Topological contact dynamics III: uniqueness of the topological Hamiltonian and C^0-rigidity of the geodesic flow

TL;DR

This paper establishes the uniqueness of topological contact Hamiltonians for topological contact isotopies, extending classical results to a broader topological setting and demonstrating applications like C^0-rigidity of geodesic flows.

Contribution

It proves the unique correspondence between topological contact isotopies and their Hamiltonians, generalizing classical smooth results to topological contact dynamical systems.

Findings

Uniqueness of topological contact Hamiltonian for isotopies

C^0-rigidity of geodesic flows on Riemannian manifolds

Transformation laws for topological contact systems

Abstract

We prove that a topological contact isotopy uniquely defines a topological contact Hamiltonian. Combined with previous results from [MS11], 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.