). Let ϕ 1 be a function on PV α such that ϕ 1 (V α,η ) = ϕ(η). Since J satisfies hypothesis of Lemma 2.2.8, this Lemma implies

|ϕ(η) -ϕ(η )| d α (η, η ) = |ϕ 1 (V α,η ) -ϕ 1 (V α,η )| d(V α,η , V α,η ) ≤ |Lip PVα ϕ 1 | < δ -O(1) ≤ δ -O(1) v α ,
which is G1 assumption (2.4.1).

For general ϕ, it remains to verify G4 assumption (2.4.4). There exists a simple root α such that |θ α | = |θ|. Since ϕ α satisfies G4 assumption and |∂ α ϕ| = |∂ α ϕ α | by Lemma 2.2.32, the function ϕ also satisfies G4 assumption.

Finally, we verify the term c γ (r) and |r| ∞ .

Lemma 2.4.26. For 0 < γ ≤ 1, let f, τ be two γ-Hölder functions on a compact metric space X. Then

c γ (τ f ) ≤ c γ (τ ) f | suppτ ∞ + |τ | ∞ c γ (f | suppτ ).
The proof of Lemma 2.4.26 is elementary. Recall that r(η) = f (gη) f (hη)e (σ(g,η)+σ(h,η)) α∈Π τ α .

For the infinity norm, due to (g, h) ∈ G n, , we have

|r| ≤ e | |( κ(g) + κ(h ) ≤ e | |(2 σµ + )n ≤ |θ| | |C 2 (2 σµ + ) . Take | | small enough, then |r| ∞ ≤ |θ| 1 /2 .
For the term c γ (r), we only need to verify that each term in the formula of r has a bounded c γ value. Due to Lemma 2.4.26, we only need to verify the c γ value on X g,h .

• Since the action of g on X g,h is contracting, by Lemma 2.2.13, we have

c γ (f (g•)| X g,h ) ≤ c γ (f )(Lip g| X g,h ) γ ≤ (|θ|βδ -2 ) γ .
Due to (2.4.52), we have log β = -n min α∈Π ασ µ < -n/C 2 ≤ -log |θ|. Therefore c γ (f (g•)| X g,h ) ≤ δ -O(1) .

• Due to |e a -e b | ≤ max{e a , e b }|a -b| γ for all a, b in R and 0 ≤ γ ≤ 1, by Lemma 2.2.13,

c γ (e σ(g,•) | X g,h ) ≤ e | | κ(g) (Lip σ(g, •)| X g,h ) γ ≤ e | |( σµ + )n+ γn | | γ .
Hence when | | is small enough depending on σ µ , we obtain c γ (e σ(g,•) | X g,h ) ≤ δ -O(1) .

• In c γ (τ α ), the only term we need to be careful about is τ δ (4δ(V 4χα-α,η , F ραg,ραh ) -4δ). By Lemma 2.2.14, we have d(V 4χα-α,η , V 4χα-α,η ) d(η, η ). Hence the c γ value of this term is also bounded by δ -O(1) .

The proof is complete.

Exponential error term

In this section, we will prove that the speed of convergence in the renewal theorem is exponential using our result on the spectral gap. (Proposition 2.4.22) Recall X = PV and we have defined a renewal operator R as follows: For a positive bounded Borel function f on X × R, a point x in X and a real number t, we set

Rf (x, t) = +∞ n=0 G f (gx, σ(g, x) -t)dµ * n (g).
Recall P z is the transfer operator defined by P z f (x) = G e zσ V (g,x) f (gx)dµ(g). Using the analytical Fredholm theorem, we summarize the property of P z .

Proposition 2.4.27. With the same assumption as in Theorem 2.1.4, for any γ > 0 small enough, there exists η > 0 such that when | z| < η, the transfer operator P z is a bounded operator on C γ (X) and depends analytically on z. Moreover there exists an analytic operator U (z) on a neighbourhood of | z| < η such that the following holds for | z| < η

(I -P z ) -1 = 1 σ V,µ z N 0 + U (z),
where N 0 is the operator defined by N 0 f = X f dν V . There exists C > 0 such that for

| z| ≤ η U (z) C γ →C γ ≤ C(1 + | z|) 2γ . (2.4.67)
This is generalization of Proposition 1.4.1 in Chapter 1 and [Boy16, Theorem 4.1], and the proof is exactly the same. The main difference is that the spectral radius of P z is bounded below 1 in a strip of imaginary line (except at 0), due to Proposition 2.4.22. From this we have the analytic continuation of U (z) to the strip and the bound of the operator norm of U (z). Now, we give the precise statement and the proof of Theorem 2.1.4.

Proposition 2.4.28. With the same assumption as in Theorem 2.1.4, there exists > 0 such that for f ∈ C ∞ c (R), we have

Rf (x, t) = 1 σ V,µ ∞ -t f (u)du + e -|t| O(e |suppf | (|f | L 1 + |f | L 1 )).
Proof. By the same computation as in Proposition 1.4.5 in Chapter 1 and [Boy16, Prop. 4.14], we have

Rf (x, t) = 1 σ V,µ ∞ -t f (u)du + lim s→0 + 1 2π e -itξ f (ξ)U (s + iξ)1(x)dξ,
where f is the Fourier transform of f given by ξ = e iξu f (u)du. Hence, we only need to control the error term.

This was defined in [BQ16, Definition 11.1] and was verified for the action on the flag variety in [START_REF] Benoist | Random Walks on Reductive Groups[END_REF]Lemma 13.5]. For the projective space PV , the same proof also works.

For the γ norm, let x, y in X and g in G e zσ(g,x) f (gx) -e zσ(g,y) f (gy) = (e zσ(g,x) -e zσ(g,y) )f (gx) + e zσ(g,y) (f (gx) -f (gy)) One term is controlled by (2.4.48), the other term is due to (µ, γ) contraction (2.5.1). Therefore when a small enough, there exists (we take the same constant C 1 ). Therefore

Let A n = | G e zσ(
ρ 1 < 1 such that A n ≤ C 1 ρ n 1 c γ (f ), where C 1 > 0.
c γ (P n z f ) ≤ C 1 ρ n 1 c γ (f ) + |b| γ C 1+(|a|+γ)n 1 |f | ∞ (2.5.2)
We want the term C (|a|+γ)n 1

does not depend on γ. Fix n large enough such that C 1 ρ n 1 = ρ 2 < 1. For natural number N , iterate (2.5.2) N times and use (2.4.48). We have

c γ (P nN z f ) ≤ ρ 2 c γ (P n(N -1) z f ) + |b| γ C 1+(|a|+γ)n 1 |P n(N -1) z f | ∞ ≤ ρ 2 c γ (P n(N -1) z f ) + |b| γ C 1+(|a|+γ)n 1 |f | ∞ C |a|n(N -1) 1 ≤ c γ (f )ρ N 2 + |b| γ C 1+(|a|+γ)n 1 |f | ∞ C |a|nN 1 1 -ρ 2 C -|a|n 1 ≤ c γ (f )ρ N 2 + O n (|b| γ C |a|nN 1 )|f | ∞ (2.5.3)
Given m ∈ N, we can write m = nN + r with r ∈ [0, n -1]. Therefore by (2.5.3) (2.5.2)

c γ (P m z f ) = c γ (P nN +r f ) ≤ ρ N 2 c γ (P r z f ) + O n (|b| γ C |a|nN )|P r z f | ∞ ≤ ρ N 2 (C 1 ρ r 1 c γ (f ) + |b| γ C (1+(a+γ)r) 1 |f | ∞ ) + O n (|b| γ C |a|m 1 )|f | ∞ .
By setting ρ = ρ

|P (m+1)N z f | 2 ∞ ≤ C |a|mN |P mN |P N z f | 2 | ∞ ≤ C |a|mN ( |P N z f | 2 dν + ρ mN |P N z f | 2 C γ ) ≤ C |a|mN e -2 N/C 1 + ρ mN (C 1+|a|N (1 + |b| γ ) + Cρ N |b| γ ) 2
(2.5.4)

So we can choose m large such that ρ mN |b| 2γ = ρ mC ln |b| |b| 2γ < 1. This m is only depend on γ, C and ρ. By continuity of a we obtain the equality for infinity norm. That is when m is large enough and a is small enough depending on m we have |P

(m+1)N z f | 2 ∞ |b| -3 , where 3 > 0
For γ norm, we use (2.4.50) for (P N z , P

N z f ) and (P

(m+1)N z , f ) c γ (P (m+2)N z f )/|b| γ ≤ C |a|N |P (m+1)N z f | ∞ + ρ N c γ (P (m+1)N z f )/|b| γ ≤ C |a|N |P (m+1)N z f | ∞ + ρ N (C 1+|a|mN |b| γ + ρ mN |b| γ )/|b| γ
Then, when |b| is large enough and a is small enough, we have

|P (m+2)N z f | γ,b ≤ |b| -4
(2.5.5) (where we should use (2.5.4) with m replaced by m + 1).

Let N 1 = (m + 2)N = (m + 2)C 1 ln |b|. Given n, we can write n = dN 1 + r with 0 ≤ r < N 1 . By (2.5.5), (2.4.48), (2.4.50)

|P n z f | γ,b ≤ |b| -4 d |P r z f | γ,b ≤ |b| -4 d C 1+|a|r ≤ C|b| 4 ρ n , where ρ = |b| -4 /N 1 C |a| = e - 4 (m+2)C 1 C |a| .
The result follows by taking |a| small enough.

Equivalence of distances

Definition 2.5.2. Let (X, d) be a metric space. Let d be another metric on X. We say that d, d are equivalent metrics if there exist c, C > 0 such that for all

x 1 , x 2 in X cd(x 1 , x 2 ) ≤ d (x 1 , x 2 ) ≤ Cd(x 1 , x 2 ).
Recall that P 0 is the homogeneous space G/A e N , on which the compact group K acts simply transitively. We will define three distances on P 0 . Due to the fact that P 0 is homeomorphic to K, a distance on P 0 is also a distance on K and we will continue our argument on K. Let k, k be two points in K.

• d 0 (k, k ) = sup α∈Π kv α -k v α / √ 2
, where v α is a unit vector in V α with highest weight. This is also the distance induced by the embedding of P 0 to Π α∈Π SV α .

• d 1 (k, k ) = k-k , where • is a K invariant norm on the space of (m+1)×(m+1) square matrices M m+1 (R) ⊃ K.

• d 2 (k, k ) is the distance induced by the bi-invariant Riemannian metric on K.

We can easily verify that they are distances.

Lemma 2.5.3. The three distances d 0 , d 1 and d 2 on P 0 are equivalent.

Proof. First we observe that the three distances are left K invariant. It is sufficient to prove the equivalence for k equal to the identity e. Fix small depending on K. Let B be the neighbourhood of e given by {k ∈ K|d 1 (k, e) < }. Then B c is a compact subset of K. Consider the function f i,j (k) = (k,e) for k ∈ B c and i, j ∈ {0, 1, 2}. Then f i,j is a positive continuous function B c . The compactness of B c implies that it has positive minimum on B c . Hence there exists c i,j > 0 such that for k outside of B d i (k, e) ≥ c i,j d j (k, e).

d i (k,e) d j
Finally, we only need to consider a small neighbourhood of the identity. We take small such that the exponential map at e is bi-Lipschitz. Suppose that k = exp(tZ) with Z a unit vector in k and t > 0. Then

d 1 (k, e) = e -exp(tZ) t = d 2 (k, e).
Due to d 0 (k, e) = max α∈Π kv α -v α / √ 2 k -e = d 1 (k, e), it remains to prove that d 0 is not small. We can decompose Z as

Z = α∈R + c α K α .
There exists α o ∈ R + such that c αo 1. Since v α is a vector of highest weight, for a positive root

β let A α,β := dρ α (K β )v α = dρ α (Y β )v α .
Consider the representation of s β = {Y β , X β , H β } sl 2 . Due to the classification of the representations of sl 2 , Lemma 2.5.4. The vector A α,β is non zero if and only if χ α (H β ) > 0.

only need to verify that if d(η, η ) ≤ ξ -0 and η, η satisfies that

| v 0 , v | ≥ v /C 0 for v in V χ,η and V χ,η , then |ϕ(η) -ϕ(η )| ≤ ξ 0 /2 α∈Π θ α d(V α,η , V α,η ),
for ξ large enough. Replacing (2.4.43) by the following lemma, we conclude that G1 assumption is always verified if ξ is large enough.

Lemma 2.5.7. Let C 0 , C 1 > 0 and let η, η be two points in

P such that d(η, η ) ≤ 1/(C 1 C 0 ) and | v 0 , v | ≥ v /C 0 for v in V χ,η and V χ,η .
Then with C 1 large enough depending on the norm, we have

|ϕ(η) -ϕ(η )| ≤ CC m 0 α∈Π θ α d(V α,η , V α,η ),
where C only depends on the group G and the norm on V .

Proof. The main idea is to take derivative on P, and prove that in every direction the result is true. We will first prove the directions given by positive roots. The structure of Sym 2 (∧ 2 V ) gives us a formula, that is for

v 1 , v 2 , w 1 , w 2 , w 3 in V v 1 ∧ v 2 , w 1 ∧ w 2 = v 1 ∧ v 2 , w 3 ∧ v 1 , w 2 w 1 -v 1 , w 1 w 2 v 1 , w 3 . (2.5.6)
In order to simply the notation, we write

Y 1 , • • • , Y m for Y α 1 , • • • , Y αm .
The structure of Lie algebra gives us that for a vector

v in V v ∧ Y 1 • • • Y k v = Y 1 • • • Y k-1 (v ∧ Y k v) - I Y I v ∧ Y I c v,
(2.5.7)

where

I = {j 1 , • • • , j l } is a nonempty subset of {1, • • • , k -1}, I c is the complement of I in {1, • • • , k} and Y I = Y j 1 • • • Y j l with j 1 < • • • < j l .
Let e 1 be the unit vector in V with highest weight. We claim that if | v 0 , e 1 | ≥ 1/C 0 , then for J ⊂ {1, • • • , m}, we have

| v 0 ∧ u 0 , e 1 ∧ Y J e 1 | ≤ CC |J| 0 i∈J θ α i .
(2.5.8)

We make an induction on k = |J|. By symmetry, it is sufficient to prove the claim for

Y J = Y 1 • • • Y k . For k = 1, due to e 1 ∧ Y 1 e 1 ∈ q 2χ-α 1 (∧ 2 V ), we have | v 0 ∧ u 0 , e 1 ∧ Y 1 e 1 | ≤ q 2χ-α 1 (v 0 ∧ u 0 ) = θ α 1 .
Suppose that (2.5.8) holds for all the integer less than k -1. Then by (2.5.7),

v 0 ∧ u 0 , e 1 ∧ Y 1 • • • Y k e 1 = v 0 ∧ u 0 , Y 1 • • • Y k-1 (e 1 ∧ Y k e 1 ) - I v 0 ∧ u 0 , Y I e 1 ∧ Y I c e 1 Due to Y 1 • • • Y k-1 (e 1 ∧ Y k e 1 ) ∈ q 2χ-α k (∧ 2 V )
, the first term is controlled by θ α k . The other term, due to I = ∅, using (2.5.6) with w 3 = e 1 , we have

| v 0 ∧ u 0 , Y I e 1 ∧ Y I c e 1 | = | v 0 ∧ u 0 , e 1 ∧ v 0 , Y I c e 1 Y I e 1 -e 1 , Y I e 1 Y I c e 1 v 0 , e 1 | ≤ C 0 (| v 0 ∧ u 0 , e 1 ∧ Y I e 1 + v 0 ∧ u 0 , e 1 ∧ Y I c e 1 |)
Since the length of I and I c are less than k, by the hypothesis of induction, we have the claim for k.

The choice of Y β for a positive root β is fixed in Section 2.2 and we have

Y α 1 +•••+α k = C α 1 ,••• ,α k [Y α 1 , [Y α 2 , • • • , [Y α k-1 , Y α k ] • • • ]] with a constant C α 1 ,••• ,α k . By the claim, Lemma 2.5.8. Let β be a positive root. If | v 0 , e 1 | ≥ 1/C 0 , then | v 0 ∧ u 0 , e 1 ∧ Y β e 1 | ≤ CC m 0 α∈Π,α≤β θ α ,
where C only depends on G and the norm on V . In particular, for

Z = β∈R + c β K β | v 0 ∧ u 0 , e 1 ∧ Ze 1 | ≤ CC m 0 α∈Π θ α β≥α,β∈R + |c β |.
This is almost the directional derivative of G1. For η = kη 0 , η = k η 0 in P, we can find a unit vector Z in the Lie algebra k such that k = k exp(tZ) with t d(η, η ). Let γ(s) = k exp(sZ)η 0 for 0 ≤ s ≤ t. Then by the Newton-Leibniz formula,

|ϕ(η) -ϕ(η )| ≤ t 0 |∂ s ϕ(γ(s))|ds.
Let k s = k exp(sZ). By the same computation of Lemma 2.4.14, we have (2.5.9)

∂ s ϕ(γ(s)) = ∂ Z ϕ(γ(s)) = v 0 ∧ u 0 , k s e 1 ∧ k s Ze 1 v 0 , k s e 1 = k -1 s v 0 ∧ k -1 s u 0 , e 1 ∧ Ze 1 k -1 s v 0 , e 1 . Due to k s e 1 -e 1 d(γ(s), γ(0)) ≤ d(η, η ) ≤ 1/(C 0 C 1 ), with C 1 large enough, we have | k -1 s v 0 , e 1 | = | v 0 , k s e 1 | ≥ | v 0 , e 1 | -k s e 1 -e 1 ≥ 1/(2C 0 ) Due to q 2χ-α (k -1 s (v 0 ∧ u 0 )) = q 2χ-α (v 0 ∧ u 0 ) = θ α ,
For Z in k, let Z α = β≥α,β∈R + c β K β . Due to t small, we only need to consider distance is a small neighbourhood. Hence by the fact that vectors of different weights are orthogonal, we have (Y β v α is nonzero for every positive root β ≥ α due to Lemma 2.5.4)

d(V α,η , V α,η ) = d(v α , exp(tZ)v α ) exp(tZ)v α -v α t Zv α = t Z α v α t Z α .
Therefore, combined with (2.5.9)

α∈Π θ α d(V α,η , V α,η ) ≥ α∈Π tθ α Z α ≥ t α∈Π θ α β≥α,β∈R + |c β | ≥ 1 CC m 0 |ϕ(η) -ϕ(η )|.
The proof is complete.

