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

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/ahl.87
Classification: 37C20, 37D40, 53C22, 20F65
Keywords: Geodesic flow, structural stability, closing lemma, conjugate points, quasi-convex space, Gromov hyperbolic space
Rifford, Ludovic 1; Ruggiero, Rafael 2

1 Université Côte d’Azur, CNRS, Inria, Labo. J.-A. Dieudonné, UMR CNRS 7351, Parc Valrose 06108 Nice, Cedex 2, (France)
2 Departamento de Matemática, Pontifícia Universidade Católica do Rio de Janeiro, Rua Marqués de São Vicente 225, Gávea, Rio de Janeiro, RJ, (Brazil), 22453-900
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Rifford, Ludovic; Ruggiero, Rafael. On the stability conjecture for geodesic flows of manifolds without conjugate points. Annales Henri Lebesgue, Volume 4 (2021), pp. 759-784. doi : 10.5802/ahl.87. http://archive.numdam.org/articles/10.5802/ahl.87/

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