Portal de Periódicos da CAPES

Stability properties of divergence-free vector fields

2012; Taylor & Francis; Volume: 27; Issue: 2 Linguagem: Inglês

10.1080/14689367.2012.655710

1468-9375

Célia Ferreira,

Geometric Analysis and Curvature Flows

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 fieldAnosov vector fielddominated splittingstructurally stable vector fieldheterodimensional cycleAMS Subject Classfications: Primary: 37D2037D3037C27Secondary: 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.