If we continue to exploit the additive structure, that is to say replacing µ k by ν = (µ k ) * r , the r-times additive convolution of µ k , then the Fourier transform of ν can have arbitrary large decay rate.

The Fourier transform detects the additive structure. But our measure µ k has the multiplicative structure. The decay of Fourier transform means that the additive and multiplicative structures are hard to coexist, the sum-product philosophy.

The projective non concentration means the projection of the measure µ on every one dimensional linear subspace Rv satisfies a non concentration assumption (the case of R).

The case n = 1 is due to Bourgain [START_REF] Bourgain | The discretized sum-product and projection theorems[END_REF]Lemma 8.43]. The main ingredient of the proof of Fourier decay is the discretized sum-product estimates in R n . The sum-product estimate roughly says that if the set does not concentrate in small balls, then under addition or multiplication the size of the set will become robustly larger than the initial set.

For δ > 0 and a bounded set A in a metric space E, let N δ (A) be the minimal number of closed balls of radius δ needed to cover A. In a metric space, we say that a set A is ρ away from a set B if A is not contained in the ρ neighborhood of B, that is there exists x in A such that d(x, B) ≥ ρ. In (R * ) n , we note id the identity element (1, • • • , 1) ∈ (R * ) n . In R n , we will consider maximal proper unitary subalgebras, such subalgebras are given by {x ∈ R n |x i = x j } for 1 ≤ i < j ≤ n. We say that A is ρ away from proper unitary subalgebras of R n if A is ρ away from any maximal proper unitary subalgebra of R n . Now we state the discretized sum-product estimates on R n , which is the main ingredient of the proof of Theorem 3.1.1.

Theorem 3.1.3. We will consider the action of (R * ) n on V = R n . The action is given by gv = (g 1 v 1 , . . . , g n v n ) for g in (R * ) n and v in V . There exists a neighborhood U of the identity in (R * ) n such that the following holds. Given κ > 0, σ ∈ (0, n), there exists > 0 such that for all δ > 0 sufficiently small, if A ⊂ U and X ⊂ B V (0, δ -) satisfy the following (δ, κ, σ, ) assumption:

(i)For j = 1, . . . , n ∀ρ ≥ δ, N ρ (π j (A)) ≥ δ ρ -κ , where π j denotes the projection into j-th coordinate, (ii) A is δ away from proper unitary subalgebras of R n , (iii

)For j = 1, . . . , n ∀ρ ≥ δ, N ρ (π j (X)) ≥ δ ρ -κ , (iv)N δ (X) ≤ δ -(n-σ)-. Then N δ (X + X) + sup a∈A N δ (X + aX) ≥ δ -N δ (X).
Remark 3.1.4. The case n = 1 is due to Bourgain. Compared with [BG12, Prop.1], our situation does have invariant subspace under the action. Hence we put more regularity on the projection into coordinate subspaces.

The assumption (2) in Lemma 3.2.1 is the original definition of Bourgain. This assumption roughly says that the measure ν has dimension κ at scale δ to scale δ . The assumption (1) is more convenient to be proved. The smaller the parameter 1 is, the more regularity the measure ν has.

Let A be a bounded subset of R n . Let A s be the set of elements which are obtained by taking sum or multiplication of elements in A at most s times.

Lemma 3.2.2. Let A be a subset of B R n (0, K). If N δ (A + A) + N δ (A + A • A) ≤ KN δ (A), then for any integer s N δ ( A s ) ≤ K Os(1) N δ (A).
(See [START_REF] He | Discretized Sum-Product Estimates in Matrix Algebras[END_REF]Lemma 11] and [Bre11, Lemma 4.5] for more details) This lemma tells us that instead of proving that A + A or A + A • A is large, it is sufficient to prove that A s is substantially large.

Our result on the discretized sum-product estimates relies on a result of He and de Saxcé. They study sum-product phenomenon in finite dimensional linear representations of Lie groups. We will state the version we need, their theorem is much more general. Definition 3.2.3. Recall that we consider the action of (R * ) n on V = R n given by multiplication in each coordinate. Let W be a linear subspace of V such that W is not a submodule, that is there exists g in (R * ) n such that gW W . Then we call Stab (R * ) n (W ) a proper stabilizer.

Let A be a subset of (R * ) n and let X be a subset of R n . For s ≥ 1, we define A, X s to be the set of elements which can be obtained as sums, differences and products of at most s elements of A and X. For example, we have A,

X s = {±g 1,1 • • • g 1,i 1 v 1 ± • • • ± g l,1 • • • g l,i l v l | i 1 , • • • , i l , l ∈ N, i 1 + • • • i l + l ≤ s}
Proposition 3.2.4. [HdS18, Thm.2.3] Recall that we consider the action of (R * ) n on V = R n given by multiplication in each coordinate. There exists a neighborhood U of the identity in (R * ) n such that the following holds. Given 0 , κ > 0, there exist s ≥ 1 and > 0 such that for all δ > 0 sufficiently small, if A ⊂ U and X ⊂ B V (0, 1) satisfy the following (δ, κ, ) assumption:

(i) For j = 1, . . . , n ∀ρ ≥ δ, N ρ (π j (A)) ≥ δ ρ -κ ,
where π j denotes the projection into j-th coordinate, (ii) A is δ away from proper stabilizers, (iii) X is δ away from coordinate subspaces.

Then, B V (0, δ 0 ) ⊂ A, X s + B V (0, δ).

We will use the ring structure of R n . Recall that for a subset A of (R * ) n , which is also a subset of R n , we define A s as A, X s with X = A. As a corollary of Proposition 3.2.4, we have Proposition 3.2.5. There exists a neighborhood U of the identity in (R * ) n such that the following holds. Given κ > 0, 0 > 0, there exist > 0 and s > 0 such that, for δ sufficiently small, if A is a subset of U satisfies the following (δ, κ, ) assumption:

(i)For j = 1, . . . , n ∀ρ ≥ δ, N ρ (π j (A)) ≥ δ ρ -κ , where π j denotes the projection into j-th coordinate, (ii)A is δ away from maximal proper unitary subalgebras.

Then we have

B R n (0, δ 0 ) ⊂ A s + B R n (0, δ).
Proof. Take X = A -A. We can shrink U to ensure that X ⊂ U -U ⊂ B R n (0, 1). Then we claim that A, X satisfies (δ, κ, 2 /κ) assumption of Proposition 3.2.4. Assumption (i) of Proposition 3.2.4 is the same as Assumption (i) of this proposition. For assumption (iii) of Proposition 3.2.4, take ρ = δ 2 /κ . Then

N ρ (π j (X)) ≥ N ρ (π j (A)) ≥ δ ρ -κ = δ -> 1.
Hence, X is δ 2 /κ away from coordinate subspaces. The assumption(iii) in Proposition 3.2.4 is satisfied.

It remains to verify Assumption (ii) of Proposition 3.2.4. We need to change the point of view. The set G = (R * ) n ⊂ R n is seen as subsets of Aut(V ) ⊂ End(V ), the automorphism group and the endomorphism ring of V . The main point is that in the case of R n , proper stabilizers are contained in the subalgebras. In other words, let W be a subspace of V which is not a G-submodule. Then the proper stabilizer satisfies

Stab G (W ) = G ∩ Stab R n (W ) = G ∩ {a ∈ R n |a • W ⊂ W }.
By definition, Stab G W is a proper subgroup of G. The fact that Stab R n (W ) is a unitary subalgebra of R n implies that Stab R n (W ) must be a proper unitary subalgebra of R n . Hence, the assumption (ii) of Proposition 3.2.4 is automatically satisfied.

Applying Proposition 3.2.4 with κ, 0 implies that there exists s 1 such that

B R n (0, δ 0 ) ⊂ A, X s 1 + B R n (0, δ),
when is small enough. The observation that

A, X s 1 = A, A -A s 1 ⊂ A 2s 1 implies the result.
As a byproduct, using Lemma 3.2.2, we have the following version of discretized sum-product estimates in R n . Proposition 3.2.6. There exists a neighborhood U of the identity in (R * ) n such that the following holds. Given κ > 0, σ ∈ (0, n), there exists > 0 such that for all δ > 0 sufficiently small, if A ⊂ U satisfies the following:

(i) For j = 1, . . . , n

∀ρ ≥ δ, N ρ (π j (A)) ≥ δ ρ -κ ,
where π j denotes the projection into j-th coordinate, (ii) A is δ away from proper unitary subalgebras of R n , (iii) N δ (A) ≤ δ -σ-. Then

N δ (A + A) + N δ (A + A • A) ≥ δ -N δ (A).
We deduce Proposition 3.2.6 from Lemma 3.2.2 and Proposition 3.2.5. The proof is exactly the same as the proof of [He, Theorem 2]. We include its proof for completeness.

Proof of Proposition 3.2.6. Suppose that the result fails. For every > 0 there exists A satisfying the hypothesis of Proposition 3.2.6 but

N δ (A + A) + N δ (A + A • A) < δ -N δ (A).
We will reach a contradiction when is small enough depending only on κ, σ and R n .

Then by Lemma 3.2.2 and assumption (ii) of Proposition 3.2.6, for every integer s, we have

N δ ( A s ) ≤ δ -Os( ) N δ (A) ≤ δ -Os( )-σ .
(3.2.1)

On the other hand, A also satisfies the assumptions of Proposition 3.2.5. Given 0 > 0, there exist 1 > 0 and integer s such that if ≤ 1 , then

B R n (0, δ 0 ) ⊂ A s + B R n (0, δ). Therefore N δ ( A s ) ≥ N δ (B R n (0, δ 0 )) = δ n(-1+ 0 ) (3.2.2)
If we take 0 sufficiently small such that n(1 -0 ) > σ, and take sufficiently small such that

n(1 -0 ) > O s ( ) + σ, then (3.2.1) contradicts (3.2.2).
This version is not sufficient to imply the decrease of Fourier transform of multiplicative convolution of measures. We will introduce more tools of additive combinatorics to obtain a stronger form of discretized sum-product estimates.

Remark 3.3.2. The first assumption that the L 2 norm is not small means that the measure is not too smooth. Because if the measure is already smooth, then the convolution can not make the measure more smooth. This assumption should be compared with the assumption (iv) in Theorem 3.1.3, where we need that the covering number of the set is not too large.

By definition and (3.3.1), ν δ 2 2 ≤ ν δ ∞ ν δ 1 ≤ δ κ--n . Hence κ + σ 1 ≤ + n, that is the non concentration assumption gives a upper bound of L 2 norm. Another explication of the L 2 norm is in Lemma 3.3.14.

Remark 3.3.3. The non concentration assumption here is stronger than the non concentration in Theorem 3.1.3. This is because we need to make multiplication in the proof. The projective non concentration assumption is stable under multiplication and addition. But the non concentration assumption in Theorem 3.1.3 is not.

The hypothesis of projective non concentration can be weakened to (i) non concentration on coordinate subspaces and (ii) away from linear subspaces. Please see Remark 3.3.11. But the assumption needed in Theorem 3.1.1 is projective non concentration. Hence we write the same assumption here for simplicity. The step where we really need a projective non concentration is explained in Remark 3.3.18. Remark 3.3.4. When n equals 1, this is due to Bourgain [Bou03] [Bou10]. It roughly says that under multiplicative and additive convolution the Hölder regularity of a measure will increase, that is given κ > 0 there exists > 0 such that if for all x in R and r > 0, we have ν(B(x, r)) ≤ r κ , then ν * ν 2 (B(x, r)) ≤ r κ+ . With this observation, Bourgain gave a quantitative proof of the Erdös-Volkmann ring conjecture [Bou03, Section 4].

Instead of using the original approach in [START_REF] Bourgain | On the Erdős-Volkmann and Katz-Tao ring conjectures[END_REF] [Bou10], we will follow the approach used for proving L 2 -flattening in the case of simple Lie groups, using dyadic decomposition to simplify the argument, developed by Bourgain and Gamburd (see [START_REF] Bourgain | On the spectral gap for finitely-generated subgroups of su (2)[END_REF], [START_REF] Benoist | A spectral gap theorem in simple lie groups[END_REF], [START_REF] Boutonnet | Local spectral gap in simple lie groups and applications[END_REF] for example). We introduce an approximation by dyadic level sets. Definition 3.3.5. Let {D i } i∈I be a family of subsets of R n . We call {D i } i∈I an essentially disjoint union, if each point x in R n is covered by at most C different D i , where C is a fixed constant only depending on R n .

Lemma 3.3.6. [LS15][BISG17, Lemma A.4] Let ν be a Borel probability measure on R n . Let C be a maximal δ-separated set of R n . Let C 0 = {x ∈ C|0 < ν 2δ (x) ≤ 1} and C i = {x ∈ C|2 i-1 < ν 2δ (x) ≤ 2 i } for i ≥ 1. For i ≥ 0, let X i = ∪ x∈C i B(x, δ). Then X i is empty if i ≥ O(log 1 δ ), and we have (1) ν δ i≥0 2 i 1 X i and i>0 2 i 1 X i ν 3δ .
(2) X i is an essentially disjoint union of balls of radius δ, for each i ≥ 0.

(π -1 j B R (a, ρ)) δ -O( ) sup a∈R ν(π -1 j B R (a, ρ)) ≤ δ -O( ) ρ κ , Therefore by b -1 o ≤ δ -O( ) and Lemma 3.2.7, N ρ (π j (b -1 o B 1 )) ≥ δ O( ) N ρ (π j (B 1 )) ≥ ν 1 (B 1 ) sup a∈R ν 1 (π -1 j B R (a, ρ)) ≥ δ O( ) ρ -κ . ( 3
of R n have a form {x ∈ R n |x i = x j } with i = j. Let f ij (x) = x i -x j for x ∈ R n . By (3.3.10) we know that |(b o ) i |, |(b o ) j | ≥ δ O( ) . By (3.3.18), ν 1 {x|f ij (b -1 o x) ∈ B R (0, ρ)} ≤ δ -O( ) ν{x|f ij (b -1 o x) ∈ B R (0, ρ)}.
This is an estimate of being away from linear subspace. If we take the vector w with its i-th, j-th coordinate equal to (b o ) -1 i , -(b o ) -1 j , and other coordinates equal to zero, and let v = w/ w , then

f ij (b -1 o x) = w, x .
Hence projective non concentration (3.3.1) for v implies that

ν{x|f ij (b -1 o x) ∈ B R (0, ρ)} ≤ ν(π -1 v B R (0, δ -O( ) ρ)) ≤ δ -O( ) ρ κ . Hence b -1 o B 1 is δ O( ) away
= 1, • • • , n, N ρ (π j (X o )) ≥ N δ (X o ) sup a∈R N δ (X o ∩ π -1 j B R (a, ρ)) δ O( ) N δ (X i ) sup a∈R N δ (X i ∩ π -1 j B R (a, ρ)) δ O( ) ρ -κ .
Assumption (iv): By (3.3.13),

N δ (X o ) N δ (X i ) δ -n+σ 1 -O( ) .
When δ is small enough such that δ ≤ 1/2, the inequalities with Landau notation can be replaced by ≥ or ≤ with augmenting O( ).

The end of the proof of the L 2 -flattening lemma: Let C 1 and C 2 be given in (3.3.17) and Lemma 3.3.10, respectively. Suppose that C 2 ≥ C 1 (we can always augment C 2 in Lemma 3.3.10. The larger C 2 is, the easier the assumption is). Applying Theorem 3.1.3 with A = b -1 o B 1 and X = X o , when is sufficiently, we have

N δ (X o + X o ) + sup b∈b -1 o B 1 N δ (X o + bX o ) ≥ δ -C 2 N δ (X o ). Due to C 2 ≥ C 1 , we have δ -C 2 N δ (X o ) ≥ δ -C 1 N δ (X o ),
(+, A × bA) ≥ K -1 |A| 3/2 |B| 3/2 . Then there exist x in B and A ⊂ A, B ⊂ x -1 B with |A | ≥ K -O(1) |A| and |B | ≥ K -O(1) |B| such that for all b ∈ B , |A + b A | ≤ K O(1) |A |.
The main point is to find A which is uniform for b. This is accomplished by using the pigeonhole principle. For more details, please see [START_REF] Green | Sum-product phenomena in F_p: a brief introduction[END_REF]Prop. 4.1] or the following proof.

Proof of Proposition 3.3.9. We follow closely the proof of [Gre09, Proposition 4.1]. Since B and B -1 are bounded by K, if we multiply a set by an element in B, then Lemma 3.2.7 implies that we only lose some power on K, which does not change the result. That means for b in B and a subset X of R n , we have

K -O(1) N δ (bX) ≤ N δ (X) ≤ K O(1) N δ (bX)
Hence, we will not write the comparison of N δ (A) with N δ (bA) for bounded set A. They have the same size.

For every b ∈ B, using additive Balog-Szemerédi-Gowers theorem (Proposition 3.2.17), we have

X b × X b ⊂ X × X such that N δ (X b + bX b ) ≤ K O(1) N δ (X) 1/2 N δ (X ) 1/2 (3.3.20) and N δ (X b ) ≥ K -O(1) N δ (X), N δ (X b ) ≥ K -O(1) N δ (X ). (3.3.21)
The result we need is a uniform version, independent of b. For this purpose, we want to find an element b o in B and a portion of B such that the intersection of X bo , X b is large for b in this portion.

