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

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

2011-06-10

arxiv.org/abs/1106.1968v2

Mathematics

Symplectic Geometry

34 pages; improved presentation and notation, fixed a few minor mistakes and typos

Scientific paper

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.

Affiliated with

Also associated with

No associations

Rating

Helicity of vector fields preserving a regular contact form and topologically conjugate smooth dynamical systems does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Helicity of vector fields preserving a regular contact form and topologically conjugate smooth dynamical systems, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Helicity of vector fields preserving a regular contact form and topologically conjugate smooth dynamical systems will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-319863