Continuous Hamiltonian dynamics and area-preserving homeomorphism group of D2

Abstract

The main purpose of this paper is to propose a scheme of a proof of the nonsimpleness of the group ${\rm Homeo}^\Omega(D^2,\del D^2)$ 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 $\Cal: {\rm Diff}^\Omega(D^1,\del D^2) \to \R$ to a homomorphism $\overline \Cal: {\rm Hameo(}D^2,\del D^2) \to \R$ to that of the vanishing of the basic phase function $f_{\underline{\mathbb F}}$, a Floer theoretic graph selector constructed in \cite{oh:jdg}, 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 ${\rm Hameo(}D^2,\del D^2)$ is the group of Hamiltonian homeomorphisms introduced by M\"uller and the author \cite{oh:hameo1}. 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 Hamiton-Jacobi equation.