Stability properties of divergence-free vector fields

Abstract

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.

Topological stability for conservative systems

“…The next result was proved recently by Ferreira [9] and is a generalization of a three-dimensional theorem by the authors [6]. [9].)…”

Hyperbolicity and stability for Hamiltonian flows

“…This result was recently generalized in [13] for a d-dimensional closed manifold, d 4. We point out that the proof in [9] could not be trivially adapted to higher dimensions.…”

“…Consequently, using volume-preserving arguments the authors were able to prove the existence of a dominated splitting for the linear Poincaré flow and then the hyperbolicity. The main novelties of the proof in [13] are the use of a new strategy to prove the absence of singularities and the adaptation of an argument of Mañé in [17] to obtain hyperbolicity from a dominated splitting, which follows easily in dimension 3. The key ingredient in the proof is the following dichotomy for C 1 -divergence-free vector fields: a periodic orbit of large period either admits a dominated splitting of a prescribed strength or can be turned into a parabolic one by a C 1 -small perturbation along the orbit.…”

Hamiltonian suspension of perturbed Poincaré sections and an application

“…In fact, any divergence-free vector field defined on a 3-dimensional closed manifold can be C 1 -approximated in the same class by a vector field either Anosov or with a homoclinic tangency associated to a hyperbolic closed orbit [4]. This was recently generalized in [9] for a d-dimensional closed manifold, d ≥ 4: any divergence-free vector field can be C 1 -approximated by another one satisfying either one of the properties of the 3-dimensional case, or with a heterodimensional cycle. Here we address the problem of obtaining a version of [4] in the Hamiltonian context.…”