Dynamical zeta functions for Anosov flows via microlocal analysis

Abstract

The purpose of this paper is to give a short microlocal proof of the meromorphic continuation of the Ruelle zeta function for C∞ Anosov flows. More general results have been recently proved by Giulietti–Liverani–Pollicott [GiLiPo] but our approach is different and is based on the study of the generator of the flow as a semiclassical differential operator. The purpose of this article is to provide a short microlocal proof of the meromorphic continuation of the Ruelle zeta function for C∞ Anosov flows on compact manifolds: Theorem. Suppose X is a compact manifold and φt : X → X is a C∞ Anosov flow with orientable stable and unstable bundles. Let {γ} denote the set of primitive orbits of φt, with T ] γ their periods. Then the Ruelle zeta function, ζR(λ) = ∏