The Group of Hamiltonian Homeomorphisms in the L^\infty-norm

TL;DR

This paper investigates the group of Hamiltonian homeomorphisms on symplectic manifolds using the L^ty-norm, establishing that it coincides with the previously studied group defined via the L^{(1,\u221e)}-Hofer norm.

Contribution

The authors prove that the Hamiltonian homeomorphism groups defined with the L^ty-norm and the L^{(1,)}-Hofer norm are identical, confirming a conjecture and clarifying the norm dependence.

Findings

The L^ty-Hofer norm defines the same Hamiltonian homeomorphism group as the L^{(1,)}-Hofer norm.

The group of Hamiltonian homeomorphisms is independent of the choice between these two norms.

The result aligns with the known norm invariance for Hamiltonian diffeomorphisms.

Abstract

The group Hameo (M,\omega) of Hamiltonian homeomorphisms of a connected symplectic manifold (M,\omega) was defined and studied in [7] and further in [6]. In these papers, the authors consistently used the L^{(1,\infty)}-Hofer norm (and not the L^\infty-Hofer norm) on the space of Hamiltonian paths (see below for the definitions). A justification for this choice was given in [7]. In this article we study the L^\infty-case. In view of the fact that the Hofer norm on the group Ham (M,\omega) of Hamiltonian diffeomorphisms does not depend on the choice of the L^{(1,\infty)}-norm vs. the L^\infty-norm [9], Y.-G. Oh and D. McDuff (private communications) asked whether the two notions of Hamiltonian homeomorphisms arising from the different norms coincide. We will give an affirmative answer to this question in this paper.