Geodesic Anosov flows, hyperbolic closed geodesics and stable ergodicity

TL;DR

This paper establishes a characterization of Anosov geodesic flows in Finsler and Riemannian metrics on surfaces, linking hyperbolic closed geodesics with stable ergodicity.

Contribution

It proves that Anosov geodesic flows correspond to neighborhoods with hyperbolic closed geodesics and characterizes stable ergodicity for these flows.

Findings

Anosov geodesic flow iff existence of a neighborhood with hyperbolic closed geodesics

On surfaces, stable ergodicity is equivalent to being Anosov

Results extend to both Finsler and Riemannian metrics

Abstract

In this paper we show that the geodesic flow of a Finsler metric is Anosov if and only if there exists a $C^2$ open neighborhood of Finsler metrics all of whose closed geodesics are hyperbolic. For surfaces this result holds also for Riemannian metrics. This follows from a recent result of Contreras and Mazzucchelli. Furthermore, geodesic flows of Riemannian or Finsler metrics on surfaces are $C^2$ stably ergodic if and only if they are Anosov.