Abstract
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 $\frac 19$-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.
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