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 HomeoΩ(D2 , ∂D2 ) of area pre- serving homeomorphisms of the 2-disc D2. We first establish the exis- tence of Alexander isotopy in the category of Hamiltonian homeomor- phisms. This reduces the question of extendability of the well-known Calabi homomorphism Cal : DiffΩ(D1 , ∂D2) → R to a homomorphism Cal : Hameo(D2 , ∂D2 ) → R to that of the vanishing of the basic phase function fF, a Floer theoretic graph selector constructed in [9], that is associated to the graph of the topological Hamiltonian loop and its nor- malized Hamiltonian F on S2 that is obtained via the natural embedding D2 ֒→ S2. Here Hameo(D2, ∂D2 ) is the group of Hamiltonian homeo- morphisms introduced by Mu¨ller and the author [18]. We then provide an evidence of this vanishing conjecture by proving the conjecture for the special class of weakly graphical topological Hamiltonian loops on D2 via the associated Hamiton-Jacobi equation.

c 2016 Korean Mathematical Society