Helicity of vector fields preserving a regular contact form and topologically conjugate smooth dynamical systems

TL;DR

This paper calculates the helicity of volume-preserving vector fields on 3-manifolds with contact structures, extends helicity invariance to topological conjugacies, and constructs examples of topologically conjugate but not smooth conjugate Hamiltonian and contact systems.

Contribution

It advances the understanding of helicity invariants in contact and Hamiltonian dynamics, especially under topological conjugacy, and provides new examples of conjugate but non-smooth systems.

Findings

Helicity computed for vector fields preserving regular contact forms.

Helicity extends as an invariant under topological conjugation.

Constructed examples of topologically conjugate but not C^1-conjugate systems.

Abstract

We compute the helicity of a vector field preserving a regular contact form on a closed three-dimensional manifold, and improve results by J.-M. Gambaudo and \'E. Ghys [GG97] relating the helicity of the suspension of a surface isotopy to the Calabi invariant of the latter. Based on these results, we provide positive answers to two questions posed by V. I. Arnold [Arn86]. In the presence of a regular contact form that is also preserved, the helicity extends to an invariant of an isotopy of volume preserving homeomorphisms, and is invariant under conjugation by volume preserving homeomorphisms. A similar statement also holds for suspensions of surface isotopies and surface diffeomorphisms. This requires the techniques of topological Hamiltonian and contact dynamics developed in [MO07, M\"ul08b, Vit06, BS11b, BS11a, MS11]. Moreover, we generalize an example of H. Furstenberg [Fur61] of topologically conjugate but not C^1-conjugate area preserving diffeomorphisms of the two-torus to trivial T^2-bundles, and construct examples of Hamiltonian and contact vector fields that are topologically conjugate but not C^1-conjugate. Higher-dimensional helicities are considered briefly at the end of the paper.