Hyperbolicity through stable shadowing for generic geodesic flows

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 C2-robustly on the metric. We obtain analogous results using weak specification and the shadowing property allowing bounded time reparametrization.
The authors were partially supported by the Project ‘New trends in Lyapunov exponents’ (PTDC/MAT-PUR/29126/2017). MB was partially supported by FCT - ‘Fundação para a Ciência e a Tecnologia’, through Centro de Matemática e Aplicações (CMA-UBI), Universidade da Beira Interior, project UID/MAT/00212/2013. JLD was partially supported by the Project CEMAPRE - UID/MULTI/00491/2019 financed by FCT/MCTES through national funds. MJT was partially supported by the Research Centre of Mathematics of the University of Minho with the Portuguese Funds from the Fundação para a Ciência e a Tecnologia, through the Project UID/MAT/00013/2013.