Lemma 3.3.13. Let µ be a probability measure on a set B ⊂ B R n (K). Let S be a compact set of R n . Assume that for every b in B, there exists S b ⊂ S such that

|S b | ≥ K -1 |S|.
Then there exists b o in B and

B 1 ⊂ B ∩ B(b o , 1/K 2 ) such that µ(B 1 ) ≥ K -O(1)
, and for every

b in B 1 |S b ∩ S bo | ≥ K -O(1) |S|. (3.3.22)
Proof. We cover B with O(K 2n ) balls of radius 1/K 2 , written as C 1 , . . . , C j . We claim that: There exists i such that

C 2 i |S b ∩ S b |dµ(b)dµ(b ) K -O(1) |S|. (3.3.23)
By hypothesis, we have

B S 1 S b (x)dxdµ(b) = B |S b |dµ(b) ≥ K -1 |S|. (3.3.24)
By Cauchy-Schwarz's inequality The claim (3.3.23) follows. By Lemma 3.3.8, we can find C , a subset of

K 2n i C i 1 S b (x)dµ(b) 2 B 1 S b (x)dµ(b) 2 . ( 3
C 2 i , such that µ ⊗ µ(C ) K -O(1) and for all (b, b ) ∈ C |S b ∩ S b | K -O(1) |S|. (3.3.27) By Fubini's theorem, we can find a b o such that µ{b ∈ C i |(b o , b) ∈ C } K -O(1) . We let B 1 = {b ∈ C i |(b o , b) ∈ C },
then this set satisfies the measure assumption.

The δ neighborhood of a set behaves well under intersection. In order to simplify the notation, abbreviate X (δ) , X (δ) , X (3.3.28)

Remark 3.3.18. This is a step where we really need projective non concentration.

Lemma 3.3.19. Let C > 0 and r ∈ N. Let µ be a probability measure on [1/C, 1] n ⊂ R n . Let ν be defined by

ν = 1 2 (µ 2 * µ - 2 ) (r) + (µ * µ -) (r) .
We have

B(0,2δ -1 ) |ν(ξ)| 2 dξ ∼ C B(0,2δ -1 ) |μ(ξ)| 4r dξ, (3.3.41) and B(0,2δ -1 ) |μ 2 (ξ)| r dξ B(0,2δ -1 ) |ν(ξ)| 2 |ν(yξ)| 2 dν(y)dξ, (3.3.42)
where r = 8r 2 + 4r.

The proof is an elementary computation, using Fourier transform and the Hölder inequality.

Proof. The lower bound part of (3.3.41) is trivial, which is due to the definition of ν.

For a measure m on R and r ∈ N, we have a formula Integrating ξ on B(0, 2δ -1 ), we have

| m(ξ)| 4r = | (m * m -) (2r) (ξ)|. ( 3
B(0,2δ -1 ) |μ 2 (ξ)| 4r dξ ≤ y∈R n ,ξ∈B(0,2δ -1 ) |μ(yξ)| 4r dµ(y)dξ. Due to suppµ ⊂ [1/C, 1] n , we have B(0,2δ -1 ) |μ(yξ)| 4r dµ(y)dξ ≤ C n B(0,2δ -1 ) |μ(yξ)| 4r d(yξ)dµ(y) = C n B(0,2δ -1 ) |μ(ξ)| 4r dξ, which implies that B(0,2δ -1 ) |μ 2 (ξ)| 4r dξ ≤ C B(0,2δ -1 ) |μ(ξ)| 4r dξ. By (3.3.51), we have P 2 1/δ 2 (x) = P 2 1 (x/δ 2 ) 1 B(0,δ 2 ) (x). Combined with |B(0, δ 1 )| C δ -n , this implies ν * P 2 1/δ 2 (x)δ -n ≥ ν * P δ 1 (x) = ν δ 1 (x).
Together with (3.3.53), we have the other direction of (3.3.32).

The second inequality (3.3.33) follows from the same argument. By Parseval's formula

ν δ 1 * (m y ) * ν δ 1 2 2 dν(y) = |ν δ 1 (ξ)| 2 | (m y ) * ν δ 1 (ξ)| 2 dξdν(y) = |ν(ξ)| 2 |ν(yξ)| 2 | P δ 1 (y) 2 || P δ 1 (yξ)| 2 dν(y)dξ = |ν(ξ)| 2 |ν(yξ)| 2 | P 1 (δ 1 y) 2 || P 1 (δ 1 yξ)| 2 dν(y)dξ.
(3.3.54)

For y ∈ B(0, C) and ξ ∈ B(0, 2δ -1 ), we have δ 1 yξ ≤ 1. By (3.3.51), the inequality (3.3.54) implies (3.3.33).

Proof of Lemma 3.3.16. Let R = δ -1 . Consider H R,t = {ξ ∈ B(0, R)||ν(ξ)| ≥ t}, where 0 < t < 1 will be fixed later. Since ν is supported on B(0, K), the function |ν| is K Lipschitz. We have

H R,t + B 0, t 2K ⊂ H R+1, t 2 .
Hence by (3.2.4)

N t (H R,t ) K |H ( t 2K ) R,t |t -n ≤ |H R+1, t 2 |t -n .
By the definition of H R,t , Chebyshev's inequality and (3.3.37),

N t (H R,t ) t -n-1 B(0,R+1) |ν(ξ)|dξ R β t -n-1 . (3.3.55) From now on, suppose that ξ ∈ [R/2, R]. Let H ξ R,t = {x ∈ R n |xξ ∈ H R,t }. Then due to ξ ≤ R, we have xξ ≤ R for x ∈ suppµ ⊂ B(0, 1), and |ν(xξ)|dµ(x) ≤ t + µ(H ξ R,t ). (3.3.56)
We cover H R,t with balls of radius t and we also get a cover of H ξ R,t by B ξ (y, t) = {x ∈ R n |xξ ∈ B(y, t)}. Due to ξ ≥ R/2, there is at least one j ∈ {1, . . . , n} such that |ξ j | ≥ R/(2n). Therefore, we can replace B ξ (y, t) by a cylinder π -1 j B R (y, 2n/R) and we obtain

µB ξ (y, t) = µ{x ∈ R n |xξ ∈ B(y, t)} sup y∈R,j=1,...,n (π j ) * µ{x|x ∈ B R (y, 2n/R)}.
The above inequality combined with the hypothesis (3.3.36) implies

µB ξ (y, t) R -α . (3.3.57) • If λ(R d ) ≥ τ -2 κ/2 , then replace λ by λ = λ/λ(R d ). For ρ ∈ [τ -1 , τ -2 ], we have sup a∈R,v∈S n-1 (π v ) * λ (B R (a, ρ)) ≤ τ 2 κ/2 sup a∈R,v∈S n-1 (π v ) * λ(B R (a, ρ)) ≤ τ 2 κ/2 ρ κ
Due to ρ ≤ τ -2 , we have τ 2 κ/2 ρ κ ≤ ρ κ/2 . The measure λ satisfies non concentration with κ/2. By Proposition 3.4.1, we have the result.

• If λ(R d ) < τ -2 κ/2 , then we have exp(2iπ ξ, x 1 • • • x k )dλ(x 1 ) . . . dλ(x k )| ≤ τ -k 2 κ/2 .
Hence we can take = min{ 2 , k 2 κ/2}. Then we want to prove that the result holds for different measures. For z ∈ R k , let Lemma 3.4.3. Let F be a polynomial of k variables of degree less than n. Let h(F ) be the maximum of the absolutely value of the coefficients in F . Then

λ z = 1≤j≤k z j λ j . Let G(λ 1 , • • • , λ k ) = exp(2iπ ξ, x 1 • • • x k )dλ 1 (x 1 ) . . . dλ k (x k ) and F (z) = F (z 1 , • • • , z k ) = G(λ z , • • • , λ z ). Then F (z) is polynomial of k variables of degree k, and k!G(λ 1 , • • • , λ k ) is the coefficient of z 1 • • • z k in F (z). For z ∈ R k ≥0 ,
h(F ) ≤ O k,n sup z∈{0,••• ,n} k ⊂R k {|F (z)|}.
In this lemma, we define two norms on the space of polynomials of k variable of degree less than n. The inequality is due to the equivalence of norms on finite dimensional vector space. Hence

|G(λ 1 , • • • , λ k )| k h(F ) k τ -.
The proof is complete. Now we will give another version of Fourier decay of multiplicative convolution, which releases the assumption on the support of λ j .

Chapter 4

Finiteness of Small Eigenvalues of Geometrically Finite Rank one Locally Symmetric Manifolds

Let M be a geometrically finite rank one locally symmetric manifolds. We prove that the spectrum of the Laplace operator on M is finite in a small interval which is optimal.

Introduction

Let M be a complete Riemannian manifold. The Laplace operator ∆ acts on the compactly supported smooth functions and admits an extension to an unbounded selfadjoint operator on L 2 (M ). The study of the spectrum of the Laplace operator on geometrically finite hyperbolic manifolds was started by Lax and Phillips in [START_REF] Lax | The Asymptotic Distribution of Lattice Points in Euclidean and Non-Euclidean Spaces[END_REF]. The motivation is to give an exponential error term in the asymptotic distribution of orbits for discrete subgroups of the group of isometries of M . They proved that on a geometrically finite real hyperbolic manifold of dimension n, the intersection of the interval (-(n -1) 2 /4, 0] and the spectrum consists of at most finitely many eigenvalues. Geometrical finiteness means that the quotient manifold has a fundamental domain which is a finitely sided polyhedron.

For general cases (see [START_REF] Hamenstädt | Small eigenvalues of geometrically finite manifolds[END_REF]), Hamenstädt proved that on geometrically finite rank one locally symmetric manifolds Γ\X, where

X = H n F for F = R, C, H or H n F = H 2
O , the intersection of the interval [-δ(X) 2 /4 + χ, 0] and the spectrum also consists of at most finitely many eigenvalues, where χ is any positive number and the exponent of growth δ(X) is (n + 1) dim R F -2. In this case, the applications to counting problems were given in [START_REF] Kim | Counting, mixing and equidistribution of horospheres in geometrically finite rank one locally symmetric manifolds[END_REF]. Inspired by the method in [START_REF] Lax | The Asymptotic Distribution of Lattice Points in Euclidean and Non-Euclidean Spaces[END_REF], we generalize the result of Lax and Phillips to the case of rank one locally symmetric manifolds.

Theorem 4.1.1. Let M = Γ\X be a geometrically finite rank one locally symmetric manifold. Then the intersection of the spectrum of the Laplace operator and the critical interval (-δ(X) 2 /4, 0] consists of finitely many eigenvalues of finite multiplicities.

Remark 4.1.2. 1. If M has infinite volume, the result is optimal in the following sense. Under the same assumption as in Theorem 4.1.1, the interval (-∞, -δ(X) 2 /4] is contained in the essential spectrum of ∆. This may be proved as in [START_REF] Borthwick | Spectral theory of infinite-area hyperbolic surfaces[END_REF]Prop.7.2]. Together with our result, this implies that the essential spectrum of ∆ is exactly (-∞, -δ(X) 2 /4] when the volume is infinite.

2. In the convex cocompact complex hyperbolic case, the meromorphic extension of the resolvent is already known from [START_REF] Mazzeo | Meromorphic extension of the resolvent on complete spaces with asymptotically constant negative curvature[END_REF]. The finiteness of the spectrum in the critical interval is a consequence of the meromorphic extension, as in [Bor07, Prop.7.3]. The main idea is that an eigenvalue of the Laplace operator in the critical interval corresponds to a pole of the resolvent in [0, δ(X)/2).

3. In the convex cocompact quaternion hyperbolic case, the finiteness of the discrete spectrum is due to [START_REF] Bunke | The spectrum of Kleinian manifolds[END_REF]. Later, they published a paper [START_REF] Bunke | Scattering theory for geometrically finite groups[END_REF] to treat the geometrically finite rank one case. But they made an additional assumption for quaternion case and they didn't treat the Cayley hyperbolic case. Our result is new in these two cases.

Unless otherwise stated we assume that the manifolds, the Riemannian metrics and the functions are C ∞ smooth.

Estimates for the spectrum on Riemannian manifolds

Barta's trick

Here we will use Barta's trick, a way to estimate the bottom of the spectrum of the Laplace operator by one good function (see [START_REF] Roblin | Critical exponent and bottom of the spectrum in pinched negative curvature[END_REF]Lem.3.3]). Recall that for every complete Riemannian manifold M , the bottom of the spectrum of the Laplace operator λ 0 (M ) equals the infimum of the Rayleigh quotients R(f ) = ∇f 2 dvol f 2 dvol , over all nonzero smooth functions on M with compact support. Hence it is natural to consider the quantity ( ∇f 2 -λf 2 )dvol. Lemma 4.2.1. Let u, ϕ be two real smooth functions on a Riemannian manifold M , the support of u being compact. We have

∇(ϕu) 2 dvol = ϕ 2 ∇u 2 dvol -u 2 ϕ∆ϕdvol. (4.2.1) Please see [RT15, Lemma 3.3] Proposition 4.2.2.
Let M be a Riemannian manifold. Let f be a real smooth function on M with compact support. Assume that for some λ > 0 there exists a smooth function ϕ such that ϕ > 0 and -∆ϕ ≥ λϕ. Then

∇f 2 dvol -λ f 2 dvol ≥ ϕ 2 ∇(f /ϕ) 2 dvol.
where g ij is the inverse matrix of the matrix of metric. By definition, we have

h(x, t) = | det(g x,t )|.
Since ϕ is a function only depending on t and the formula of metric implies g tj = 0 when j = t, we have

∆ϕ = 1 | det(g)| ∂ t ( | det(g)|∂ t ϕ) = 1 h ∂ t (h∂ t ϕ) = ∂ tt ϕ + ∂ t h∂ t ϕ h . (4.2.5) Due to ∂ t h ≥ 2λ(1 -c(t))h and ∂ t ϕ = -λϕ < 0, we have -∆ϕ ≥ -∂ tt ϕ -2λ(1 -c(t))∂ t ϕ = λ 2 (1 -2c(t))ϕ.
Applying Lemma 4.2.1 with this ϕ and u = f /ϕ, we obtain the result.

The Lax-Phillips inequality

Compared with [START_REF] Hamenstädt | Small eigenvalues of geometrically finite manifolds[END_REF], where she fully used the information from the spectrum of the Laplace operator on each component and the negative curvature, the key new input in our article is the observation that the particular form of the Laplace operator gives us more information. A version on R is as follows: Proposition 4.2.6. Let ∆ = ∂ tt be the standard Laplace operator on R. For every C > 0, the positive spectrum of ∆ + Ce -|t| on L 2 (R) is finite. This is a classical result in spectral theory (see [START_REF] Edward | Spectral theory and differential operators[END_REF]Thm 8.5.1] for a similar version, which says that the positive spectrum is at most countable). In Davies' book, the result comes from a compact perturbation. Another way to prove this type of result is to use the following observation, because the exponential function e -|t| has a very rapid decay. We will give a proof of Proposition 4.2.6 in Section 4.3. Lemma 4.2.7. For every C 0 > 0, there exist a compact interval U and C 1 > 0 such that for every real smooth, compactly supported function f on R we have

∇f 2 + C 1 U f 2 ≥ C 0 e -|t| f (t) 2 dt. (4.2.6)
We call (4.2.6) the Lax-Phillips inequality (LPI). This is a consequence of the following elementary lemma. Lemma 4.2.8. If g is a real smooth bounded function on R, then for all r in R ≥0 we have The proof is complete.

∞ r g(t) 2 e -t dt ≤ 2 g(r) 2 e -r + ∞ r g (t) 2 (1 + t -r)e -t dt . ( 4
Proof of LPI (Lemma 4.2.7). We divide R into R ≥0 and R <0 . By symmetry, we only need to prove the inequality on R ≥0 . Let T be a large number such that e T ≥ C 0 . Then by monotonicity, we have (1 + t -T )e -t ≤ 1/C 0 for all t ≥ T . Integrating (4.2.7) on [T, T + 1] with g = f , we have This idea is already utilized in Theorem 3.3 of [START_REF] Lax | The Asymptotic Distribution of Lattice Points in Euclidean and Non-Euclidean Spaces[END_REF]. The error term e -|x| f 2 is not artificial, which will appear naturally when we estimate the spectrum on the complement of the convex core.

Corollary 4.2.9. Under the same assumption as in Proposition 4.2.4, suppose that c(t) = e -t . If there exists C 0 > 0 such that e 2λt /C 0 ≤ h(t) ≤ C 0 e 2λt for t ∈ R, then for every C > 0, there exist a compact interval I and a positive constant C 1 such that the following holds. For every smooth, compactly supported function f on L × R and every compact subset K of L, we have

( ∇f 2 -λ 2 f 2 )dvol + C 1 K×I f 2 dvol ≥ C K×R e -|t| f 2 dvol.
Proof. Let f 1 = f e λt . Due to the quasi-warped product structure of the Riemannian metric, This formula together with Proposition 4.2.4 implies the result.

∇f 1 2 = ∇ f 1 2 + ∂ t f 2 1 ≥ ∂ t f

Finiteness of the spectrum

Proposition 4.3.1. Let M be a complete Riemannian manifold. If for some smooth bounded function c(x) ≥ 0, there exists a compact subset U with smooth boundary and , C U > 0 such that the following holds. For any compact subset V there is V > 0 such that for all complex valued function f ∈ C ∞ c (M ) we have This is a "baby case" of the main result of this manuscript, whose proof will also follow from Proposition 4.3.1. We will establish (4.3.1) for geometrically finite rank one locally symmetric manifolds in Section 4. Proposition 4.3.3. Let M be a complete Riemannian manifold. Define the space H 1 0 (M ) as the completion of C ∞ c (M ) under the norm

( ∇f 2 -c(x)|f | 2 )dvol + C U U |f | 2 dvol ≥ U ∇f 2 dvol + V V |f | 2 dvol.
f H 1 = f L 2 + ∇f L 2 .
Let the domain of the Laplace operator be

D = {f ∈ H 1 0 (M )| ∆f ∈ L 2 (M )}.
Then ∆ : D ⊂ L 2 → L 2 is a self-adjoint operator.

( Proof. Because of inequality (4.3.1), we can define a bounded restriction map from H to H 1 (U ) and compose it with the injection ι from H 1 (U ) to L 2 (U ), that is

S : H → H 1 (U ) ι -→ L 2 (U ).
Let S * be the adjoint of S, then S * S : H → H is a self-adjoint operator. For f ∈ H, we have

F (S * Sf, f ) = U |f | 2 dvol = 1 C U K(f ).
E(f ) = F (f ) -K(f ) ≥ 1 2 F (f ).
The proof is complete.

