Anosov Flows and Dynamical Zeta Functions
Abstract
We study the Ruelle and Selberg zeta functions for $\Cs^r$ Anosov flows, $r > 2$, on a compact smooth manifold. We prove several results, the most remarkable being: (a) for $\Cs^\infty$ 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.
 Publication:

arXiv eprints
 Pub Date:
 March 2012
 DOI:
 10.48550/arXiv.1203.0904
 arXiv:
 arXiv:1203.0904
 Bibcode:
 2012arXiv1203.0904G
 Keywords:

 Mathematics  Dynamical Systems;
 37C30