Comptes Rendus
no. 1
Dynamical systems
On the density of singular hyperbolic three-dimensional vector fields: a conjecture of Palis
[Sur la densité de l'hyperbolicité singulière pour les champs de vecteurs en dimension trois : une conjecture de Palis]

Présenté par : Étienne Ghys

Sylvain Crovisier1 ; Dawei Yang2
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.

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 C1. 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.

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 C1-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.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2014.10.015
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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