Proof of Proposition 4.3.1. By definition,

f F ≤ f L 2 + ∇f L 2 = f H 1 .
Thus we can extend the injection C ∞ c (M ) → H to an application j : H 1 0 (M ) → H. By (4.3.1), for any compact subset V of M , we have F (f ) ≥ V V |f | 2 for f in C ∞ c (M ). Therefore j is injective and can be seen as an inclusion.

Since ∆ is self-adjoint and c is bounded, the operator T = ∆ + c(x) is also selfadjoint. Since ( c ∞ + 1)Id -T is positive, by using Proposition 4.

Rank one locally symmetric manifolds

We study the spectrum on cusps and complement of convex sets in this section, which will be used in Section 4.5 to obtain global result.

Real rank one globally symmetric spaces

Real rank one globally symmetric spaces are usually classified into four families:

H n R = SO o (1, n)/SO(n), H n C = P U (1, n)/U (n), H n H = P Sp(1, n)/(Sp(1) × Sp(n)
) and an exceptional one, the Cayley hyperbolic plane H 2 O = F 4 /Spin(9). For the first three types, we have a uniform treatment by projective models, in which the metric and the curvature can be computed explicitly (see for example [START_REF] George D Mostow | Strong Rigidity of Locally Symmetric Spaces[END_REF], [START_REF] Pansu | Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un[END_REF], [START_REF] Quint | An overview of Patterson-Sullivan theory[END_REF]). But the Cayley hyperbolic plane needs different model.

The space H n F is a Riemannian manifold with pinched curvature between -1 and -4. The ideal boundary of the symmetric space H n F , denoted by ∂H n F , is the set of equivalent classes of geodesic rays. From now on, abbreviate H n F , ∂H n F , H n F ∪ ∂H n F to X, X I , X c . Equip X c with the topology such that X c becomes a compact manifold with boundary. The space X is homeomorphic to an open ball of dimension n dim R F, and X c X, that is (ν, y) → ν • c(log y). We also give a differential structure to X c . Locally the compactification is obtained by adding the hyperplane {y = 0}. Locally, this also gives a differential structure to the manifold with boundary. If we fix another point in X I , and give a differential structure by the same procedure, then by the compatibility of the differential structures, we have a differential structure on X c .

Discrete subgroups

Let G be the group of isometries of the symmetric space X, and let Γ be a discrete subgroup of G. The group Γ may have torsion. In the geometrically finite case, the group Γ is always finitely generated [Bow95, Prop5.5.1]. We can use a result of Selberg [START_REF] Selberg | On discontinuous groups in higher-dimensional spaces[END_REF], passing to a normal subgroup Γ which is of finite index in Γ and has no torsion. Since the spectrum of the Laplace operator on a finite covering space contains the spectrum of the original one, we suppose that Γ is without torsion.

Fix a point x in X, and let Γ • x be the closure of the orbit Γ • x in X c . The limit set of the group Γ is defined by

Λ(Γ) = X I ∩ Γ • x.
This definition of the limit set is independent of the choice of x. Let Ω = X I -Λ(Γ). Since the actions of Γ on X ∪ Ω and X are properly discontinuous [Bow95, Prop3.2.6], we set M c (Γ) = Γ\(X ∪ Ω), M (Γ) = Γ\X.

These are a manifold with boundary and a rank one locally symmetric manifold, respectively.

In order to study the spectrum, we define an energy form, which will be used throughout the paper. Let

E(f ) = M (Γ) ( ∇f 2 -2 f 2 )dvol (4.4.3) for f ∈ C ∞ c (M (Γ))
, where Γ is a discrete subgroup of G and = δ(X)/2 = ((n + 1) dim R F -2)/2 is half the exponent of growth.

Cusps

Definition 4.4.1 (Parabolic subgroup). Let Π be a subgroup of G. We call Π parabolic if the set of fixed points of Π in X c consists of a unique point ξ in X I , and Π preserves setwise every horosphere based at ξ.

Let Π be a discrete parabolic subgroup with fixed point ∞, a point in X I . We use the horospherical model N × A → X introduced in Section 4.4.1. Then Π is a subgroup of M N . Recall that M is the subgroup of K which preserves setwise every horosphere based at ∞ and the metric g t on N . The part N of M N acts as translation on N and M acts as rotation. Let Π\N be the quotient space of N under the action of Π. The quotient manifold satisfies M (Π) (Π\N ) × R as a topologic space.

Remark 4.4.6. The first inequality in (4.4.6) has already been used in [Ham04, Lemma 2.3] without proof. For completeness, a proof is given here.

Remark 4.4.7. This lemma is a consequence of the negative curvature and the convexity. We do not have a better inequality ∂ t h(t) ≥ 2 h(t). For example, in H 2 , let D be the r-neighbourhood of a geodesic, then we can compute explicitly that h(x, t) = cosh(t + r)/ cosh r.

The proofs of these two lemmas, which use the standard computations for Jacobi fields, will be given later. By Lemma 4.4.5, we have where v, u are two vectors in the tangent space T x S. By the convexity of S, the second fundamental form II S is positive definite at every x in S. Fix a point x in S. By (4.4.2), starting from the outer unit normal vector n(x), we can find an orthonormal basis {n(x), (Y j ) 1≤j≤m , (Y k ) m+1≤k≤m+q } of T x M such that R(n(x), Y j , n(x)) = -Y j , R(n(x), Y k , n(x)) = -4Y k , where m = (n-1) dim R F, q = dim R F-1.

∂ t h ≥ 2 (tanh t)h ≥ 2 (1 -2e -t )h.
Let B be the matrix representation of II S with the basis {(Y j ) 1≤j≤m , (Y k ) m+1≤k≤m+q } of T x S. Lemma 4.4.9. With the same assumption as in Lemma 4.4.5, we have h(x, t) = det cosh tId m 0 0 cosh 2tId q + B sinh tId m 0 0 1 2 sinh 2tId q . (4.4.10)

Proof. There exists a local chart on S defined by (φ, U ) φ : R m+q ⊃ U → S such that φ(0) = x and ∂ ∂u i φ(0) = Y i . This particular choice of local chart implies that du is the volume element at x. Let φ(u, t) = exp φ(u) (tn(φ(u))

and Ũ = U × R ≥0 . Then ( φ, Ũ ) is a local chart of M . For every fixed u, the curve t → φ(u, t) is a geodesic starting from φ(u) with tangent vector n(φ(u)). Fix all u w to 0 except w = i. Then the map H : R 2 → M defined by (u i , t) → φ(u i , t), is a variation of geodesic. Let J i (t) be the Jacobi field defined by J i (t) = ∂ ∂u i H(0, t). The volume element, in a local chart, can be written as det g ∂ ∂x i , ∂ ∂xw dx. In our case, the volume element at φ(0, t) is det(g(J i (t), J w (t)))dudt. Therefore by definition, we have h(x, t) = det(g(J i (t), J w (t))).

Let Y i (t), n(t) be the images of Y i and n(x) under the parallel transport along the geodesic t → φ(0, t) with t ≥ 0. They also form an orthonormal basis. By Lemma 4.4.3, the vectors J i (t) are orthogonal to n(t). We decompose J i (t) with respect to Y i (t), that is J i (t) = Σ w a iw (t)Y w (t), and write A(t) = (a iw (t)) 1≤i,w≤m+q . Then A(0) = Id m+q and the matrix A (0) equals B, the matrix representation of the second fundamental form. This is because by Schwarz's theorem, we have

a iw (0) = ∂ t g(J i (t), Y w (t))| t=0 = g D dt J i (0), Y w = g D ∂ ∂u i ∂ ∂t , Y w = g(D Y i n(x), Y w ).
Because J i are Jacobi fields, they satisfy the Jacobi equation: D 2 dt 2 J i (t) + R(n(t), J i (t))n(t) = 0. (4.4.11)

The map R(n(t), •)n(t) is a linear map on the orthogonal complement of n(t) in the tangent space T φ(0,t) M . By [Hel79, IV, Thm1.3], in locally symmetric spaces the curvature tensor R is invariant under parallel transport. Hence in our choice of the orthonormal basis {Y i (t)} 1≤i≤m+q , the linear map R(n(t), •)n(t) can be represented by the matrix diag{Id m , 4Id q }. From (4.4.11), we have

A (t) = A(t) Id m 0 0 4Id q .
The solution is determined by the initial conditions. Due to A(0) = Id and A (0) = B, it is A(t) = cosh tId m 0 0 cosh 2tId q + B sinh tId m 0 0 1 2 sinh 2tId q . Hence h(x, t) = det(g(J i (t), J w (t))) 1/2 = det(A(t)), which implies the result.

For computing the determinant, we need a lemma. Proof. It is elementary that the sum of two symmetric positive semidefinite matrices has determinant no less than the determinant of each one. We only need to transform our matrix to a symmetric matrix. We have (4.4.12)

det
Since B is positive semidefinite, using Lemma 4.4.10, we have h(t) ≥ cosh m t cosh q 2t ≥ e 2 t /C 1 . The upper bound of h(t) is due to the upper bound on the second fundamental form, that is to say B is bounded.

The derivative in the scalar part of (4.4.12), cosh m t cosh q 2t, gives us (m tanh t + 2q tanh 2t)h(t) ≥ (m + 2q) tanh t h(t) = 2 tanh t h(t).

It remains to prove the positivity of the derivative of the determinant part of (4.4.12), which is the sum of derivatives in every column. Since all the terms are similar, we need only to show that the derivative in the first column is non negative. The derivative of tanh t is 1/ cosh 2 t, and we multiply the first column with tanh t cosh 2 t to recover the original column. The determinant of the derivative of the first column becomes Then J(t) is a variation of geodesic, and it is a Jacobi field, which is determined by its value and derivative at 0. We have J(0) = ∂ ∂u i φ(u, 0)| u=0 = ∂ ∂u i φ(0), which is a tangent vector of S at x. Hence J(0) is normal to n(x). Hence J(t) is normal to the tangent vector of the geodesic. This is true for all i, and the result follows.

1

Geometrically finite manifolds

We return to the study of the spectrum of the whole manifold. We give a topological decomposition of geometrically finite manifolds. In order to describe it, we use standard cusp regions. For a subset S of M c , we use S to denote its topological interior, that is S = S -Sc ∩ S. The reader who is only interested in convex cocompact manifolds (geometrically finite manifolds without cusps) can skip Section 4.5.1 and go directly to Section 4.5.2. Our goal in this section is to obtain the Lax-Phillips inequality, then Theorem 4.1.1 follows by Proposition 4.3.1.

Standard cusp regions

The manifold with boundary M c may have some cusps. To analyse the structure of cusps, we introduce the concept of a topological end. Definition 4.5.1. Let T be a differential manifold. An end e is a function which assigns every compact subset K of T a non empty connected component of T \K, such that if K ⊂ K , then e(K) ⊃ e(K ).

Let K 1 ⊂ K 2 ⊂ . . . be an ascending sequence of compact subsets whose interiors cover T . We call e(K i ) a system of neighbourhoods for the end e.

A neighbourhood for the end e is an open subset U such that U ⊃ e(K n ) for some n.

Proposition 4.5.2. [Bow95, Proposition 4.4] If Π is a discrete parabolic subgroup of G, then M c (Π) has precisely one topological end. Moreover, we can find a system of neighbourhoods for the end consisting of geodesically convex submanifolds of M c (Π).

Following [START_REF] Bowditch | Geometrical finiteness with variable negative curvature[END_REF], we define a standard cusp region (with boundary). This definition is not explicitly written, but it is implicitly defined in [Bow95, Sec.5.1]. For more details on the real hyperbolic case, we refer to [START_REF] Bowditch | Geometrical Finiteness for Hyperbolic Groups[END_REF]Sec. We have a corollary of Proposition 4.5.2. Proposition 4.5.4. Let Γ be a discrete subgroup of G, and let O be a standard cusp region of M c (Γ). There exists a smaller standard cusp region O such that O is contained in the interior of O.

Proof. By Definition 4.5.3, the cusp region O is isometric to E and the complement of the interior of E is a compact subset K of M c (Π). Denote by f the isometric map from O to E. By Proposition 4.5.2, M c (Π) has only one end and we have a larger compact subset K 1 , whose interior contains K and whose complement is geodesically convex. Let E 1 = K c 1 and let O be the corresponding subset of O, that is O = f -1 (E 1 ).

For a symmetric space of real rank one, we use the half space model introduced at the end of Section 4.4.1, that is a diffeomorphism from N × R >0 to X. Recall that Π is a parabolic subgroup of G which fixes ∞. It is a subgroup of M N , which acts isometrically on N equipped with the metric g t . We have M c (Π) (Π\N ) × R ≥0 as topological spaces Then our standard cusp region is isometric to the complement of an open relatively compact subset in M c (Π), with additional convex condition on the interior. (see figure 4.5.1.) Definition 4.5.5 (Maximal rank). Let Π be a discrete parabolic subgroup of G. We call Π a subgroup of maximal rank if the quotient Π\N is compact.

A cusp region is said to be maximal rank, if the corresponding discrete parabolic subgroup is of maximal rank Remark 4.5.6. We explain our definition as follows. The rank of a nilpotent group is defined to the sum of the rank of its central series.

For real hyperbolic case, Π is a discrete subgroup of Isom(R n-1 ) = O(n -1) R n-1 . By Bieberbach's theorem (see for example [Bow93, Theorem 2.2.5]), the group Π is virtually abelian. The rank of Π is defined to be the rank of its maximal normal abelian subgroup. Hence when Π attends maximal rank, the quotient space is compact.

For rank one symmetric spaces, by Margulis' Lemma, the discrete parabolic subgroup Π is virtually nilpotent. As in [CI99, Lemma 3.4], for a virtually nilpotent discrete subgroup Π < M N , we can find a subgroup Π 1 < Π of finite index which is nilpotent, and there exists a subgroup Π 2 < N which is isomorphic to Π 1 and satisfies Π 1 •x = Π 2 •x for some x in N . This means that the Π 1 , Π 2 -orbits of x are the same in N . Let N 2 be the Zariski closure of Π 2 . Then the rank of Π is the same as the rank of N 2 . When Π attends maximal rank, N 2 coincides with N . Then Π 2 is a cocompact subgroup of N because N is nilpotent. Due to Π 2 • x = Π 1 • x, every point in N has a bounded distance to the orbit Π 1 • x. Hence Π 1 \N is compact, so is Π\N . Proposition 4.5.7 (Maximal rank). Let Γ be a discrete subgroup of G. Let O be a standard cusp region of M c (Γ) with maximal rank. Then we can find a smaller cusp region O 1 , which is isomorphic to the quotient of a horoball by a discrete parabolic subgroup of Γ.

Proof. By Definition 4.5.3, there is a discrete subgroup Π of Γ, such that O is isometric to a closed subset E of M c (Π), whose interior is a neighbourhood for the end of M c (Π). By Definition 4.5.2 and Proposition 4.5.2, the complement of E in M c (Π) is relatively compact. Under the half space model, we can suppose that E c ⊂ (Π\N ) × [0, 1]. Let B be the horoball, which is homeomorphic to N × R ≥1 . Then the quotient Π\B is a subset of E. Due to maximal rank, the quotient Π\N is compact. Hence M c (Π) -Π\ B (Π\N ) × [0, 1] is compact. The quotient Π\B is geodesically convex. Let O 1 be the preimage of Π\B in O under the isometric map from O to E. The proof is complete.

In later proof, for cusps of maximal rank, we will always take the quotient of horoball as a standard cusp region.

A good partition of unity

Definition 4.5.8 (Geometrical finiteness). A discrete subgroup Γ in G is called geometrically finite, if M c (Γ) is the union of a compact set and a finite number of standard cusp regions O i for 1 ≤ i ≤ k, that is to say M c (Γ) -1≤i≤k Oi is compact. This definition is not explicitly written in [START_REF] Bowditch | Geometrical finiteness with variable negative curvature[END_REF], but is given in the discussion after [Bow95, Def. F1]. (See also [START_REF] Bowditch | Geometrical Finiteness for Hyperbolic Groups[END_REF]Def.(GF1)] for the real hyperbolic case. In [START_REF] Bowditch | Geometrical Finiteness for Hyperbolic Groups[END_REF], Bowditch explained the equivalence of the definition in the introduction and Definition 4.5.8 for the real hyperbolic case.) By [Bow95, Lemma 6.2], if Γ is geometrically finite, then there exist standard cusp regions O i for 1 ≤ i ≤ k, such that O i are pairwise disjoint. For the purpose of the exposition, we can limit our consideration to the case that there is at most one cusp; the results hold and the methods of proof work for general cases.

For a real number r > 0, we define the r-neighbourhood of a set Q in X by N r (Q) = {x ∈ X|d(x, Q) ≤ r}. Let W r = N r (hull(Λ(Γ)) ∩ X) be the r-neighbourhood of the convex hull of the limit set Λ(Γ). Let C(M ) be the convex core defined by C(M ) = Γ\(hull(Λ(Γ)) ∩ X). Let C r (M ) = Γ\W r be the r-neighbourhood of the convex core. One problem here is that the boundary of C r (M ) may not be C ∞ -smooth, but is only C 1,1 -smooth (see for instance [START_REF] Walter | d'erreur exponentiel pour le théorème de renouvellement dans le cadre des produits de matrices aléatoires[END_REF] or [START_REF] Federer | Curvature measures[END_REF]). To overcome this difficulty, we use a result of [START_REF] Parkkonen | On strictly convex subsets in negatively curved manifolds[END_REF]Prop.6]. (In the statement of Proposition 6 in [START_REF] Parkkonen | On strictly convex subsets in negatively curved manifolds[END_REF], they do not have a Γ-invariant condition. But if we start from a Γ-invariant set, their method automatically gives us a Γ-invariant set.) We can find a closed convex subset W with C ∞ smooth boundary such that W 1 ⊂ W ⊂ W 3/2 and W is also Γ-invariant. Let D = Γ\W . Then D ⊂ C2 (M ).

Let O be a standard cusp region of the unique cusp in M c . We have a smaller standard cusp region O such that O ⊂ O.

