The $ {C}^{0}$ general density theorem for geodesic flows

Mário Bessa; Maria Joana Torres

doi:10.1016/j.crma.2015.03.012

Dynamical systems

The

C^{0}

general density theorem for geodesic flows
[Le théorème de densité de Pugh

C^{0}

pour les flots géodésiques]

Présenté par : Jean-Christophe Yoccoz

Mário Bessa ¹ ; Maria Joana Torres ²

¹ CMA-UBI, Departamento de Matemática, Universidade da Beira Interior, Rua Marquês d'Ávila e Bolama, 6201-001 Covilhã, Portugal
² CMAT, Departamento de Matemática e Aplicações, Universidade do Minho, Campus de Gualtar, 4700-057 Braga, Portugal

Comptes Rendus. Mathématique, Volume 353 (2015) no. 6, pp. 545-549.

Résumé
Abstract

Étant donnée une variété riemannienne compacte sans bord, nous démontrons un théorème de densité $C^{0}$ -générique pour les flots géodésiques et, plus précisément, nous prouvons qu'il existe une partie $C^{0}$ -résiduelle de l'ensemble des flots géodésiques continus, telle que tout flot dans cette partie admet un ensemble dense d'orbites périodiques.

Given a closed Riemannian manifold, we prove the $C^{0}$ -general density theorem for continuous geodesic flows. More precisely, we prove that there exists a residual (in the $C^{0}$ -sense) subset of the continuous geodesic flows such that, in that residual subset, the geodesic flow exhibits dense closed orbits.

Reçu le : 2014-05-21
Accepté le : 2015-03-02
Publié le : 2015-04-04

DOI : 10.1016/j.crma.2015.03.012

Affiliations des auteurs :

Mário Bessa ¹ ; Maria Joana Torres ²

¹ CMA-UBI, Departamento de Matemática, Universidade da Beira Interior, Rua Marquês d'Ávila e Bolama, 6201-001 Covilhã, Portugal
² CMAT, Departamento de Matemática e Aplicações, Universidade do Minho, Campus de Gualtar, 4700-057 Braga, Portugal

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Mário Bessa; Maria Joana Torres. The $ {C}^{0}$ general density theorem for geodesic flows. Comptes Rendus. Mathématique, Volume 353 (2015) no. 6, pp. 545-549. doi : 10.1016/j.crma.2015.03.012. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2015.03.012/

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Assis Azevedo; Davide Azevedo; Mário Bessa; Maria Torres The closing lemma and the planar general density theorem for Sobolev maps, Proceedings of the American Mathematical Society, Volume 149 (2021) no. 4, p. 1687 | DOI:10.1090/proc/15352
Mário Bessa A Note on Expansiveness and Hyperbolicity for Generic Geodesic Flows, Mathematical Physics, Analysis and Geometry, Volume 21 (2018) no. 2 | DOI:10.1007/s11040-018-9271-7
Mário Bessa; Maria Joana Torres; Paulo Varandas On the periodic orbits, shadowing and strong transitivity of continuous flows, Nonlinear Analysis, Volume 175 (2018), p. 191 | DOI:10.1016/j.na.2018.06.002

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