Anosov flows and dynamical zeta functions

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 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 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.