We Since M c inherits the differential structure from X ∪Ω, the covering is about a differential manifold with boundary. We can find a smooth partition of unity subordinate to this cover, written as { φ1 , φ2 , φ3 }, which is smooth on the boundary. Here M 1 , M 2 may intersect the ideal boundary M I = M c -M . Our covering has the advantage that M 3 is compact, and M 1 , M 2 have quasi-warped product Riemannian structure.

Te

  2ikx dν(x), pour k ∈ Z. Théorème (Theorem 1.1.1, Theorem 2.1.2). Soit µ une mesure de probabilité borélienne Zariski dense sur SL 2 (R) avec un moment exponentiel fini. Soit ν la mesure µstationnaire sur T. Alors il existe > 0 tel que |ν(k)| = O(|k| -).

b 2 b 1 e

 1 Lemma 1.3.8. For b 1 < b 2 and λ nonzero, we have| iλ exp(-u) du| ≤ 2(e b 1 + e b 2 ) |λ| .

  |φ(gx) -φ(gy) -sign(x, y, x m g )φ (gx)d a (gx, gy)| ≤ |φ | ∞ e -4(t-s 1 ) . (1.5.6) By equality sin d a (gx, gy) = d(gx, gy), we have |d a (gx, gy) -d(gx, gy)| = O(d(gx, gy) 3 ).

For ζ =

  kζ o ∈ P, by definition, we have V * α,ζ = kV * α,ζo . For η ∈ P and ζ ∈ P, we set δ(η, ζ) = min α∈Π δ(V α,η , V * α,ζ ).

0 ∂

 0 ) we have ϕ(gz 2 ) -ϕ(gz 1 ) = u 0 dϕ gγ(s) dg γ(s) K α ds = u 0 dϕ gγ(s) dg γ(s) Y α ds = u 0 dϕ gγ(s) exp(-ασ(g, γ(s)))Y α ds = u α ϕ gγ(s) e -ασ(g,γ(s)) ds.

  g,y) (f (gx)-f (gy)) d(x,y) γ dµ * n (g)| and B n = | G (e zσ(g,x) -e zσ(g,y) )f (gx) d(x,y) γ dµ * n (g)|. By Cauchy-Schwarz's inequalityA n ≤ c γ (f ) G e aσ(g,y) d(gx, gy) γ d(x, y) γ dµ * n (g) ≤ c γ (f )G e 2aσ(g,y) dµ * n (g)

  Since |e c -e d | ≤ (2 max(e c , e d )) 1-γ (max(e c , e d )|c -d|) γ for c, d in C, we have|e zσ(g,x) -e zσ(g,y) | d(x, y) γ ≤ (2e |a|κ(g) ) 1-γ (e |a|κ(g) |z|Lip(σ(g, •))) γ ≤ 2e |a|κ(g)+γκ 0 (g)|b| γ , where κ 0 (g) is the Lipschitz norm of σ(g, •) and κ 0 (g) ≤ C κ(g) by [BQ16, Lemma 13.1]. Then by the hypothesis of finite exponential moment and Hölder's inequality, we have B n ≤ |b| γ |f | ∞ C (|a|+γ)n 1

  by Lemma 2.5.8, we have|∂ s ϕ(γ(s))| ≤ CC m

1

  S b (x)dµ(b)dx 2 /|S| ≥ K -O(1) |S|. (3.3.26) Rewrite the left hand side of (3.3.25) and integrate it with respect to the Lebesgue measure on S. Combined with (3.3.26) we have i C 2 i |S b ∩ S b |dµ(b)dµ(b ) K -O(1) |S|.

  b to Y, Y , Y b , Y b . By (3.2.4) we have N δ (X) ∼ |Y |δ -n .

  .3.43) By the multiplicative structure of R n , we have |μ 2 (ξ)| = e 2iπ ξ,xy dµ(x)dµ(y) = μ(yξ)dµ(y) . (3.3.44) By the Hölder inequality, |μ 2 (ξ)| 4r ≤ |μ(yξ)| 4r dµ(y).

  we have |F (z)| ≤ |z| k τ -, where |z| = 1≤j≤k |z j |, by using the result of the first part with λ = 1 |z| λ z .

  The result follows by taking U = [0, T + 1] and C 1 = 2C 0 .

  (4.3.1) Then the positive spectrum of the operator T = ∆ + c(x) consists of at most finitely many eigenvalues of finite multiplicities. Proof of Proposition 4.2.6 from Proposition 4.3.1. From Lemma 4.2.7, we have f (t) 2 dt. Take C 0 = 4C and c(x) = Ce -|x| . Then c(x) satisfies (4.3.1). Therefore, Proposition 4.2.6 follows from Proposition 4.3.1.

  5. It remains to prove Proposition 4.3.1. Recall some results in spectral theory: Definition 4.3.2. Let H be a complex Hilbert space, T be a linear operator with the domain of definition D, which is dense in H. We call T self-adjoint if its adjoint T * equals T on D and the domain of definition of T * satisfies D * = D, where D * is defined as D * = {f ∈ H|∃C f > 0 such that ∀g ∈ D, |(f, T g)| ≤ C f |g| H }.

(

  See[START_REF] Lax | Functional Analysis[END_REF] 32 Thm1] for more details.) For f ∈ C ∞ c (M ), defineK(f ) = C U U |f | 2 dvol, E(f ) = ( ∇f 2 -c(x)|f | 2 )dvol,and F (f ) = E(f )+K(f ). Inequality (4.3.1) implies that F is a positive definite quadratic form on C ∞ c (M ). We define a Hilbert space H as the completion ofC ∞ c (M ) with respect to the norm | • | F , written as H = C ∞ c (M ) F . For an open subset V of M , we define H 1 (V ) as the completion of C ∞ (V ) with respect to the norm f 2 H 1 (V ) = V |f | 2 + V ∇f 2 .Proposition 4.3.5. With the same assumption as in Proposition 4.3.1, there exists a subspace H 1 of finite codimension in H, on which E is positive definite.

  By the Rellich theorem [Tay10a, Chapter 4, Proposition 4.4] and the smoothness of the boundary of U , the injection ι is compact. Therefore S * S is a compact self-adjoint operator. The set of eigenvalues has a unique accumulation point 0. In a subspace of finite codimension in H, we have|K(f )| = C U |F (S * Sf, f )| ≤ 1 2 |F (f )|. Therefore

  3.4 with T andH = L 2 (M ), we have H + ⊂ D ⊂ H 1 0 (M ) ⊂ H.For a nonzero element u in H + , we haveE(u) = M ( ∇u 2 -c(x)|u| 2 )dvol = M -(T u)ūdvol = -(T u, u) < 0.Proposition 4.3.5 implies H 1 ∩ H + = {0}, therefore H + is of finite dimension. Due to Proposition 4.3.4, the operator T is self-adjoint on H + , hence the positive spectrum of T is finite and each element is an eigenvalue of finite multiplicity.

  2.4 and Corollary 4.2.9 with S × R ≥0 , (Proposition 4.2.4 deals with manifolds L × R, but the proof of L × R ≥0 is exactly the same) we have Lemma 4.4.8. Let E be as in (4.4.3). With the same assumptions as in Lemma 4.4.5, for a functionf ∈ C ∞ c (exp ⊥ (S × R ≥0 )), we have E(f ) ≥ ( ∇(e t )f ) 2 e -2 t -4 2 e -t f 2 )dvol. (4.4.8) Given C > 0, there exist a compact set I ⊂ R ≥0 and a constant C 1 > 0 such that for all compact set K in S E(f ) + C 1 exp ⊥ (K×I) f 2 dvol ≥ C exp ⊥ (K×R ≥0 )e -t f 2 dvol. (4.4.9) It remains to prove Lemma 4.4.3 and Lemma 4.4.5. Proof of Lemma 4.4.5. The main idea is to compute the density function h by the second fundamental form. The second fundamental form of S at x is defined to be the symmetric form II S : T x S × T x S → R, II S (v, u) = g(D v n(x), u),

  Lemma 4.4.10. Let D be a diagonal matrix with nonnegative entries, let B be a symmetric positive semidefinite matrix, and let B 11 B 12 B 21 B 22 be a block partition of B such that B 11 , B 22 are square matrices. Then for all λ 1 , λ 2 > 0 we have det D + λ 1 B 11 λ 2 B 12 λ 1 B 21 λ 2 B 22 ≥ det(D).

  For the derivative, by Schwarz's theorem we haveJ (0) = D dt J(t)| t=0 = D ∂ ∂t

Figure 4

 4 Figure 4.5.1: A standard cusp region

  can cover the geometrically finite manifold with three open setsM c = O ∪ (D c -O ) ∪ ( C2 (M ) -O ), (4.5.1)where D c is the complement in M c . For the simplicity of the notation, we writeM 1 = O, M 2 = D c -O and M 3 = C2 (M ) -O .

  Proposition 4.5.9. The set M 3 is relatively compact in M .Proof. By Definition 4.5.8, we have thatM c -O is compact in M c . Therefore C 2 (M )-O is compact in M c . It is also contained in M , hence it is a compact subset of M .

  of Abbreviations and Symbols in Chapter 2 χ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . δ(η, ζ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . δ(x, y) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A (z 1 , z 2 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . P z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . q λ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V α,η . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V χ,η . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

	x M g
	y m

δ α (η, ζ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . η M g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . γ 1,2 (g) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

γ(g) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ν V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . P 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ∂ α ϕ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . m(g, g ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . σ V,µ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ζ m g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B m g (r) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . b M g (r) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B m V,g (r) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . b M V,g (r) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C γ (X) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . c γ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . d(η, η ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . d(x, x ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . d 0 (k, k ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . d α (η, η ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . d g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

  Lemma 2.2.12. For g in G and η in P,

	.2.14)
	Using Lemma 2.2.5 and (2.2.14), we deduce the following two lemmas from Lemma 2.2.7
	and Lemma 2.2.8

  By Lemma 2.2.22, it is sufficient to prove the case that m(k, k ) = e.

		By Lemma
	2.2.23,
	m(k g , ghk) = m( -1 g , hk).	(2.2.20)
	Denote kη

o , k η o by η, η . Then by Lemma 2.2.13, we have hη, hη

∈ b M h (δ) ⊂ B m g (2δ). Hence by d(hη, hη ) < 2δ ≤ δ(hη, ζ m g ) = δ(hη, -1 g ζ o )

and Lemma 2.2.24, we have

m( -1 g , hk) = m( -1 g , hk )m(hk, hk ). (2.2.21)

The main point here is to prove the following lemma. Lemma 2.2.26. Under the same assumption as in Lemma 2.2.25, we have m(hk, hk ) = m(k, k ). Combined with (2.2.20) and (2.2.21), the proof is complete. Proof of Lemma 2.2.26. Without loss of generality, suppose that m(k, k ) = e. Due to kη 0 ∈ B m h (δ), we can chose a h in the Cartan decomposition h = k h a h h such that m( -1 h , k) = e. By Lemma 2.2.24, the hypothesis that δ(kη o , -1 h

  αr,η and l = l r,η . Due to the hypothesis, Lemma 2.2.34 and Lemma 2.2.36, we can find η j+1 in the same α r -circle of η j such that d αr (gη j+1 , gη ) = d(ρ αr gV αr,η j+1 , ρ αr gV αr,η ) ≤ δ -2 γ 1,3 (ρ αr g) ≤ δ -2 βe -αrκ(g) . (2.2.28) Hence by (2.2.27) and (2.2.28), d αr (gη j+1 , gη j ) = d αr (gη j+1 , gη) = d αr (gη, gη )

  where h i has distribution µ * n . Due to Lemma 2.2.53, with a loss of exponentially small measure, we can suppose that X i are bounded by C = e 2 n , where 2 = /(2d).Due to (2.3.1), we have v, Y n (h, η) = p d v, E d X(n, h, η) . PNC asks exactly that the probability that E d X is close to a hyperplane is small. By (2.3.2), PNC on dimension d follows from SNC on dimension d.By (2.3.3), SNC on dimension d follows from PNC on dimension d.Remark 2.3.8. We explain that SNC implies the stronger form of SNC, which will be used later. Let O(d) be the orthogonal group in dimension d. The stronger form of SNC says that for any (ρ 1

  ×d , it follows from Remark 2.3.8 that SNC implies this stronger form.) By Lemma 2.2.52, the set that h d+1 is not (n, 2 , ζ m g ) good and h d is not η good have exponentially small measure. Hence

	M ≤

good dµ * n (h d+1 )

  Using the regularity of stationary measure (2.2.34) with V = W α , the irreducible subrepresentation of ∧ 2 Sym 2 V α with the highest weight, we have

	ν{η ∈ P|δ(V 4χα-α,η , F ραg,ραh ) < δ}	e -c n .	(2.4.62)
	Using the regularity of stationary measure (2.2.34) with V = V α , we obtain
	ν{η ∈ P|V α,η ∈ B m h (δ) ∪ B m g (δ)}	e -c n .	(2.4.63)
	Hence by (2.4.61)-(2.4.63), we have
	ν(X c g,h )	e -c n = |θ| -c /C 2 .	(2.4.64)
	For (g, h) in G n, , by (2.4.60) and (2.4.64)
				.4.54)
	By large deviation principle (Proposition 2.2.46, Proposition 2.2.47), the set G c n, has exponentially small µ * 2n measure, that is
	µ(G c n, )	δ c .	(2.4.55)
	By f ∞ ≤ 1 and (2.4.48), we have
	|A g,h | 2 dµ * n (g)dµ * n (h) ≤ |P n 2 1| 2 ∞ ≤ C 4n .	(2.4.56)

  from proper subalgebra. Assumption (iii) (non concentration of X o ): By (3.3.15) and (3.3.14) we have for ρ ≥ δ and j

  The only place where we need a stronger non concentration than non concentration on coordinate subspaces is in the proof of Lemma 3.3.10, when we check assumption (ii) of Theorem 3.1.3. In this step, we need a property of being away from a linear subspace.It remains to prove Proposition 3.3.9. We first state a similar version on F

	which contradicts (3.3.17). The
	proof is complete.
	Remark 3.3.11.

p Proposition 3.3.12. [Bou09, Thm.C] [Gre09, Prop. 4.1] Let K > 1. Let A ⊂ F p and B ⊂ F * p be two sets. If for all b in B, we have ω

  ∇ is the gradient on L × {t}. Substituting (4.2.10) into LPI (Lemma 4.2.7), with the constant C 2 = C 2 0 (C + 2λ 2 ), implies that there exist I compact, C 1 > 0 such that Using dvol = h(x, t)dtdx and e 2λt /C 0 ≤ h(t) ≤ C 0 e 2λt , we obtain

	∇f 1 (x, t) 2 dt + C 1	f 1 (x, t) 2 dt ≥ C 2	e -|t| f 2 1 (x, t)dt, everywhere on L. (4.2.11)
	R		I		R
	Integrating (4.2.11) over K with respect to dx, we get
	∇f 1	2 dxdt + C 1	f 2 1 dxdt ≥ (C + 2λ 2 )	e -|t| f 2 1 dxdt.
	K×R			K×I	K×R
	∇f 1	2 e -2λt dvol + C 1	f 2 dvol ≥ (C + 2λ 2 )
	K×R			K×I
					2 1 ,	(4.2.10)

where K×R e -|t| f 2 dvol.

  See [Tay10b, Sec.8.2] for more details.) Proposition 4.3.4. If T is a self-adjoint operator on a Hilbert space H, then there is a decomposition H = H + ⊕ H -⊕ ker T such that T preserves the decomposition, T is self-adjoint on H + , H -and T is positive, negative on H + , H -, respectively. If there exists λ > 0 such that λId-T is positive, then H + in the above decomposition is actually in D, the domain of definition.

  D + λ 1 B 11 λ 2 B 12 λ 1 B 21 λ 2 B 22 Id q .Since D is diagonal, the first matrix in the right-hand side is again symmetric. We havedet D + λ 1 B 11 λ 2 B 12 λ 1 B 21 λ 2 B 22 ≥ det(D).Return to the proof of Lemma 4.4.5. Let B 11 B 12 B 21 B 22 be the block partition of B such that B 11 , B 22 are square matrices of order m, q. By (4.4.10) h(x, t) = (cosh t) m (cosh 2t) q det Id m+q + tanh tB 11

	= det (D λ 2 The proof is complete. λ -1 1 Id m 0 0 λ -1 2 Id q + B 11 B 12 B 21 B 22 λ 1 Id m ) 0	0
	tanh tB 21	tanh 2t 2 B 12 tanh 2t 2 B 22	.

  Proof of Lemma 4.4.3. Use the same notation as in the proof of Lemma 4.4.9. Let

	J(t) =	∂ ∂u i	H(0, t) =	∂ ∂u i	φ(u, t)| u=0 .
	tanh t cosh 2 t	det	0 0 Id m+q-1 0	+	tanh tB 11 tanh tB 21	tanh 2t 2 B 12 tanh 2t 2 B 22	,
	which is nonnegative by Lemma 4.4.10.
	It remains to prove Lemma 4.4.3.

Remerciements

We call T positive if for every nonzero f in D, we have (T f, f ) > 0.

Finiteness of the spectrum

Lemma 2.2.36. The line l r,η is the image of the α r -circle of η in PV αr .

Proof. Due to l r,kη 0 = kl r,η 0 and the left K invariance of the set of α circles, it is sufficient to consider η 0 . The line l r,η 0 is generated by two points V χα r = Re 1 ∧ • • • ∧ e r-1 ∧ e r and V χα r -αr = Re 1 ∧ • • • ∧ e r-1 ∧ e r+1 in PV αr . By Lemma 2.2.15, this is exactly the image of α r circle containing η 0 in PV αr .

Definition 2.2.37. Let (η 0 , η 1 , . . . , η k ) be a sequence of points in P. We call it a chain if any consecutive elements η i , η i+1 are in the same α-circle for some α ∈ Π, and we write α(η i , η i+1 ) for this simple root.

By the structure of the root system of G, we have Lemma 2.2.38. We can separate Π into a disjoint union Π 1 and Π 2 such that for α, α in the same atom Π j , α + α is not a root.

Let l 1 = #Π 1 and l 2 = #Π 2 .

By Lemma 2.2.15

Lemma 2.2.39. Let (η 0 , . . . , η l ) be a chain and let α be a simple root. If the set of simple roots appearing in the chain does not contain α, then the image of the chain in PV α is a single point, that is V α,η j = V α,η 0 , ∀j = 1, . . . , l. Now, we state our main result of this part, which will be used in the main approximation (Proposition 2.4.11).

