Three-dimensional conservative star flows are Anosov

Abstract

A divergence-free vector field satisfies the star property if any divergence-free vector field in some $C^1$-neighborhood has all the singularities and all closed orbits hyperbolic. In this article we prove that any divergence-free star vector field defined in a closed three-dimensional manifold is Anosov. Moreover, we prove that a $C^1$-structurally stable three-dimensional conservative flow is Anosov.