Hyperbolicity through stable shadowing for generic geodesic flows

Abstract

We prove that the closure of the closed orbits of a generic geodesic flow on a closed Riemannian n ≥ 2 dimensional manifold is a uniformly hyperbolic set if the shadowing property holds C 2-robustly on the metric. We obtain analogous results using weak specification and the shadowing property allowing bounded time reparametrization.

Expansiveness and Hyperbolicity in Convex Billiards

“…It is well known that in certain classes of conservative dynamical systems, the robusteness of certain properties ensures some kind of hyperbolicity. Examples include expansiveness [20], ergodicity [24], transitivity [2], shadowing [4,5,6], weak shadowing [4] and topological stability [4].…”