On shadowing and hyperbolicity for geodesic flows on surfaces

TL;DR

This paper investigates conditions under which geodesic flows on closed surfaces exhibit hyperbolic behavior, focusing on robust shadowing properties and their implications for hyperbolicity.

Contribution

It establishes that robust shadowing properties imply hyperbolic sets for geodesic flows, extending results to weaker properties like weak shadowing and specification.

Findings

Robust shadowing implies hyperbolic sets in geodesic flows.

Weak shadowing and specification also lead to hyperbolicity.

Results differ from previous Hamiltonian system approaches.

Abstract

We prove that the geodesic flow on closed surfaces displays a hyperbolic set if the shadowing property holds C2-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.