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The main purpose of this paper is to propose a scheme of a proof of thenonsimpleness of the group $Homeo^\Omega(D^2,\partial D^2)$ of area preservinghomeomorphisms of the 2-disc $D^2$. We first establish the existence ofAlexander isotopy in the category of Hamiltonian homeomorphisms. This reducesthe question of extendability of the well-known Calabi homomorphism ${\rmCal}:Diff^\Omega(D^1,\partial D^2) \to \mathbb R$ to a homomorphism$\overline{{\rm Cal}}:Hameo(D^2,\partial D^2) \to \mathbb R$ to that of thevanishing of the basic phase function $f_{\underline{\mathbb F}}$, a Floertheoretic graph selector previously constructed by the author, that isassociated to the graph of the topological Hamiltonian loop and its normalizedHamiltonian $\underline{F}$ 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 ofHamiltonian homeomorphisms introduced by M\"uller and the author. We thenprovide an evidence of this vanishing conjecture by proving the conjecture forthe special class of \emph{weakly graphical} topological Hamiltonian loops on$D^2$ via a study of the associated Hamilton-Jacobi equation.

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