Stability properties of divergence-free vector fields

A divergence-free vector field satisfies the star property if any divergence-free vector field in some C1-neighborhood has all singularities and all periodic orbits hyperbolic. In this paper 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 C1-structurally stable divergencefree vector field can be approximated by an Anosov divergence-free vector field. Moreover, we prove that any divergence-free vector field can be C1-approximated by an Anosov divergence-free vector field, or else by a divergence-free vector field exhibiting a heterodimensional cycle.