Banach spaces adapted to Anosov systems

Details Banach spaces adapted to Anosov systems Banach spaces adapted to Anosov systems

2004-05-14

arxiv.org/abs/math/0405278v2

Mathematics

Dynamical Systems

26 pages

Scientific paper

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^\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 SRB measure, the variance for the CLT, the rates of decay for smooth observable, etc.).

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