Geodesic Anosov flows, hyperbolic closed geodesics and stable ergodicity

Gerhard Knieper; Benjamin H. Schulz

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arXiv · 2202.05084

Geodesic Anosov flows, hyperbolic closed geodesics and stable ergodicity

Gerhard Knieper·Benjamin H. Schulz

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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.

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BibTeX RIS

Gerhard Knieper, Benjamin H. Schulz. 2022-02-10. Geodesic Anosov flows, hyperbolic closed geodesics and stable ergodicity. https://arxiv.org/abs/2202.05084

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