Three-dimensional conservative star flows are Anosov

Mário Bessa; Jorge Rocha

doi:10.3934/dcds.2010.26.839

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2010, Volume 26, Issue 3: 839-846. Doi: 10.3934/dcds.2010.26.839

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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

Received: January 2009

Revised: October 2009

Published: September 2010

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Abstract

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.

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Mathematics Subject Classification: Primary: 37D30; Secondary: 37C75.

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