Banach spaces adapted to Anosov systems

Abstract

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

Characterization of weak convergence of Birkhoff sums for Gibbs-Markov maps

“…In this paragraph, we describe a general spectral theorem extending the results of [KL99] to the case of several derivatives. A very similar result has been proved in [GL06], but with slightly stronger assumptions that will not be satisfied in the forthcoming application to Gibbs-Markov maps (in particular, [GL06] requires (3.16) below to hold for 0 ≤ i < j ≤ N , instead of 1 ≤ i < j ≤ N ). Let us also mention [HP08] for related results.…”

“…The required characteristic expansion is obtained in some cases using classical perturbation theory as in [AD01b], but other tools are also required in other cases: weak perturbation theory [KL99,GL06,HP08] and interpolation spaces [BL76]. Finally, Appendix A describes another application of our techniques, to the speed in the central limit theorem.…”

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“…This transversality condition is proved to be a generic one in the last section. In Section 3, we introduce some norms on the space of C r functions on S 1 × R and prove a Lasota-Yorke type inequality for them, imitating the argument in the recent paper [3] of C. Liverani and the second named author with slight modification. Section 4 is the core of this paper, where we prove a Lasota-Yorke inequality involving the W s norm and the norm introduced in Section 3.…”