Stability properties of divergence-free vector fields

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.