Lemma 2.2.40. Let η, η be two points in P and let g be in G. If for α ∈ Π 1 , δ(V α,η , y m ρα(g) ), δ(l α,η , y m ∧ 2 ρα(g) ) > δ, for α ∈ Π 2 , δ(V α,η , y m ρα(g) ), δ(l α,η , y m ∧ 2 ρα(g) ) > δ.

Then we can find two chains (η = η 0 , η 1 , . . . , η l 1 ) and (η = η 0 , η 1 , . . . , η l 2 ) such that d(gη j , gη j+1 ) = d α (gη j , gη j+1 ) = d α (gη, gη ) + O(δ -2 βe -ακ(g) ),

(2.2.25)

where α = α(η j , η j+1 ) ∈ Π 1 and different j correspond to different roots; similarly for η . We also have that for all α ∈ Π d α (gη l 1 , gη l 2 ) ≤ βe -ακ(g) δ -2 , (2.2.26)

where β is the gap of g, that is β = γ(g) = max α∈Π {e -ακ(g) }.

Lemma 2.2.43 (Furstenberg). Let µ be a Zariski dense Borel probability measure on G. There exists a unique µ-stationary probability measure ν on the flag variety and its images in the projective spaces PV are the unique µ-stationary probability measures when V is an irreducible representation of G.

See [START_REF] Furstenberg | Boundary theory and stochastic processes on homogeneous spaces[END_REF], [START_REF] Benoist | Random Walks on Reductive Groups[END_REF]Proposition 10.1] for more details. In order to distinguish stationary measures on different spaces, we use ν V to denote a µ-stationary measure on PV .

Definition 2.2.44. Let µ be a Zariski dense Borel probability measure with exponential moment on G. The Lyapunov constant σ µ is defined as the average of the Iwasawa cocycle σ µ := G×P σ(g, η)dµ(g)dν(η).

Lemma 2.2.45. Let µ be a Zariski dense Borel probability measure with exponential moment on G. Then the Lyapunov constant σ µ is in a ++ , the interior of the Weyl chamber. Equivalently, for any simple root α, we have α(σ µ ) > 0.

The maximal positivity of Lyapunov constant in Lemma 2.2.45 is due to Guivarc'h-Raugi [START_REF] Guivarc | Frontiere de Furstenberg, propriétés de contraction et théoremes de convergence[END_REF] and Goldsheid-Margulis [START_REF] Gol | Lyapunov indices of a product of random matrices[END_REF]. See [START_REF] Benoist | Random Walks on Reductive Groups[END_REF]Corollary 10.15] for more details. Lemma 2.2.45 will be used to show that the action of G on P is contracting in Section 2.4.2, where the contraction speed is give by β = sup α∈Π {e -ασµ }.

In following proposition, we give the large deviation principle for the Cartan projection. We keep the assumption that µ is a Zariski dense Borel probability measure on G with a finite exponential moment. Proposition 2.2.46. For every > 0 there exist C, c > 0 such that for all n ∈ N and η ∈ P we have

(2.2.29)

See [START_REF] Benoist | Random Walks on Reductive Groups[END_REF]Thm 13.17] for more details.

Proposition 2.2.47. If (ρ, V ) is an irreducible representation of G, then for every > 0 there exist C, c such that for all x in PV and y in PV * and n ≥ 1 we have µ * n {g ∈ G| δ(x, y m g ) ≤ e -n } ≤ Ce -c n , (2.2.30)

See [BQ16, Prop 14.3] for more details. Attention, we need ρ to be proximal in Proposition 2.2.47. Here the representation is automatically proximal due to the splitness of G.

Proposition 2.2.48. For every > 0 there exist C, c such that for all η, η in P and n ≥ 1 we have

.2.32)

Fix h d+1 . By (2.3.6), F is a function on v j for 1 ≤ j ≤ m. Recall that v j are vectors in V j,η . Let

Now, we want to explain how to realize F 0 as a linear functional.

Lemma 2.3.14. Let F be a multi homogeneous polynomial of degree q = (q 1 , • • • , q d+1 ) ∈ (N) d+1 . Then F 0 (v 1 , • • • , v m ) := F ((X 1 ) 2 , • • • , (X d+1 ) 2 ) v j 2q j is a linear functional F 1 on the space V 0 = 1≤j≤m (Sym 2 V j ) ⊗q j , where X j is defined in (2.3.7).

Proof. Since F is a multi homogeneous polynomial, it is sufficient to prove that every monomial in F has the same property. By Definition 2.3.12, a monomial of F is of the form

with n ij ∈ N and 1≤i≤d+1 n ij = q j . The term Π v j 2q j is used to compensate v j in the denominator of X i j in (2.3.6). Now, by multiplying v j , we can view X i j as gh i v j with some coefficient. By (2.3.6) and ghv j 2 = ghv j , ghv j , the function (X i j ) 2 is a linear functional on Sym 2 V j . Hence Π 1≤i≤d+1 (X i j ) 2n ij is a linear functional on (Sym 2 V j ) ⊗q j . This is because if we have two linear functionals f 1 and f 2 on W 1 and W 2 , then f 1 f 2 is the linear functional on W 1 ⊗ W 2 given by f 1 f 2 (w 1 ⊗ w 2 ) = f 1 (w 1 )f 2 (w 2 ). Then by the same reason, the monomial Π i,j (X i j ) 2n ij is a linear functional on V 0 . In order to express the linearity of F 0 , we rewrite

where v j is in V j,η and F 1 is understood as a linear functional on V 0 .

Proof of Lemma 2.3.10. Recall β = max α∈Π e -ασµn . Let δ = e -2 n , where the constant 2 will be determined later depending on . We suppose that n is large enough such that δ ≤ 1/2. Because for small n, WNC can be obtained by enlarging the constant C.

Step 1: We take into account of measures. We want to reduce the condition of WNC on A n d to F , which is essentially a linear functional. For this purpose, we will bound the measure of small A n d by the measure of small F .

Lemma 2.3.15. Let f 1 , f 2 be two Borel measurable functions on a locally compact Hausdorff space X and m be a Borel probability measure on X. Then for c > 0

In order to control F/A n d (h d+1 , η), we take h d+1 which is η good, that means for every i in {1,

When | | is small enough depending on , by (2.4.54), (2.4.55) and (2.4.56)

(2.4.57)

We compute the major term, that is (g, h) in G n, . We want to use Theorem 2.1.7 to control this part with ϕ = θ(σ(g, η) -σ(h, η)) and a suitable r. For applying Theorem 2.1.7, we need that ϕ is (C, r) good, which will be accomplished by multiplying smooth cutoffs. The most important is G2 assumption (2.4.2), which will be verified with the help of Lemma 2.4.18. Hence we want that r vanishes when η does not satisfy Lemma 2.4.18 (2).

Let X g,h,α be the subset of P, defined as the set of elements which satisfy Lemma 2.4.18 (2) with

where

where δ α is defined to be δ α (η, ζ m g ) = δ(V α,η , y m ρα(g) ). The choice of τ α is sophisticate. We only need to keep in mind that they come from Lemma 2.4.18. Then e i|θ|ϕ r(η) equals e zσ(g,η)+zσ(h,η) f (gη)f (hη) on X g,h .

Lemma 2.4.25. Let 0 , 1 be given by Theorem 2.1.7. Let (g, h) be in G n, . With small enough depending on 0 and | | small enough depending on and 1 , for ϕ, r defined in (2.4.58) and (2.4.59) we have that ϕ is (|θ| 0 , r) good and c γ (r) ≤ |θ| 0 , |r| ∞ ≤ |θ| 1 /2 . By Lemma 2.4.25, we can fix a value of and the functions ϕ and r|θ| -1 /2 satisfy the condition in Theorem 2.1.7. (Theorem 2.1.7 still holds when r is a complex function) Hence Theorem 2.1.7 implies

(2.4.60)

The difference between A g,h and e i|θ|ϕ(η) r(η)dν(η) is bounded by

(2.4.68) By (2.4.67), (2.4.68) and the dominant convergence theorem, we have

(2.4.70) By (2.4.68), we have

(2.4.71) Combining (2.4.69), (2.4.70) and (2.4.71), we have the result.

Appendix

Two classic proofs in Section 2.4.4

In order to simply the notation, we abbreviate , θ to a, b.

Proof of (2.4.50). We need an idea of Guivarc'h Definition 2.5.1. We call the action of G on X is (µ, γ) contracting, if there exist

(2.5.1)

Now take β = α o . Since {χ α , α ∈ Π} is a basis of a * , there exists a simple root α such that χ α (H αo ) = 0. Hence by the fact that vectors of different weights are orthogonal, we have

Then we have d 0 (k, e) d 2 (k, e). The proof is finished.

Recall the definition of the sign function m of Section 2.2.5.

Lemma 2.5.5.

We have

Proof. Suppose that the angle between kv α and

The assumption d 0 (z, z ) ≤ 1 is equivalent to that for every simple root α, the angle θ is less than π/2, which is equivalent to m(z, z ) = e due to Lemma 2.2.19.

If m(z, z ) = e, then for every simple root α, the angle θ is less than π/2. Hence sin θ = 2 sin θ 2 cos θ 2 ≥ √ 2 sin θ 2 , which implies the result.

Corollary 2.5.6. The K-invariant Riemannian distance on P is equivalent to the distance defined in (2.2.12).

Proof. By P = P 0 /M and since the group M is a subgroup of K which preserves the distance, let d 2 also be the quotient Riemannian distance on P. By the same argument of the proof as in Lemma 2.5.3, it is sufficient to prove on a small neighbourhood of η 0 . For any two points η, η in this small neighbourhood, we can find z, z in P 0 such that π(z) = η, π(z ) = η and d 2 (z, z ) = d 2 (η, η ). Due to d 2 (z, z ) small, we see that d 0 (z, z ) is less than 1. Hence by Lemma 2.5.5, we have m(z, z ) = e and then

By Lemma 2.5.3, we have

The proof is complete.

Here we give a proof of G1 assumption (2.4.1) in the proof of Theorem 2.1.1 (Section 2.4.3).

Recall that V is an irreducible representation of G with a norm and with highest weight χ, and v 0 , u 0 are two unit vectors in V and θ α = q 2χ-α (v 0 ∧ u 0 ) for simple root α. Recall that ϕ(η) = u 0 ,v v 0 ,v for a nonzero vector v in V χ,η and η ∈ P. By (2.4.42), we

Chapter 3

Discretized Sum-product and Fourier decay in R n

Introduction

The purpose of this manuscript is to generalize a result of Bourgain to R n . This result deals with the Fourier decay of the multiplicative convolution of Borel probability measures on R.

If E is a metric space, we write B E (x, r) for a close ball centered at x of radius r. Vectors in R n are seen as column vectors. The product structure on R n is given by coordinate, that is for

For a Borel probability measure on R n , let µ k be the k-times multiplicative convolution of µ.

Theorem 3.1.1. Given κ 0 > 0, there exist , 1 > 0 and k ∈ N such that the following holds for δ > 0 small enough. Let µ be a probability measure on [1/2, 1] n ⊂ R n which satisfies (δ, κ 0 , ) projective non concentration assumption, that is Compared with the projective non concentration in Theorem 3.1.1, the assumption here is weaker. In multiplicative convolution, we need additionally that µ is not trapped in any affine subspace.

From the discretized sum-product theorem to the Fourier decay of multiplicative convolution can be found in [START_REF] Bourgain | The discretized sum-product and projection theorems[END_REF]. The analogue result for finite fields is established in [START_REF] Bourgain | Estimates for the Number of Sums and Products and for Exponential Sums in Fields of Prime Order[END_REF]. See also [START_REF] Green | Sum-product phenomena in F_p: a brief introduction[END_REF], where he gave a really clear treatment of the sum-product phenomenon in F p . The proof of Theorem 3.1.1 from Theorem 3.1.3 will be given in Section 3.3.

Notation

We will make use of some classic notation: For two real valued functions A and B, we write A = O(B), A B or B A if there exists constant C > 0 such that |A| ≤ CB, where C only depends on the ambient space. We also write

Discretized sum-product estimates in R n

The non concentration assumption in Theorem 3.1.1 is a little different from that in [START_REF] Bourgain | The discretized sum-product and projection theorems[END_REF], but the two assumptions are equivalent up to constants. Lemma 3.2.1. Let 1 > δ > 0. Let ν be a Borel probability measure on R. We have two non concentration assumptions.

(1) (δ, κ 1 , 1 ) For ∀ρ ≥ δ, we have ν(B(a, ρ)) ≤ δ -1 ρ κ 1 .

(2) (δ, κ, ) For ρ ∈ [δ, δ ], we have ν(B(a, ρ)) ≤ ρ κ . Then (2)(δ, κ, ) implies (1)(δ, min κ, 1, ) and if κ 1 > 2 1 , we have that (1)(δ, κ 1 , 1 ) implies (2)(δ, κ 1 /2, 2 1 /κ 1 ).

Proof.

(2) ⇒ (1) For ρ < δ , it is obvious. For ρ > δ , we use the trivial bound

Hence (2) implies that (1) holds for ( 1 , κ 1 ) = ( , min{κ, 1}).

(1) ⇒ (2) We want to find ( , κ) such that (2) holds. Let ρ = δ t . That means

Due to κ 1 > 2 1 , we can take ( , κ) = (2 1 /κ 1 , κ 1 /2).

Basics of discretized sets

Before proving our results, we recall some elementary and known results in the discretized setting. Let δ > 0 be the scale. Let K ≥ 2 be a roughness constant. Two quantities bounded by a polynomial of K is considered as equivalent.

Lemma 3.2.7. Let f be a K-Lipschitz function from R n to R n . Let A be a bounded subset of R n . We have

(3.2.3) Definition 3.2.8. For a bounded subset A of R n , we denote by A (δ) the δ-neighborhood of A, given by

Lemma 3.2.9. Let A be a bounded subset of R n . Let à be a maximal δ-separated subset of A, that is different elements of à have distance at least δ and à is maximal for inclusion.

where |A| denotes the volume of A and # Ã denotes the number of elements of Ã.

Definition 3.2.10 (Ruzsa distance). Let A, B be two bounded subsets of R n . We define the Ruzsa distance of A, B at scale δ by

This is not a real distance. It measures the additive structure of A and B.

Lemma 3.2.11 (Ruzsa triangular inequality). Let A, B, C be three bounded subsets of

(3.2.5)

The above inequality (3.2.5) is roughly a triangular inequality for the Ruzsa distance d δ .

Lemma 3.2.12 (Plünnecke-Ruzsa inequality). Let A, B be two bounded subsets of R n . If N δ (A + B) ≤ KN δ (B), then for k, l in N we have

In [START_REF] He | Orthogonal projections of discretized sets[END_REF], He explains how to deduce the discretized version from the discrete version of the above two lemmas. For the discrete version, please see [START_REF] Tao | Additive combinatorics[END_REF]. The main ingredient of proof is the Ruzsa covering lemma. Definition 3.2.13. Let A, B be two bounded subsets of R n . We define the doubling constant of A at scale δ by

Lemma 3.2.14 (Ruzsa calculus). Let A, B, C be three bounded subsets of R n . Then

The proofs are direct applications of the Ruzsa triangular inequality and the Plünnecke-Ruzsa inequality. For the discrete version, please see [START_REF] Tao | Additive combinatorics[END_REF] and the second note of Green in [Gre]. The first and second statements says that the Ruzsa distance is symmetric and transitive. The Ruzsa calculus will be used to prove Proposition 3.3.9 (Additive-Multiplicative Balog-Szemerédi-Gowers theorem).

The additive energy: the discrete case

We first introduce the additive energy in the discrete case. Let A, B be two finite sets in an abelian group G. We define the additive energy ω(+, A × B) as the number of the quadruplet (a, b, a , b

We also have a formulation with 2 norm

where the measure in defining 2 norm is the counting measure. From the definition, by Young's inequality, we have

where |A| denotes the number of elements in A. The additive energy is important because it reflects the additive structure of A and B.

which is robustly large with respect to the optimal value of ω(+, A × B) (3.2.7). (See [START_REF] Tao | Additive combinatorics[END_REF] and [START_REF] Green | Sum-product phenomena in F_p: a brief introduction[END_REF] for more details).

The additive energy: the continuous case

We now define the discretized version of the additive energy. On a Cartesian product X × Y of metric spaces, we use the distance defined by

where x, x are in X and y, y are in Y .

Definition 3.2.15 (Energy of a map). Let X, Y be two metric spaces, and let ϕ be a Lipschitz map from X to Y . For a subset C of X, the energy of ϕ at scale δ is defined by

(3.2.9) Lemma 3.2.16. Let ϕ be a K-Lipschitz map from R m to R n , and let C be a bounded subset of R m . Then (i) We have

We have a formulation with L 2 norm (see [BISG17, Appendix A.1] for example. This is also the discretized version of (3.2.6).) We have an inequality

(3.2.12) Lemma 3.2.16 (i) implies that

This means that when two sets A, B have additive structure then the additive energy is relatively large. The additive energy is powerful when combined with the following proposition, a partial converse to (3.2.14), which says that if two sets have relatively large additive energy, then there exist large subsets which have additive structure. Proposition 3.2.17 (Balog-Szemerédi-Gowers). [Tao08, Theorem 6.10] Let A, B be two bounded subsets of R n such that

Then there exist subsets A , B of A, B such that

and

Sum-product estimates in R n

We first state the discrete version of the growth under a ball.

Lemma 3.2.18. [Gre09, Lemma 3.1] Let p be a prime number. If X is a subset of F p , then sup

The proof is by calculating the additive energy in two ways. Suppose that the result does not hold, then the additive energy ω(+, X × aX) is large for every a in F p . But the sum of the additive energy ω(+, X × aX) with respect to a in F p is small, which gives the contradiction.

