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Anosov flows and dynamical zeta functions. (English) Zbl 1418.37042

Summary: We study the Ruelle and Selberg zeta functions for \(C^r\) Anosov flows, \(r > 2\), on a compact smooth manifold. We prove several results, the most remarkable being (a) for \(C^\infty\) flows the zeta function is meromorphic on the entire complex plane; (b) for contact flows satisfying a bunching condition (e.g., geodesic flows on manifolds of negative curvature better than \(\frac{1}{9}\)-pinched), the zeta function has a pole at the topological entropy and is analytic in a strip to its left; (c) under the same hypotheses as in (b) we obtain sharp results on the number of periodic orbits. Our arguments are based on the study of the spectral properties of a transfer operator acting on suitable Banach spaces of anisotropic currents.

MSC:

37C30 Functional analytic techniques in dynamical systems; zeta functions, (Ruelle-Frobenius) transfer operators, etc.
37C40 Smooth ergodic theory, invariant measures for smooth dynamical systems
37D20 Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.)
37B40 Topological entropy
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