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

Geodesic Anosov flows, hyperbolic closed geodesics and stable ergodicity