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Korean Mathematical Society
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ISSN(Print) 0304-9914 ISSN(Online) 2234-3008

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J. Korean Math. Soc. 2008; 45(6): 1769-1784

Printed November 1, 2008

Copyright © The Korean Mathematical Society.

The group of Hamiltonian homeomorphisms in the $L^\infty$-norm

Stefan M\"uller

Korea Institute for Advanced Study

Abstract

The group ${\rm Hameo} \, (M,\omega)$ 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 ${\rm Ham} \, (M,\omega)$ 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.

Keywords: Hamiltonian homeomorphism, $L^\infty$-Hofer norm, $L^{(1,\infty)}$-Hofer norm, Hamiltonian topology

MSC numbers: 53D05, 53D35

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