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

Gouëzel, Sébastien; Liverani, Carlangelo

Ergodic Theory Dyn. Syst. 26, No. 1, 189-217 (2006).

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)

Keywords:

spectral properties; Ruelle-Perron-Frobenius operator; Anosov map; classes of functions with high smoothness; transfer operator; stability; deterministic perturbations; random perturbations; differentiability results