On shadowing and hyperbolicity for geodesic flows on surfaces

Abstract

Abstract. We prove that the geodesic flow on closed surfaces displays a hyperbolic set if the shadowing property holds C 2 -robustly on the metric. Similar results are obtained when considering even feeble properties like the weak shadowing and the specification properties. Despite the Hamiltonian nature of the geodesic flow, the arguments in the present paper differ completely from those used in [5] for Hamiltonian systems.

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].…”

“…The abstract general result for dissipative systems was obtained in [7,Corollary 2.19]. For Hamiltonian and geodesic flows see [3, section 3.1], [13, section 8] and [5]. In brief terms, assuming the non-dominance hypothesis, we can rotate the solutions along the closed orbits in order to mix different expansion rates.…”

Hyperbolicity through stable shadowing for generic geodesic flows

“…The projection of the orbits of the flow into the manifold are the geodesics and there is a one-to-one correspondence between closed orbits and closed geodesics. The case of stability of shadowing, weak shadowing and weak specification for 2-dimensional closed manifolds is dealt in [6]. For geodesic flows on surfaces, invariant KAM tori around elliptic orbits split the phase space and spoil the possibility of having those properties.…”

“…This is not the case in higher dimensions as KAM tori are not hypersurfaces in energy level sets. Therefore, one has to improve the arguments in [6] to obtain a multidimensional result.…”