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

TL;DR

This paper explores the structure of the area-preserving homeomorphism group of the 2-disc, establishing key isotopy results and linking the Calabi homomorphism to Floer theory, aiming to prove its nonsimpleness.

Contribution

It introduces Alexander isotopy in Hamiltonian homeomorphisms and relates the Calabi homomorphism extendability to a Floer-theoretic phase function, advancing understanding of the group's algebraic properties.

Findings

Established Alexander isotopy for Hamiltonian homeomorphisms.

Linked Calabi homomorphism extendability to a Floer-theoretic phase function.

Proved the vanishing conjecture for weakly graphical loops via Hamilton-Jacobi analysis.

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.