Some properties of Hamiltonian homeomorphisms on closed aspherical surfaces

TL;DR

This paper extends Schwarz's classical results on Hamiltonian diffeomorphisms to the $C^0$-setting on closed aspherical surfaces, revealing new properties of fixed points, action growth, and group structures of Hamiltonian homeomorphisms.

Contribution

It generalizes Schwarz's theorem to $C^0$-Hamiltonian homeomorphisms on surfaces and explores implications for fixed points, action width, and group torsion properties.

Findings

Fixed points set of nontrivial Hamiltonian homeomorphisms is disconnected.

Action width grows at least linearly under iteration.

Groups of Hamiltonian homeomorphisms are torsion free.

Abstract

On closed symplectically aspherical manifolds, Schwarz proved a classical result that the action function of a nontrivial Hamiltonian diffeomorphism is not constant by using Floer homology. In this article, we generalize Schwarz's theorem to the $C^0$-case on closed aspherical surfaces. Our methods involve the theory of transverse foliations for dynamical systems of surfaces inspired by Le Calvez and its recent progresses. 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. Furthermore, we obtain that the growth of the action width of a Hamiltonian homeomorphism increases at least linearly, and that the group of Hamiltonian homeomorphisms of $\mathbb{T}^2$ and the group of area preserving homeomorphisms isotopic to the identity of $\Sigma_g$ ($g>1$) are torsion free, where $\Sigma_g$ is a closed orientated surface with genus $g$. Finally, we will show how the $C^1$-Zimmer's conjecture on surfaces deduces from $C^0$-Schwarz's theorem.