Some properties of Hamiltonian homeomorphisms on closed aspherical surfaces

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.