TY - JOUR

T1 - On the group of symplectic homeomorphisms

AU - Banyaga, Augustin

N1 - Copyright:
Copyright 2008 Elsevier B.V., All rights reserved.

PY - 2008/8

Y1 - 2008/8

N2 - Let (M, ω) be a closed symplectic manifold. We define a Hofer-like metric d on the identity component Sym (M, ω)0 in the group Symp (M, ω) of all symplectic diffeomorphisms of (M, ω). Unlike the Hofer metric on the group Ham (M, ω) of Hamiltonian diffeomorphisms, the metric d is not bi-invariant. We show that the metric topology τ defined by d is natural (i.e. independent of the choice involved in its definition). We define the symplectic topology as a blend of the Hofer-like topology τ and the C0-topology. We use it to construct a subgroup SSympeo (M, ω) of the group Sympeo (M, ω) of all symplectic homeomorphisms, containing the group Hameo (M, ω) of Hamiltonian homeomorphisms (introduced by Oh and Muller). If M is simply connected SSympeo (M, ω) coincides with Hameo (M, ω). Moreover its commutator subgroup [SSympeo (M, ω), SSympeo (M, ω)] is contained in Hameo (M, ω). To cite this article: A. Banyaga, C. R. Acad. Sci. Paris, Ser. I 346 (2008).

AB - Let (M, ω) be a closed symplectic manifold. We define a Hofer-like metric d on the identity component Sym (M, ω)0 in the group Symp (M, ω) of all symplectic diffeomorphisms of (M, ω). Unlike the Hofer metric on the group Ham (M, ω) of Hamiltonian diffeomorphisms, the metric d is not bi-invariant. We show that the metric topology τ defined by d is natural (i.e. independent of the choice involved in its definition). We define the symplectic topology as a blend of the Hofer-like topology τ and the C0-topology. We use it to construct a subgroup SSympeo (M, ω) of the group Sympeo (M, ω) of all symplectic homeomorphisms, containing the group Hameo (M, ω) of Hamiltonian homeomorphisms (introduced by Oh and Muller). If M is simply connected SSympeo (M, ω) coincides with Hameo (M, ω). Moreover its commutator subgroup [SSympeo (M, ω), SSympeo (M, ω)] is contained in Hameo (M, ω). To cite this article: A. Banyaga, C. R. Acad. Sci. Paris, Ser. I 346 (2008).

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U2 - 10.1016/j.crma.2008.06.011

DO - 10.1016/j.crma.2008.06.011

M3 - Article

AN - SCOPUS:49449093705

SN - 1631-073X

VL - 346

SP - 867

EP - 872

JO - Comptes Rendus Mathematique

JF - Comptes Rendus Mathematique

IS - 15-16

ER -