Abstract. We study the spectral properties of the Ruelle-Perron-Frobenius operator associated to an Anosov map on classes of functions with high smoothness. To this end we construct anisotropic Banach spaces of distributions on which the transfer operator has a small essential spectrum. In the C ∞ case, the essential spectral radius is arbitrarily small, which yields a description of the correlations with arbitrary precision. Moreover, we obtain sharp spectral stability results for deterministic and random perturbations. In particular, we obtain differentiability results for spectral data (which imply differentiability of the Sinai-Ruelle-Bowen measure, the variance for the central limit theorem, the rates of decay for smooth observable, etc.). IntroductionThe study of the statistical properties of Anosov systems dates back almost half a century [1] and many approaches have been developed to investigate various aspects of the field (the most historically relevant one being based on the introduction of Markov partitions [2, 8, 24, 31]). At the same time the types of question and the precision of the results have progressed over the years. In the last few years the emphasis has been on strong stability properties with respect to various types of perturbation [4], dynamical zeta functions and the related smoothness issue (see [9,14,26]). In the present paper we present a new approach, improving on the previous partial and still unsatisfactory method by Blank et al that allows one to obtain easily an array of results (many of which are new) and we hope will reveal an even larger field of applicability. Indeed, the ideas in [7] have already been applied with success to some partially hyperbolic situations (flows) [19] and we expect them to be applicable to the study of dynamical zeta functions.The basic idea is inspired by the work on piecewise expanding maps, starting with [12,15] and the many others that contributed subsequently (see [4] for a review

Abstract. We study the dynamics of the Teichmüller flow in the moduli space of Abelian differentials (and more generally, its restriction to any connected component of a stratum). We show that the (Masur-Veech) absolutely continuous invariant probability measure is exponentially mixing for the class of Hölder observables. A geometric consequence is that the SL(2, R) action in the moduli space has a spectral gap.

We generalize a method developed by Sarig to obtain polynomial lower bounds for correlation functions for maps with a countable Markov partition. A consequence is that LS Young's estimates on towers are always optimal. Moreover, we show that, for functions with zero average, the decay rate is better, gaining a factor 1/n. This implies a Central Limit Theorem in contexts where it was not expected, e.g.x + Cx 1+α with 1/2 α < 1. The method is based on a general result on renewal sequences of operator, and gives an asymptotic estimate up to any precision of such operators.

In the setting of abstract Markov maps, we prove results concerning the convergence of renormalized Birkhoff sums to normal laws or stable laws. They apply to onedimensional maps with a neutral fixed point at 0 of the form x + x 1+α , for α ∈ (0, 1). In particular, for α > 1/2, we show that the Birkhoff sums of a Hölder observable f converge to a normal law or a stable law, depending on whether f (0) = 0 or f (0) = 0. The proof uses spectral techniques introduced by Sarig, and Wiener's Lemma in non-commutative Banach algebras.

25 pages v2: minor revision v3: published versionInternational audienceWe prove the almost sure invariance principle for stationary R^d--valued processes (with dimension-independent very precise error terms), solely under a strong assumption on the characteristic functions of these processes. This assumption is easy to check for large classes of dynamical systems or Markov chains, using strong or weak spectral perturbation arguments

Completing a strategy of Gouëzel and Lalley [GL11], we prove a local limit theorem for the random walk generated by any symmetric finitely supported probability measure on a non-elementary Gromov-hyperbolic group: denoting by R the inverse of the spectral radius of the random walk, the probability to return to the identity at time n behaves like CR −n n −3/2 . An important step in the proof is to extend Ancona's results on the Martin boundary up to the spectral radius: we show that the Martin boundary for R-harmonic functions coincides with the geometric boundary of the group. In an appendix, we explain how the symmetry assumption of the measure can be dispensed with for surface groups.

Compact locally maximal hyperbolic sets are studied via geometrically defined functional spaces that take advantage of the smoothness of the map in a neighborhood of the hyperbolic set. This provides a self-contained theory that not only reproduces all the known classical results but gives also new insights on the statistical properties of these systems.

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