The continuous version uses a Fubini type argument to study the growth under a ball in

Lemma 3.2.19. Given κ > 0, σ ∈ (0, n), there exists > 0 such that for δ sufficiently small the following holds. Let X be a bounded subset of R n such that for j = 1, . . . n

Remark 3.2.20. We follow closely the proof of [He, Theorem 3]. To prove the stronger version, we need another lemma, which is a reducible version of [START_REF] He | Discretized Sum-Product Estimates in Matrix Algebras[END_REF]Prop.29]. The proof is essentially the same as the irreducible version, with the estimate of small balls replaced by thin cylinders.

This can be proved by the following standard argument. Choose a maximal subset

For all b in R, by (3.2.15) and hypothesis

Let µ be the normalized Lebesgue measure on B R n (id, 1/2) with total mass 1, and let a be a random variable following the law of µ. Define ϕ a : R n × R n → R n by ϕ a (x, y) = x + ay.

which is also

.

By the Jensen inequality on the function t

.

(3.2.17)

Therefore it is sufficient to give a bound that E(ω δ (ϕ a , X × X)) δ N δ (X) 3 . By Lemma 3.2.16 (ii), letting X be a maximal δ-separated subset of X, we have

where a is contained in B R n (id, 1/2) and K = 2. Let ρ be a parameter to be fixed later. We distinguish two cases

• Otherwise, the number of pairs (y, y ) such that min j |y j -y j | < ρ can be bounded using (3.2.16) and (3.2.4)

(3.2.20) Moreover, we have for all x, y, y ∈ X,

When is sufficiently small, we have E(ω δ (ϕ a , X × X)) N δ (X) 3 δ , which finishes the proof.

Before proving Theorem 3.1.3, we need to introduce S δ , the set of "good elements". Let A be a bounded subset of R n . Let

The following lemma says that S δ (A, K) has a "ring structure".

).

(See [He, Lemma 30] and [BKT04, Proposition 3.3] for more details)

Proof of Theorem 3.1.3. The idea is to use Proposition 3.2.5 to force A to grow to a fat ball. Then Lemma 3.2.19 implies the growth of regularity under the action of a fat ball. Assume that the result fails. That is for every > 0, there exist A, X satisfying the assumptions of Theorem 3.1.3 such that A ⊂ S δ (X, δ -).

(3.2.22)

We will reach a contradiction when is small enough depending on κ, σ. By Proposition 3.2.5, for every 0 > 0, there exist s ∈ N and 1 > 0 depending only on 0 and κ, such that if

(3.2.23) By Lemma 3.2.21 (ii) with K = δ -and (3.2.22), we have

By Lemma 3.2.21 (i) with K = δ -O(s) and (3.2.24), (3.2.23)

By Lemma 3.2.21 (iii) with K = δ -O(s) -0 , a = δ 0 2 id and (3.2.25), we have

Again by Lemma 3.2.21 (ii), using product and (3.2.25), (3.2.26), we obtain

By Lemma 3.2.19, there exists 2 > 0 depending only on σ and κ, such that when < 2

Taking 0 sufficiently small, and then taking sufficiently small such that O(s) + O( 0 ) < 2 , we get a contradiction from (3.2.27) (3.2.28)

The proof is complete.

Application to multiplicative convolution of measures

Notation: For a measure µ on R n , let µ -be the symmetry of µ, that is µ -(E) = µ(-E) for any Borel set E of R n . Let µ (r) be the r-times additive convolution of µ. Recall that µ k is the k-times multiplicative convolution of µ. For an element x in R n , we write x j for its j-th coordinate, j = 1, 2, • • • , n. We use the norm induced by the standard scalar product on R n , that is to say for

All vectors x, ξ in R n are column vectors, and •, • is the inner product. For y in R n and measure ν on R n , let (m y ) * ν be the pushforward measure of µ by the multiplication action of y, that is (m y ) * ν(E) = ν(y -1 E). In order to simplify the notation, we abbreviate B R n (0, R) to B(0, R). For a function f on R n , we write f p , p = 1, 2, ∞, for its L p norm on R n .

Let

|B(0,δ)| , where | • | is the Lebesgue measure of a Borel set in R n . Let ν δ = ν * P δ , which is an approximation of ν at scale δ.

L 2 -flattening

Lemma 3.3.1 (L 2 -flattening). Given σ 1 , κ > 0, there exists = (σ 1 , κ) > 0 such that the following holds for δ small enough. Let ν be a symmetric Borel probability measure on

We also need the following inequality, which is an inverse Chebyshev's inequality. Its proof is elementary. Lemma 3.3.8. Let K > 0. Let ν be a probability measure on a measure space X. Let f be a nonnegative function on X.

Here is the main idea of the proof of L 2 -flattening: Suppose that (3.3.2) fails. By (3.2.12), we can obtain two sets with large additive energy from the convolution of its character function. Hence we can find some sets in the support of ν δ with large additive energy. Together with Balog-Szemerédi-Gowers theorem (Proposition 3.2.17), this produces two sets which violate sum-product estimates (Theorem 3.1.3).

Proof of L 2 -flattening (Lemma 3.3.1). We follow closely the proof of [BdS16, Lemma 2.5]. Proof by contradiction: Assume that the result fails. Then for every > 0, there exist δ small and a measure ν satisfying

(3.3.3)

We will reach a contradiction for sufficiently small. Lemma 3.3.6, (3.3.3) and Cauchy-Schwarz's inequality imply

(y).

There must exist i, j such that

With the same argument as in [BISG17, Appendix A.2], we can conclude that i, j > 0.

If j > 0, by Lemma 3.3.6, then 2 j 1 X j 1 ν 3δ 1 = 1. Therefore, for j ≥ 0 and y ≤ δ -, by Young's inequality

which contradicts to (3.3.4) if is sufficiently small with respect to σ. Similarly, we obtain j > 0.

Therefore, Lemma 3.3.6 implies

, and similarly for j,

where the last inequality is due to Lemma 3.3.7. Hence by Young's inequality, for every y in the support of ν

(3.3.6) where det y is the determinant of y seen as an endomorphism of R n , that is

Then we take a set B such that for every y in B we have that

2 and (3.3.4), (3.3.6) we have

We verify that X i , X j and B satisfy some natural assumptions. Take y in B. By (3.3.4) and Young's inequality, we have Next, (3.3.5) and (3.3.9) also imply

We have

Since X i is an essentially disjoint union of δ balls, we have

. By (3.3.11) and (3.3.12) we have

(3.3.13) By Lemma 3.3.6(1), the projective non concentration and (3.3.13) for ρ ≥ δ, a ∈ R and

This means that X i inherits non concentration from ν.

We calculate additive energy. By (3.2.12) we have

Then for every y in B, by (3.3.7), (3.3.5), (3.3.11) and (3.3.13)

We can use the following proposition, which is a uniform version of the Balog-Szemerédi-Gowers theorem, inspired by the version on finite field F p due to Bourgain.

Proposition 3.3.9 (Additive-Multiplicative Balog-Szemerédi-Gowers theorem). Let K > 2 be the roughness constant, let X, X , B be bounded subsets of R n in B(0, K), with B -1 bounded by K (if b ∈ B then |b j | ≥ 1/K for j = 1, . . . n), and let µ be a Borel probability measure on B. If for every b ∈ B we have

Then there exist

Take K = δ O( ) , µ = 1 ν(B) ν| B , X = X i and X = X j . By (3.3.10), the set B satisfies the assumption in Proposition 3.3.9. Take B(1, 2r) ⊂ U as in Theorem 3.1.3 with the group G = (R * ) n , V = R n . Proposition 3.3.9 implies that for δ small enough that δ ≤ r there exist C 1 > 0,

(3.3.17) Lemma 3.3.10. There exists C 2 > 0. These sets b -1 o B 1 , X o satisfy the (δ, κ, σ 1 , C 2 ) assumption of Theorem 3.1.3 when δ is small enough. Proof. By Proposition 3.3.9, the set

Due to (3.3.20) and (3.3.21), we have

together with (3.3.29), we obtain

The proof concludes by Ruzsa calculus. By Lemma 3.2.14(1) and (3.3.20), we have

.

By (3.3.31) and (3.3.28), we have

. By Lemma 3.2.14(3), we have X bo ≈ K O(1) X b and X bo ≈ K O(1) X b , the latter implies bX bo ≈ K O(1) bX b . Therefore by Lemma 3.2.14(2)

We get

The proof is complete.

Proof of the Fourier decay of multiplicative convolutions

Using L 2 -flattening (Lemma 3.3.1), we give a proof of Theorem 3.1.1. The strategy is to apply L 2 -flattening to

We need a lemma which explains the connection of ν δ 2 and the Fourier transform of ν Lemma 3.3.14. Let δ > 0, C > 1 and let δ 1 = 2δ/C. Let ν be a Borel probability measure on R n with support in B(0, C). We have

The proof of Lemma 3.3.14 will be given at the end of this section.

Recall that µ k is the k-times multiplicative convolution of µ. We have

For k, r ∈ N, let σ k,r be the real number defined by

where µ k,r = (µ k * µ - k ) * r . The remainder of the proof is to control σ k,r , divided into two steps. We first prove that if σ k,r is not sufficiently small, then L 2 -flattening (Lemma 3.3.1) reduces the value of σ k,r . When σ k,r is sufficiently small, the Hölder regularity of µ enables us to finish the proof. This can be understood that if a measure µ satisfies non concentration assumption, then after sufficient multiplicative and additive convolutions, the sum-product phenomenon implies that µ k,r is much more smooth.

Proof of Theorem 3.1.1. Let

where (κ 1 /2, κ 0 ) is given in L 2 -flattening (Lemma 3.3.1).

Reducing the value: We have a consequence of L 2 -flattening (Lemma 3.3.1), whose proof will be given later.

Lemma 3.3.15. Under the assumption of Theorem 3.1.1, if σ k,r ≥ κ 1 , then for δ small enough depending on k, r, we have σ 2k,r ≤ σ k,r -, where r = 8r 2 + 4r.

Sufficient regularity:

We have a higher dimensional version of [Bou10, Theorem 7], which says that if two measures have sufficient Hölder regularity, then the multiplicative convolution of these two measures has power decay in its Fourier transform. Lemma 3.3.16. Let α > β > 0 and δ > 0. Let µ be a measure on B(0, 1) such that for j = 1, . . . , n sup

Let K > 2 be a parameter. Let ν be a compactly supported measure on B(0, K) such that

The proof of Lemma 3.3.16 is classic and will be given at the end of Section 3.3.2 for completeness.

If σ 1,1 ≥ κ 1 , iterating Lemma 3.3.15 several times implies that σ k,r < κ 1 , where k, r only depend on κ 1 .

We will now apply Lemma 3.3.16 to a well-chosen measure. Take (µ k * µ - k ) (r) as ν, α = κ 0 -, β = κ 1 and τ = α-β n+2 . For ξ ∈ [δ -1 /2, δ -1 ], by Hölder's inequality and Lemma 3.3.16,

When δ is small enough, this yields (3.1.2) with

where the last two inequalities are due to (3.3.35) and r only depends on κ 0 . Now we will prove Lemma 3.3.15, where we use the L 2 -flattening (Lemma 3.3.1).

Proof of Lemma 3.3.15. Fix k, r and set

This is the key construction of this proof. The measure ν is the bridge to connect µ 2k and µ k . We summarize the properties of ν in the following lemma.

Lemma 3.3.17. The measure ν satisfies (δ/r, κ 0 , 2 ) projective non concentration assumption when δ is sufficient small depending on k, r.

Proof. Projective non concentration property is invariant under addition. That is if a probability measure m satisfies projective non concentration, then m * m also satisfies projective non concentration for any probability measure m . The reason is the following calculation. By Fubini's theorem, we have

Hence we can drop the additive convolution, and for ρ ≥ δ 1 , we have

(3.3.39) The property that the support of µ is contained in [1/2, 1] n and the projective non concentration of µ imply the left hand side of (3.3.39) is less than

where we have used rρ ≥ rδ 1 = δ for projective non concentration and the last inequality holds for δ small enough depending on k, r. Then (3.3.1) follows from (3.3.39) and (3.3.40). The measure ν satisfies non concentration with (κ 0 , 2 ) at scale δ 1 .

Therefore (3.3.41) follows from

|μ(ξ)| 4r dξ.

By (3.3.44), Hölder's inequality and (3.3.43)

.

By the Plancherel theorem and Hölder's inequality, the above inequality becomes

Therefore, by (3.3.45) and (3.3.43) Combined with (3.3.46), we obtain (3.3.42).

Lemma 3.3.17 and Lemma 3.3.19 enable us to decrease the parameter σ k,r by L 2flattening (Lemma 3.3.1).

We return to the proof of Lemma 3.3.15. By (3.3.41) and the hypothesis σ k,2r ≥ κ 1 , we have

(3.3.47)

Due to suppν ∈ [-r, r] n and (3.3.47), taking C = r in Lemma 3.3.14, we have

When δ is small enough depending on k, r, κ 1 , we have

Lemma 3.3.17 implies that ν satisfies assumption of L 2 -flattening lemma with σ 1 = κ 1 /2, κ = κ 0 at scale δ 1 . Also notice that (3.3.35) implies 2 ≤ (κ 1 /2, κ 0 ). Then L 2 -flattening (Lemma 3.3.1) implies

Using Lemma 3.3.17, we obtain

Therefore we have

with some constant C k,r > 0. For δ small enough, it follows that δ 2k,r ≤ σ k,2r -.

It remains to prove Lemma 3.3.14 and Lemma 3.3.16.

Proof of Lemma 3.3.14. Recall that δ = 2Cδ 1 . We observe that the Fourier transform of P δ satisfies P δ (ξ) = P δ (x)e i ξ,x dx = P 1 (x/δ)δ -n e i ξ,x dx = P 1 (δξ).

Due to P 1 (ξ) = B(0,1) e i ξ,x dx|B(0, 1)| -1 = B(0,1) cos( ξ, x )dx|B(0, 1)| -1 , we see that P 1 is positive for ξ ∈ B(0, 1).

(3.3.51)

We are going to prove (3.3.32). By (3.3.51), we have P 1 (δ 1 ξ) 1 for ξ in B(0, 1/δ 1 ), which implies

For the other direction of (3.3.32), let δ 2 = 2δ = 4Cδ 1 . Due to 1/δ 2 + 1/δ 2 = 1/δ, we have P 1/δ P 1/δ 2 * P 1/δ 2 , which implies

Therefore by (3.3.55) and (3.3.57)

If we take t = R -α-β n+2 , then the result follows from (3.3.56) and (3.3.58).

Appendix

The main purpose of the Appendix is to give a version of Theorem 3.1.1 (Proposition 3.4.4) for its application to the random product of matrices.

In the application, we need to vary the measure. Using the same idea as in [BD17, Propostion 3.2], we have a version for several different measures (Proposition 3.4.2). The measures appearing in the random product of matrices are not compactly supported, hence we will relax the assumption on support in Proposition 3.4.4. Proposition 3.4.1. Fix κ > 0. Then there exist k ∈ N, > 0 depending only on κ 1 such that the following holds for τ large enough. Let λ be a Borel probability measure on

(3.4.1)

Proof. By Lemma 3.2.1, Theorem 3.1.1 implies the result.

We state a version with different measures.

Proposition 3.4.2. Fix κ > 0. Then there exist k ∈ N, > 0 depending only on κ 1 such that the following holds for τ large enough. Let λ 1 , . . . λ k be Borel measures on [ 1 2 , 1] n ⊂ R n with total mass less than 1. Assume that for all ρ ∈ [τ -1 , τ -] and j = 1, . . . , k

Proof. The proof is the same as the argument in [BD17, Propostion 3.2]. For completeness, we give a ketch here. We first verify that if the mass of the measure λ is less than 1, the result also holds. Let 2 be given by Proposition 3.4.1 when the regular exponent equals κ/2. We distinguish two cases Proposition 3.4.4. Fix κ 0 > 0. Let C 0 > 0. Then there exist 2 and k ∈ N depending only on κ 0 such that the following holds for τ large enough depending on C 0 , κ 0 . Let λ 1 , . . . λ k be Borel measures on R n supported in ([-τ 3 , -τ -3 ] ∪ [τ -3 , τ 3 ]) n with total mass less than 1, where 3 = min{ 2 , 2 κ 0 , 1}/10k. Assume that for all ρ ∈ [τ -2 , τ -2 ] and j = 1, . . . , k

Then for all ς ∈ R n , ς ∈ [τ 3/4 , τ 5/4 ] we have

Remark 3.4.5. The proof is tedious, but the idea is clear. If the non concentration assumption is valid in some large range, then there is some place to rescale a little the measure and the result still holds. We only need to find some exponent 3 carefully.

Proof. It is sufficient to prove the case that suppλ j ∈ [τ -3 , τ 3 ] n . Because we can divide each measure into λ j = m∈(Z/2Z) n λ m j , where λ m j is the unique part of λ j whose support is in the same orthant as m and we identify

We know that the support of measure (m j ) * λ j is in [τ -3 , τ 3 ] n . Hence by the result of the case suppλ j positive, we have the result with a constant 2 nk . Let as in Proposition 3.4.2 with κ = κ 0 /2, and let 2 = /4.

], by (3.4.5) we have

for τ large enough depending on C 0 .

Summing up over l ≤ 3 log 2 τ , we have

The assumption of Proposition 3.4.2 is verified by (3.4.6) with τ replaced by τ 1 . Therefore

when τ is large enough depending on k, n, . The proof is complete.

Proof. Applying Lemma 4.2.1 with u = f /ϕ implies

The proof is complete.

Remark 4.2.3. Compared with [RT15, Proposition 3.2], we keep the last term ϕ 2 ∇(f /ϕ) 2 . This term is important and will be exploited in Section 4.2.2.

The following proposition says that for a Riemannian manifold with quasi-warped product structure, the derivative of the volume density along the vertical geodesic can give us a good control of the bottom of the spectrum. Proposition 4.2.4. Let M = L × R be a Riemannian manifold with the metric given by g = dt 2 + g(x, t), (4.2.2)

where g(x, t) is a metric on L × {t}. We call this a quasi-warped product metric. The Riemannian volume element can be written as ω = dt × h(x, t)dx, where dx is the volume element on L × {0} and h is a density function from L × R to R ≥0 . Assume that for some λ > 0 and a nonnegative function c on R, we have

Remark 4.2.5. The terminology quasi-warped product is a generalization of warped product in Riemannian geometry. Let (M 1 , g 1 ), (M 2 , g 2 ) be two Riemannian manifolds. The warped product M 1 × f M 2 is the product manifold M 1 × M 2 equipped with the warped product metric given by g = g 1 + f g 2 , where f is a positive function on M 1 .

