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 $C^\infty$ 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.


Paolo Giulietti

Universidade Federal do Rio Grande do Sul, Porto Alegre, RS, Brasil

Carlangelo Liverani

Università di Roma Tor Vergata 00133 Roma, Italy

Mark Pollicott

Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom