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September  2010, 26(3): 839-846. doi: 10.3934/dcds.2010.26.839

Three-dimensional conservative star flows are Anosov

Mário Bessa 1, and  Jorge Rocha 1,

1. 

Departamento de Matemática Pura, Universidade do Porto, Rua do Campo Alegre, 687,4169-007 Porto, Portugal

Received  January 2009 Revised  October 2009 Published  December 2009

A divergence-free vector field satisfies the star property if any divergence-free vector field in some $C^1$-neighborhood has all the singularities and all closed orbits hyperbolic. In this article we prove that any divergence-free star vector field defined in a closed three-dimensional manifold is Anosov. Moreover, we prove that a $C^1$-structurally stable three-dimensional conservative flow is Anosov.
Keywords: Dominated splitting., Divergence-free vector fields, Anosov flows.
Mathematics Subject Classification: Primary: 37D30; Secondary: 37C7.
Citation: Mário Bessa, Jorge Rocha. Three-dimensional conservative star flows are Anosov. Discrete & Continuous Dynamical Systems, 2010, 26 (3) : 839-846. doi: 10.3934/dcds.2010.26.839
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