On the stability conjecture for geodesic flows of manifolds without conjugate points
Annales Henri Lebesgue, Volume 4 (2021) , pp. 759-784.

KeywordsGeodesic flow, structural stability, closing lemma, conjugate points, quasi-convex space, Gromov hyperbolic space

### Abstract

We study the ${C}^{2}$-structural stability conjecture from Mañé’s viewpoint for geodesics flows of compact manifolds without conjugate points. The structural stability conjecture is an open problem in the category of geodesic flows because the ${C}^{1}$ closing lemma is not known in this context. Without the ${C}^{1}$ closing lemma, we combine the geometry of manifolds without conjugate points and a recent version of Franks’ Lemma from Mañé’s viewpoint to prove the conjecture for compact surfaces, for compact three dimensional manifolds with quasi-convex universal coverings where geodesic rays diverge, and for $n$-dimensional, generalized rank one manifolds.

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