Geodesic Anosov flows, hyperbolic closed geodesics and stable ergodicity

2022/02/10 by Gerhard Knieper, Knieper, Gerhard, Benjamin H. Schulz +1
Physics and Astronomy · #37J46 (Primary) #53C22 (Secondary) #Advanced Differential Geometry Research #Differential Geometry (math.DG) #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.2202.05084

openalex publication_date 2022/02/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper we show that the geodesic flow of a Finsler metric is Anosov if and only if there exists a C² 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 C² stably ergodic if and only if they are Anosov.

Related

• Anosov Geodesic Flows on Surfaces
• Robust transitivity of geodesic flows from metrics with conjugate points
• A new Laplace operator in Finsler geometry and periodic orbits of Anosov flows
• Ergodic components and topological entropy in geodesic flows of surfaces
• Minimal geodesics and integrable behavior in geodesic flows