Proof. Let ϕ(x, t) = e -λt . We will prove that -∆ϕ ≥ λ 2 (1 -2c(t))ϕ on L × R. It is sufficient to prove this inequality locally. Take a local chart on L by φ :

is homeomorphic to a closed ball of the same dimension. Let ξ be a point in X I , and let x, y be two points in X. We define the Busemann function by

where d is the distance induced by the Riemannian metric, and γ(t) is a geodesic ray asymptotic to ξ. Let o be a fixed reference point in X. The level sets of b ξ (•, o) are called horospheres based at ξ.

We introduce the horospherical model for X following [Pan89, Section 9]. Fix a point ∞ of X I . Let G = KAN be the Iwasawa decomposition with A and N fixing ∞, and K, a maximal compact subgroup, fixing o. The group A is isomorphic to R and N is a simply connected nilpotent Lie group.

Here M is also the maximal subgroup of K which commutes with A. Let N be the Lie algebra of N . The group A normalizes N , and the conjugation action of A induces an automorphism on N . We make a choice of a particular generator of A such that N admits a decomposition

where V j equals to Ker(Ad(a t ) -e jt ) for t = 0 and a t is an element in A. Moreover, V 2 is the centre of N and

and the Lie algebra structure is given by

Starting from V 1 ⊕ V 2 , we obtain two left invariant distributions W 1 , W 2 on N . Let α be the endomorphism on N , which is the differential of Ad(a t ) at t = 0, mapping vector v in V j to jv. Let e α be the induced automorphism on N . Let c(t) be the geodesic ray starting from o to ∞, that is c(t) = a t o. We have a diffeomorphism from N × A to X given by (ν, t) → ν • c(t), and the action of A reads as a t (ν, s) = (e tα ν, t + s). Let m = (n -1) dim R F and q = dim R F -1. The hyperbolic metric on X can be written as

where these left invariant metrics g t on N have, under the distribution W 1 , W 2 , a matrix of the form e 2t Id m 0 0 e 4t Id q .

We also need to calculate the Riemann curvature tensor. Let n be the unit tangent vector at o to the geodesic c(t).

We introduce a half space model. As in the horospherical model, with o in X and ∞ in X I fixed, let y = e -t . The coordinate map is replaced by a map from N × R >0 to We will apply Proposition 4.2.4 to M (Π). Since Π preserves the metric g t , we have a quotient metric on Π\N . The formula (4.4.1) implies that the metric on (Π\N ) × R is a quasi-warped product metric, and the volume element is equal to e 2 t dtdη, where dη is the volume element on (Π\N ) × {0}. So the function h in Proposition 4.2.4 equals e 2 t , and h satisfies (4.2.3) with λ = and c(t) = 0. Applying Proposition 4.2.4 and Corollary 4.2.9, we have Lemma 4.4.2. Let E be as in (4.4.3). For a function f ∈ C ∞ c ((Π\N ) × R), we have

) 2 e -2 t dvol. (4.4.4)

Given C > 0, there exist a compact interval I ⊂ R and a constant C 1 > 0 such that for all compact set K in Π\N we have

e -t f 2 dvol. (4.4.5)

Convex subsets and the normal exponential map

We need a lemma which says that in the normal exponential coordinate, the Riemannian manifold has a quasi-warped product metric. This lemma is similar to the Gauss lemma, where the hypersurface S degenerates to a point. Lemma 4.4.3. Let S be a smooth hypersurface of a Riemannian manifold M . Let exp ⊥ : S ×R ≥0 → M be the normal exponential map given by exp ⊥ (x, t) = exp x (tn(x)). Assume exp ⊥ is an embedding. Then for every x in S and s ≥ 0, the curve γ x : t → exp x (tn(x)) is normal to the hypersurfaces exp ⊥ (S × {s}). Definition 4.4.4. Let M be a complete Riemannian manifold, and let D be a closed subset of M . We call D geodesically convex if the preimage D of D in the universal cover M is convex, that is for any two points x, y in D there exists a unique minimizing geodesic contained in D which connects x, y.

Let M be a rank one locally symmetric manifold such that its universal cover has exponent of growth 2 . Let D be a geodesically convex closed subset of M with non empty interior and with smooth boundary. Let S = ∂D, and let exp ⊥ : S × R ≥0 be the outer normal exponential map given by exp ⊥ (x, t) = exp x (tn(x)). Assume that exp ⊥ is a diffeomorphism from S × R ≥0 to M -D. By Lemma 4.4.3, the metric can be written as in (4.2.2). Let h be the density function defined as in Proposition 4.2.4. Lemma 4.4.5. With the above assumption, if we have an upper bound on the second fundamental form on S, then there exists C 1 > 0 depending on the bound such that the density function h satisfies the following inequalities for every x in S, and every t ≥ 0:

The energy form

We keep the assumptions on M of Sec.4.5.2, that M is a geometrically finite locally symmetric manifold of real rank one with at most one cusp. Recall that E is the energy form defined in (4.4.3).

Lemma 4.5.10. Let f ∈ C ∞ c (M ), and let ϕ be a smooth function. We have

This is a direct consequence of Lemma 4.2.1. The following proposition says that in order to calculate the energy form on the entire manifold, it is sufficient to calculate the energy forms on M 1 , M 2 , M 3 and an error term. We want to separate the energy formula, but we need a partition of unity such that the square root of the partition function is also smooth. The exact choice of the partition function is not important and we take θ = π 2 φ1 , ϑ = π 2 φ2 and ϕ 1 = sin θ, ϕ 2 = sin ϑ. Proposition 4.5.11. For f ∈ C ∞ c (M ), we have

(4.5.2)

Proof. By Lemma 4.5.10, we have

Since θ + ϑ = π/2 outside of M 3 , we have for x ∈ M c 3 ϕ 2 ∆ϕ 2 + ϕ 1 ∆ϕ 1 = sin θ∆ sin θ + sin ϑ∆ sin ϑ

The proof is complete.

We want to prove that E(ϕ 1 f ) and E(ϕ 2 f ) are positive after adding an integral over a compact subset, and to give an estimate of the error term M c 3 ∇θ 2 f 2 .

Positivity of the energy form

In this part, we take into account the topology of the whole manifold, together with the results in standard cusp regions and the complement of convex subsets, to prove the positivity.

Proposition 4.5.12 (the Lax-Phillips inequality). Let M be a geometrically finite locally symmetric manifold of real rank one. There exist a relatively compact open set U in M with smooth boundary and a constant C U > 0 such that the following holds. For any compact set V in M there exists V > 0 such that for all complex valued function f ∈ C ∞ c (M ) we have

Proof of Theorem 4.1.1. Our main theorem of this manuscript follows from Proposition 4.5.12 and 4.3.1 with c(x) = 2 .

It remains to prove Proposition 4.5.12.

Remark 4.5.13. For the simplicity of the exposition, here we will only prove the case that M has only maximal rank cusps (M may have no cusp). For the general case, please see the appendix. The idea of the proof is the same, but the appearance of non-maximal cusps will add some technical difficulties.

It is sufficient to prove this inequality for real valued functions. The complex version is immediate by separating f = f 1 + if 2 with f 1 , f 2 real valued and using the real version for each component f 1 , f 2 .

Let ρ be the nearest point retraction from X to W . We can extend this map continuously to X ∪ Ω, such that if ξ is in Ω, then ρ(ξ) is the first point of contact of W with an expanding family of horoballs based at ξ. (See [Bow95, Lem.2.2.4] for more details.) This retraction descends to a map

which is also continuous by the openness of the covering map.

In the cusp region, recall that we will use the half space model (Π\N ) × R ≥0 for M c (Π). We write M c (Π) = (Π\N ) × R ≥0 , which means the equality holds under the coordinate map. Others equalities are similar. By definition, O is isometric to a subset of M c (Π), where Π is a discrete parabolic subgroup. Hence we can identify the two sets. By Definition 4.5.1 and Proposition 4.5.2, the complement of the cusp region O in M c (Π) is relatively compact. Hence we can suppose that (O ) c is contained in (Π\N ) × [0, 1]. (Otherwise, we can change the coordinate.) Let H be the quotient of a horoball based at ∞, defined by H = (Π\N ) × [1, ∞) ⊂ O . We define proj H to be the nearest point retraction from M c (Π) to H. For a point (η, y) ∈ M c (Π) with y < 1, the map is given by proj H (η, y) = (η, 1). The last assertion is due to Proposition 4.5.7.

For x in M , let t 1 (x) = d(x, H), t 2 (x) = d(x, D). (4.5.4) Later we will see that t 1 , t 2 are the geometric descriptions of the coordinate t in cusps and the complement of convex set. Recall that M 1 , M 2 are subsets of M c , which may intersect the ideal boundary M I .

Lemma 4.5.15. In the standard cusp region, there exist a compact set U 1 in M 1 ∩ M and a constant C 1 > 0 such that the following holds. For any compact set

The proof is complete.

Lemma 4.5.16. In the complement of the convex core, there exist a compact set U 2 in M 2 ∩ M and a constant C 2 > 0 such that the following holds. For any compact set V in M 2 ∩ M there exists V > 0 such that for all f ∈ C ∞ c (M 2 ∩ M ) we have

Proof. We first verify that D satisfies the conditions in Lemma 4.4.5.

Recall that D = Γ\W , where W is a convex subset of X with smooth boundary. By convexity, the normal exponential map, given by exp x (tn(x)), is a diffeomorphism from ∂W × R ≥0 onto X -W , and satisfies t = d(exp x (tn(x)), W ). With the help the nearest point retraction ρ, the inverse of the normal exponential map from X -W to ∂W × R ≥0 is given by x → (ρ(x), d(x, W )).

Descend to the quotient space. Let S be the boundary of D. Then the normal exponential map exp ⊥ : (x, t) → exp x (tn(x)) from S × R ≥0 to M -D is again a diffeomorphism, and

The upper bound of the second fundamental form is due to [PP12, Thm.1, Prop.6], that is the obtained convex set W has bounded second fundamental form on its boundary. Applying Lemma 4.4.8 with the set K = ρ(M 2 ), defined in Lemma 4.5.14, and the constant C = 2(4 2 + 1), there exists a bounded interval I ∈ R ≥0 such that (4.4.9) holds for U 2 = K × I. Adding (4.4.8) and (4.4.9) implies the result.

Proof of Proposition 4.5.12. In view of (4.5.2), the main problem is the term M c 3 ∇θ 2 f 2 dvol. The support of ∇θ is contained in O -O , which may not be compact. But with the hypothesis that the manifold has only maximal rank cusps, the region O -O is already relatively compact in M due to Lemma 4.5.14. Because O does not intersect the ideal boundary and the complement of O is relatively compact. Hence ∇θ 2 is a bounded function supported on O -O .

The compact set V is replaced by V ∩ M 1 and V ∩ M 2 in Lemma 4.5.15 and Lemma 4.5.16. Using Lemma 4.5.15 and 4.5.16, we obtain U 1 , U 2 . Since O -O , M 3 , U 1 , U 2 are relatively compact in M , we can find a relatively compact open set U ⊂ M with smooth boundary, which contains the four sets. By (4.5.2), (4.5.5) and (4.5.6), there exists a constant C 4 large enough such that

where we use the estimate sup x∈U ∇ϕ 2 (x) < ∞, thanks to the relative compactness of U . In the standard cusp region we have the same estimate. Therefore taking C U large enough, we have (4.5.3).

Appendix

As stated in Section 4.5.4, we will give a proof of Proposition 4.5.12 without the assumption that M has only maximal rank cusps.

Compactification and estimate at infinite

Let g be a Riemannian metric on a manifold M . We define the musical isomorphism as follows (see [START_REF] Gallot | Riemannian geometry[END_REF] for more details). For a vector X in T x M , let X b be the unique 1-form such that X b (v) = g(X, v) for every v ∈ T x M . This isomorphism gives a dual tensor field (the symmetric covariant 2-tensor fields) g * of g, and (∇f ) b = df .

We will consider the compactification of a Riemannian manifold M = L × (0, 1] with metric given by g = g 1 (x, y)/y 2 , where g 1 (x, y) is a positive definite symmetric bilinear form on T (x,y) M . Now we add y = 0 to obtain a differential manifold with boundary, called M . By definition, we have (using local coordinate vectors ( ∂ ∂x i ) 1≤i≤n and (dx i ) 1≤i≤n , g * can be written as the inverse matrix of g) g * = (g 1 /y 2 ) * = y 2 g * 1 .

(4.6.1) Suppose that g * 1 can be smoothly extended to y = 0, but g * 1 (x, 0) may degenerate to a positive semidefinite form, that means g 1 (x, y) may blow up when y → 0. Lemma 4.6.1. Assume that the Riemannian metric on L × (0, 1] satisfies the above condition. Let f be a smooth function on L × [0, 1]. For every compact subset U of L, there exists a constant C > 0 such that for any (x, y) ∈ U × (0, 1] we have ∇f (x, y) 2 ≤ Cy 2 .

Proof. By definition and (4.6.1), we have ∇f 2 = g * (df, df ) = y 2 g * 1 (df, df ).

By the smoothness of f and g * 1 on the boundary and the compactness, there exists a constant C > 0 such that ∇f 2 ≤ Cy

Manifolds with non maximal rank cusp

We will give a proof of Proposition 4.5.12 with non-maximal rank cusps. In Section 4.5.4, the assumption that M has only maximal rank cusps is only used in Lemma 4.5.15 and in the proof of Proposition 4.5.12. The proof works similarly, and we will strengthen Lemma 4.5.15, 4.5.16 and give a control of M c 3 ∇θ 2 f 2 dvol. If O is a standard cusp region of non-maximal rank, then the region (O -O ) ∩ M is not relatively compact in M . Because the cusp region O intersects the ideal boundary M I = M c -M . To overcome this difficulty, we use the compactness in M c . The fact that the partition of unity is smooth not only on M but also on M c is the key point to apply Lemma 4.6.1.

Recall that M is a geometrically finite rank one locally symmetric manifold. We have a covering M c = M 1 ∪ M 2 ∪ M 3 , where M 1 is the cusp region and M 2 is a subset of the complement of the convex core. Recall that D = Γ\W is a neighbourhood of the convex core and H is a subset of M 1 , which is the quotient of a horoball. For x in M , in (4.5.4) we have defined t 1 (x) = d(x, H), t 2 (x) = d(x, D).

Recall that ϕ 1 , ϕ 2 are two smooth functions supported on M 1 , M 2 , respectively, and

The following lemma is the key additional ingredient for the general case. Lemma 4.6.2. For every smooth function f on M 4 , there exists C f > 0, such that for all x ∈ M 4 ∩ M ∇f (x) 2 ≤ C f e -t 1 (x) ϕ 1 (x) 2 + e -t 2 (x) ϕ 2 (x) 2 .

(4.6.2)

Proof. By (4.4.1), letting y = e -t , the metric in the half space model is given by g = g y ⊕ dy 2 y 2 where g y = 1 y 2 Id m 0 0 1 y 4 Id q under some distributions W 1 and W 2 on N , which satisfies the condition in Lemma 4.6.1. Due to M 4 ⊂ M 1 ⊂ M c (Π) = (Π\N ) × R ≥0 , where Π is a discrete parabolic subgroup of Γ which preserves the metric g y on N for every y in R ≥0 , the quotient metric on M 4 also satisfies the condition in Lemma 4.6. Hence e -t 1 (x) ≤ e C K,L e -t 2 (x) for x in M 4 ∩ M . Therefore on M 4 ∩ M , by (4.6.3),(4.6.4) and (4.6.5) ∇f (x) 2 ≤ C f e -2t 1 (x) (ϕ 2 1 (x) + ϕ 2 2 (x)) ≤ C f (e -2t 1 (x) ϕ 2 1 (x) + e 2C K,L e -2t 2 (x) ϕ 2 2 (x)). The proof is complete due t 1 , t 2 ≥ 0.

We state our strengthened version of Lemma 4.5.15 and 4.5.16. Lemma* 4.5.15. In the standard cusp region, for every C > 0 there exist a compact set U 1 in M 1 ∩ M and a constant C 1 > 0 such that the following holds. For any compact set V in M 1 ∩ M there exists V > 0 such that for all f ∈ C ∞ c (M 1 ∩ M ) we have

e -t 1 f 2 dvol.

(4.6.6)

Proof. By Lemma 4.5.14, we have M 4 ⊂ L × R ≥0 . By the same argument as in the proof of Lemma 4.5.15, using Proposition 4.4.2, we have the above inequality with V f 2 replaced by L×R ≥0 e -t 1 f 2 . The desired term is due to Poincaré's inequality.

Lemma 4.6.3. For relatively compact sets V 1 , V 2 in a Riemannian manifold M , where V 1 is connected open and V 2 is a subset of V 1 with nonempty interior, there exists a positive constant , such that for all g ∈ C ∞ c (M ) we have

(4.6.7) Applying (4.6.7) with g = e t 1 f ,

The proof is complete. Lemma* 4.5.16. In the complement of the convex core, for every C > 0 there exist a compact set U 2 in M 2 ∩ M and a constant C 2 > 0 such that the following holds. For any compact set V in M 2 ∩ M there exists V > 0 such that for all f ∈ C ∞ c (M 2 ∩ M ) we have

e -t 2 f 2 dvol.

(4.6.8)

Proof. By Lemma 4.5.14, we have M 4 ⊂ M 2 ⊂ exp ⊥ (K × R ≥0 ). Then the proof is exactly the same as the proof of Lemma 4.5.16, using Lemma 4.4.8 with the constant 2(4 2 + 1) + C.

Proof of Proposition 4.5.12. Applying Lemma 4.6.2 with θ = π 2 φ3 , we get a constant C θ such that (4.6.2) holds for θ. Using Lemma * 4.5.15 and 4.5.16 with C = C θ , we obtain U 1 , U 2 . Then follow the same argument as in the proof of the special case of Proposition 4.5.12. The proof is complete.