A note on expansiveness and hyperbolicity for generic geodesic flows

Bessa, Mário

http://hdl.handle.net/10400.2/13858

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Authors

Bessa, Mário

Abstract(s)

In this short note we contribute to the generic dynamics of geodesic flows associated to metrics on compact Riemannian manifolds of dimension ≥ 2. We prove that there exists a C2-residual subset R of metrics on a given compact Riemannian manifold such that if g∈R, then its associated geodesic flow φ_g(t) is expansive if and only if the closure of the set of periodic orbits of φgt is a uniformly hyperbolic set. For surfaces, we obtain a stronger statement: there exists a C2-residual R such that if g ∈ R, then its associated geodesic flow φgt is expansive if and only if φ_g(t) is an Anosov flow.