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arXiv · 1404.5130

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

Sylvain Crovisier·Dawei Yang
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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 $C^1$-vector fields. This answers a conjecture by Palis. The argument uses an extension for local fibered flows of Mañé and Pujals-Sambarino's theorems about the uniform contraction of one-dimensional dominated bundles. Sur la densité de l'hyperbolicité singulière pour les champs de vecteurs en dimension trois : une conjecture de Palis 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.

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Sylvain Crovisier, Dawei Yang. 2014-04-21. On the density of singular hyperbolic three-dimensional vector fields: a conjecture of Palis. https://arxiv.org/abs/1404.5130

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