Anosov Flows and Dynamical Zeta Functions

TL;DR

This paper investigates the meromorphic properties and analytic continuations of Ruelle and Selberg zeta functions for Anosov flows, revealing conditions under which these functions have poles, are meromorphic, or analytic, and deriving orbit counting results.

Contribution

It establishes the meromorphic continuation of zeta functions for smooth Anosov flows and analyzes their properties under contact and bunching conditions, including spectral analysis of transfer operators.

Findings

Zeta functions are meromorphic for smooth flows.

Poles at the topological entropy for contact flows with bunching.

Sharp orbit counting results under certain conditions.

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.