On the stability conjecture for geodesic flows of manifolds without conjugate points

Rifford, Ludovic; Ruggiero, Rafael

Annales Henri Lebesgue, Volume 4 (2021), pp. 759-784.

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.

Nous étudions la conjecture de stabilité en topologie $C^2$ du point de vue de Mañé pour le flots géodésiques sans points conjugués sur les variétés compactes. La conjecture de stabilité en topologie $C^1$ pour les flots géodésiques est un problème ouvert car le $C^1$-Closing Lemma n’est pas connu dans ce contexte. Sans Closing Lemma, nous démontrons que la théorie des variétés sans points conjugués et une version récente du Lemme de Franks du point de vue de Mañé permettent d’obtenir une réponse positive à la conjecture dans le cas des variétés compactes sans points conjugués de dimension 2 et 3 ayant un revêtement universel quasi-convexe avec des rayons géodésiques divergents, ainsi que pour les variétés de dimension $n$ de rank un généralisées.

Classification: 37C20, 37D40, 53C22, 20F65
Mots-clés : Geodesic flow, structural stability, closing lemma, conjugate points, quasi-convex space, Gromov hyperbolic space

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