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

We present an original approach which allows us to investigate the statistical properties of a non-uniformly hyperbolic map on the interval. Based on a stochastic approximation of the deterministic map, this method essentially gives the optimal polynomial bound for the decay of correlations, the degree depending on the order of the tangency at the neutral fixed point.

Abstract. We extend a number of results from one dimensional dynamics based on spectral properties of the Ruelle-Perron-Frobenius transfer operator to Anosov diffeomorphisms on compact manifolds. This allows to develop a direct operator approach to study ergodic properties of these maps. In particular, we show that it is possible to define Banach spaces on which the transfer operator is quasicompact. (Information on the existence of an SRB measure, its smoothness properties and statistical properties readily follow from such a result.) In dimension d = 2 we show that the transfer operator associated to smooth random perturbations of the map is close, in a proper sense, to the unperturbed transfer operator. This allows to obtain easily very strong spectral stability results, which in turn imply spectral stability results for smooth deterministic perturbations as well. Finally, we are able to implement an Ulam type finite rank approximation scheme thus reducing the study of the spectral properties of the transfer operator to a finite dimensional problem.

Exponential decay of correlations for C 4 contact Anosov flows is established. This implies, in particular, exponential decay of correlations for all smooth geodesic flows in strictly negative curvature.

Abstract. We present a common framework to study decay and exchanges rates in a wide class of dynamical systems. Several applications, ranging form the metric theory of continuons fractions and the Shannon capacity of contrained systems to the decay rate of metastable states, are given.