A Note on Expansiveness and Hyperbolicity for Generic Geodesic Flows

作者：Bessa Mario^*

来源：Mathematical Physics, Analysis and Geometry, 2018, 21(2): 14.

DOI：10.1007/s11040-018-9271 -7

• Expansiveness
• Residual sets
• Anosov
• Geodesic flows

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 C-2-residual subset R of metrics on a given compact Riemannian manifold such that if g is an element of R, then its associated geodesic flow phi(t)(g) is expansive if and only if the closure of the set of periodic orbits of phi(t)(g) is a uniformly hyperbolic set. For surfaces, we obtain a stronger statement: there exists a C-2-residual R such that if g is an element of R, then its associated geodesic flow phi(t)(g) is expansive if and only if phi(t)(g) is an Anosov flow.

• 出版日期2018-5-22