Stability properties of divergence-free vector fields

TL;DR

This paper proves that divergence-free vector fields with the star property on Riemannian manifolds are necessarily Anosov, establishing a strong link between stability properties and hyperbolic dynamics in divergence-free systems.

Contribution

It demonstrates that divergence-free vector fields satisfying the star property are Anosov and shows that C1-structurally stable divergence-free fields can be approximated by Anosov fields.

Findings

Star property implies Anosov for divergence-free vector fields.

C1-structurally stable divergence-free fields can be approximated by Anosov fields.

Any divergence-free vector field can be C1-approximated by an Anosov field or one with a heterodimensional cycle.

Abstract

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.