Some results of Hamiltonian homeomorphisms on closed aspherical surfaces

Abstract

On closed symplectically aspherical manifolds, by using Floer homology, Schwarz proved a classical result, i.e., that the action function of a nontrivial Hamiltonian diffeomorphism is not constant. In this article, we generalize Schwarz's theorem to the C0-case on closed aspherical surfaces. Our methods are purely topological and involve the theory of transverse foliations for dynamical systems of surfaces and the recent progress inspired by Le Calvez. As an application, we prove that the contractible fixed points set (and consequently the fixed points set) of a nontrivial Hamiltonian homeomorphism is not connected. We also get a similar result for an orientation preserving nonwandering point homeomorphism of the two-sphere. In the end, we give further applications based on the C0-Schwarz Theorem.