Abundance of elliptic dynamics on conservative 3-flows

TL;DR

This paper proves that on a generic set of divergence-free 3D vector fields, either the flow is Anosov or elliptic orbits are dense, revealing the prevalence of elliptic dynamics in conservative systems.

Contribution

It establishes that elliptic dynamics are abundant in conservative 3-flows, showing a dichotomy between hyperbolic and elliptic behaviors on a residual set.

Findings

Elliptic orbits are dense in a residual set of divergence-free 3-flows.

Generic divergence-free 3-flows are either Anosov or have dense elliptic orbits.

The set of such vector fields is C1-residual in the space of divergence-free vector fields.

Abstract

We consider a compact 3-dimensional boundaryless Riemannian manifold M and the set of divergence-free (or zero divergence) vector fields without singularities, then we prove that this set has a C1-residual such that any vector field inside it is Anosov or else its elliptical orbits are dense in the manifold M.