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THE GROUP OF HAMILTONIAN HOMEOMORPHISMS IN THE L-NORM
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 Title & Authors
THE GROUP OF HAMILTONIAN HOMEOMORPHISMS IN THE L-NORM
Muller, Stefan;
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 Abstract
The group Hameo (M, ) of Hamiltonian homeomorphisms of a connected symplectic manifold (M, ) was defined and studied in [7] and further in [6]. In these papers, the authors consistently used the -Hofer norm (and not the -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 -case. In view of the fact that the Hofer norm on the group Ham (M, ) of Hamiltonian diffeomorphisms does not depend on the choice of the -norm vs. the -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;-Hofer norm;-Hofer norm;Hamiltonian topology;
 Language
English
 Cited by
1.
Helicity of vector fields preserving a regular contact form and topologically conjugate smooth dynamical systems, Ergodic Theory and Dynamical Systems, 2013, 33, 05, 1550  crossref(new windwow)
2.
Uniform approximation of homeomorphisms by diffeomorphisms, Topology and its Applications, 2014, 178, 315  crossref(new windwow)
 References
1.
H. Hofer, On the topological properties of symplectic maps, Proc. Roy. Soc. Edinburgh Sect. A 115 (1990), no. 1-2, 25-38

2.
F. Lalonde and D. McDuff, The geometry of symplectic energy, Ann. of Math. (2) 141 (1995), no. 2, 349-371 crossref(new window)

3.
Y. Oh, Chain level Floer theory and Hofer's geometry of the Hamiltonian diffeomorphism group, Asian J. Math. 6 (2002), no. 4, 579-624; Erratum to: "Chain level Floer theory and Hofer's geometry of the Hamiltonian diffeomorphism group", Asian J. Math. 7 (2003), no. 3, 447-448

4.
Y. Oh, Uniqueness of L(1,$\infty$)-Hamiltonians and almost-every-moment Lagrangian disjunction, unpublished, math.SG/0612831

5.
Y. Oh, Locality of continuous Hamiltonian flows and Lagrangian intersections with the conormal of open subsets, preprint, math.SG/0612795

6.
Y. Oh, The group of Hamiltonian homeomorphisms and topological Hamiltonian flows, preprint, math.SG/0601200

7.
Y. Oh and S. M¨uller, The group of Hamiltonian homeomorphisms and $C^{0}$-symplectic topology, to appear in J. Symplectic Geom., math.SG/0402210v4

8.
L. Polterovich, Symplectic displacement energy for Lagrangian submanifolds, Ergodic Theory Dynam. Systems 13 (1993), no. 2, 357-367

9.
L. Polterovich, The Geometry of the Group of Symplectic Diffeomorphisms, Lectures in Mathematics ETH Zurich. Birkhauser Verlag, Basel, 2001

10.
C. Viterbo, On the uniqueness of generating Hamiltonian for continuous limits of Hamiltonians flows, Int. Math. Res. Not. 2006, Art. ID 34028, 9 pp; Erratum to: "On the uniqueness of generating Hamiltonian for continuous limits of Hamiltonians flows", Int. Math. Res. Not. 2006, Art. ID 38784, 4 pp