Continuous Hamiltonian dynamics and area-preserving homeomorphism group of $D^2$

Abstract

The main purpose of this paper is to propose a scheme of a proof of the nonsimpleness of the group $Homeo^\Omega(D^2,\partial D^2)$ of area preserving homeomorphisms of the 2-disc $D^2$. We first establish the existence of Alexander isotopy in the category of Hamiltonian homeomorphisms. This reduces the question of extendability of the well-known Calabi homomorphism ${\rm Cal}:Diff^\Omega(D^1,\partial D^2) \to \mathbb R$ to a homomorphism $\overline{{\rm Cal}}:Hameo(D^2,\partial D^2) \to \mathbb R$ to that of the vanishing of the basic phase function $f_{\underline{\mathbb F}}$, a Floer theoretic graph selector previously constructed by the author, that is associated to the graph of the topological Hamiltonian loop and its normalized Hamiltonian $\underline{F}$ on $S^2$ that is obtained via the natural embedding $D^2 \hookrightarrow S^2$. Here $Hameo(D^2,\partial D^2)$ is the group of Hamiltonian homeomorphisms introduced by M\"uller and the author. We then provide an evidence of this vanishing conjecture by proving the conjecture for the special class of \emph{weakly graphical} topological Hamiltonian loops on $D^2$ via a study of the associated Hamilton-Jacobi equation.