On the density of singular hyperbolic three-dimensional vector fields: a conjecture of Palis

Sylvain Crovisier; Dawei Yang

doi:10.1016/j.crma.2014.10.015

Dynamical systems

On the density of singular hyperbolic three-dimensional vector fields: a conjecture of Palis

Presented by: Étienne Ghys

Sylvain Crovisier ¹; Dawei Yang ²

¹ CNRS – Laboratoire de mathématiques d'Orsay, Université Paris-Sud 11, 91405 Orsay, France
² School of Mathematical Sciences, Soochow University, Suzhou, 215006, PR China

Comptes Rendus. Mathématique, Volume 353 (2015) no. 1, pp. 85-88.

Abstract
Résumé

In this note we announce a result for vector fields on three-dimensional manifolds: those who are singular hyperbolic or exhibit a homoclinic tangency form a dense subset of the space of $C^{1}$ -vector fields. This answers a conjecture by Palis. The argument uses an extension for local fibred flows of Mañé and Pujals–Sambarino's theorems about the uniform contraction of one-dimensional dominated bundles.

Dans cette note, nous annonçons un résultat portant sur les champs de vecteurs des variétés de dimension 3 : ceux qui vérifient l'hyperbolicité singulière ou qui possèdent une tangence homocline forment un sous-ensemble dense de l'espace des champs de vecteurs $C^{1}$ . Ceci répond à une conjecture de Palis. La démonstration utilise une généralisation pour les flots fibrés locaux des théorèmes de Mañé et Pujals–Sambarino traitant de la contraction uniforme de fibrés unidimensionnels dominés.

Received: 2014-10-15
Accepted: 2014-10-23
Published online: 2014-11-20

DOI: 10.1016/j.crma.2014.10.015

Author's affiliations:

Sylvain Crovisier ¹; Dawei Yang ²

¹ CNRS – Laboratoire de mathématiques d'Orsay, Université Paris-Sud 11, 91405 Orsay, France
² School of Mathematical Sciences, Soochow University, Suzhou, 215006, PR China

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Sylvain Crovisier; Dawei Yang. On the density of singular hyperbolic three-dimensional vector fields: a conjecture of Palis. Comptes Rendus. Mathématique, Volume 353 (2015) no. 1, pp. 85-88. doi : 10.1016/j.crma.2014.10.015. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2014.10.015/

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