Homoclinic bifurcations and uniform hyperbolicity for three-dimensions flows

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In this paper we prove that any C1 vector field defined on a three-dimensional manifold can be approximated by one that is uniformly hyperbolic, or that exhibits either a homoclinic tangency or a singular cycle. This proves an analogous statement of a conjecture of Palis for diffeomorphisms in the context of C1-flows on three manifolds. For that, we rely on the notion of dominated splitting for the associated linear Poincaré flow.

Original languageEnglish (US)
Pages (from-to)805-841
Number of pages37
JournalAnnales de l'Institut Henri Poincare (C) Analyse Non Lineaire
Issue number5
StatePublished - 2003

All Science Journal Classification (ASJC) codes

  • Analysis
  • Mathematical Physics
  • Applied Mathematics


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