The group of Hamiltonian homeomorphisms and continuous Hamiltonian flows

Abstract

In this paper, we study the dynamical aspects of the \emph{Hamiltonian homeomorphism group} $Hameo(M,\omega )$ which was introduced by Müller and the author. We introduce the notion of autonomous continuous Hamiltonian flows and extend the well-known conservation of energy to such flows. The definitions of the Hofer length and of the spectral invariants ${\rho }_a$ are extended to continuous Hamiltonian paths, and the Hofer norm and the spectral norm $\gamma :Ham(M,\omega )\to {\mathbb{R}}_{+}$ are generalized to the corresponding intrinsic norms on $Hameo(M,\omega )$ respectively. Using these extensions, we also extend the construction of Entov-Polterovich's Calabi quasi-morphism on $S^2$ to the space of continuous Hamiltonian paths. We also discuss a conjecture concerning extendability of Entov-Polterovich's quasi-morphism and its relation to the extendability of Calabi homomorphism on the disc to $Hameo(D^2,\del D^2)$, and their implication towards the simpleness question on the area preserving homeomorphism groups of the disc $D^2$ and of the sphere $S^2$.

28 pages; All main theorems are re-stated in $C^0$ setting due to lack of the proof of uniqueness of Hamiltonian in $L^{(1,\infty )}$ setting, a section containing a discussion of a wild homeomorphism is added; (v6) the final version to appear, English improved, presentation improved by rearranging some sections and adding explanations