## Banach spaces adapted to Anosov systems.(English)Zbl 1088.37010

Summary: 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 $${\mathcal C}^\infty$$ 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.).

### MSC:

 37D20 Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.) 37C30 Functional analytic techniques in dynamical systems; zeta functions, (Ruelle-Frobenius) transfer operators, etc. 46F05 Topological linear spaces of test functions, distributions and ultradistributions 47B38 Linear operators on function spaces (general)
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