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 $Home{o}^{\Omega }(D^2,\partial D^2)$ 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 $\mathrm{Cal}:Dif{f}^{\Omega }(D^1,\partial D^2)\to \mathbb{R}$ to a homomorphism $\overline{\mathrm{Cal}}:Hameo(D^2,\partial D^2)\to \mathbb{R}$ 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 Müller 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.

32 pages, Comments welcome; v2) 35 pages, an error in the proof of Theorem 1.8 corrected, a new section 9 added, many typos corrected and overall presentation improved

References in corpus (2)

• Locality of continuous Hamiltonian flows and Lagrangian intersections with the conormal of open subsets
• Geometry of generating functions and Lagrangian spectral invariants

Cited by in corpus (1)

• Topological extension of Calabi homomorphism and non-simpleness of the area-preserving homeomorphism group of two-disc