Journal of Symplectic Geometry
Volume 14 (2016)
Topological contact dynamics III: Uniqueness of the topological Hamiltonian and $C^0$-rigidity of the geodesic flow
Pages: 1 – 29
We prove that a topological contact isotopy uniquely defines a topological contact Hamiltonian. Combined with previous results from “Topological contact dynamics I: symplectization and applications of the energy-capacity inequality” [Stefan Müller and Peter Spaeth, Adv. Geom., 15, 2015, no. 3, 349–380], this generalizes the classical one-to-one correspondence between smooth contact isotopies and their generating smooth contact Hamiltonians and conformal factors to the group of topological contact dynamical systems. Applications of this generalized correspondence include $C^0$-rigidity of smooth contact Hamiltonians, a transformation law for topological contact dynamical systems, and $C^0$-rigidity of the geodesic flows of Riemannian manifolds.