2011/06/10 by Müller, Stefan, Spaeth, Peter
#37C15 #53D10 #57M50 #57R17 #Dynamical Systems (math.DS) #FOS: Mathematics #Symplectic Geometry (math.SG)
paper · doi:10.48550/arxiv.1106.1968
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 É. 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, Mül08b, Vit06, BS11b, BS11a, MS11]. Moreover, we generalize an example of H. Furstenberg [Fur61] of topologically conjugate but not C1-conjugate area preserving diffeomorphisms of the two-torus to trivial T2-bundles, and construct examples of Hamiltonian and contact vector fields that are topologically conjugate but not C1-conjugate. Higher-dimensional helicities are considered briefly at the end of the paper.