Stability properties of divergence-free vector fields

作者：Ferreira Celia^*

来源：Dynamical Systems, 2012, 27(2): 223-238.

DOI：10.1080/14689367.2012.655710

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

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.

• 出版日期2012