Franks' lemma for $\mathbf{C}^2$-Mañé perturbations of Riemannian metrics and applications to persistence

Abstract

We prove a uniform Franks' lemma at second order for geodesic flows on a compact Riemannian manifold and apply the result in persistence theory. Our approach, which relies on techniques from geometric control theory, allows us to show that Mañé (i.e., conformal) perturbations of the metric are sufficient to achieve the result.

Non-degeneracy of closed orbits for generic potentials

“…In this statement, we could replace "weakly open" by "open" using [12]. However, this is much harder to prove and not necessary for our present study.…”