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Comptes Rendus Mathématique
Volume 353, n° 1
pages 85-88 (janvier 2015)
Doi : 10.1016/j.crma.2014.10.015
Received : 15 October 2014 ;  accepted : 23 October 2014
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
 

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

Abstract

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  -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.

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Résumé

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  . 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.

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1  S. Crovisier was partially supported by International Balzan Foundation, Balzan Research Project of J. Palis.
2  D. Yang was partially supported by NSFC 11271152 and ANR project DynNonHyp BLAN08-2313375.


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