Stability properties of divergence-free vector fields

Abstract

A divergence-free vector field satisfies the star property if any divergence-free vector field in some C ¹-neighbourhood has all singularities and all closed orbits hyperbolic. In this article, we prove that any divergence-free vector field defined on a Riemannian manifold and satisfying the star property is Anosov. It is also shown that a C ¹-structurally stable divergence-free vector field is Anosov. Moreover, we prove that any divergence-free vector field can be C ¹-approximated by an Anosov divergence-free vector field, or else by a divergence-free vector field exhibiting a heterodimensional cycle.

Keywords:

• divergence-free vector field
• Anosov vector field
• dominated splitting
• structurally stable vector field
• heterodimensional cycle

AMS Subject Classfications:

• Primary: 37D20
• 37D30
• 37C27
• Secondary: 37C10

Acknowledgements

I would like to thank my supervisors, Mário Bessa and Jorge Rocha, whose encouragement, suggestions and guidance enabled me to develop this work. I also thank the financial support of the Fundaçã o para a Ci ência e a Tecnologia (scholarship SFRH/BD/33100/2007) and the partial support of the Mathematics Center of the University of Porto (CMUP) and of the Fundação para a Ciência e a Tecnologia project PTDC/MAT/099493/2008.