Article
The Group of Hamiltonian Homeomorphisms and C0-Symplectic Topology
Journal of Symplectic Geometry (2007)
• , University of Wisconsin
• , Korea Institute for Advanced Study
Abstract
The main purpose of this paper is to carry out some of the foundational study of C0-Hamiltonian geometry and C0-symplectic topology. We introduce the notions of the strong and the weak {\it Hamiltonian topology} on the space of Hamiltonian paths, and on the group of Hamiltonian diffeomorphisms. We then define the {\it group} Hameo(M,ω) and the space Hameow(M,ω) of {\it Hamiltonian homeomorphisms} such that Ham(M,ω)⊊Hameo(M,ω)⊂Hameow(M,ω)⊂Sympeo(M,ω) where Sympeo(M,ω) is the group of symplectic homeomorphisms. We prove that Hameo(M,ω) is a {\it normal subgroup} of Sympeo(M,ω) and contains all the time-one maps of Hamiltonian vector fields of C1,1-functions. We prove that Hameo(M,ω) is path connected and so contained in the identity component Sympeo0(M,ω) of Sympeo(M,ω). In the case of an orientable surface, we prove that the {\it mass flow} of any element from Hameo(M,ω) vanishes, which in turn implies that Hameo(M,ω) is strictly smaller than the identity component of the group of area preserving homeomorphisms when M≠S2. For the case of S2, we conjecture that Hameo(S2,ω) is still a proper subgroup of Homeoω0(S2)=Sympeo0(S2,ω).
Keywords
• Hamiltonian Homeomorphisms,
• Homeomorphisms,
• Mathematics
Disciplines
Publication Date
2007
DOI
10.4310/JSG.2007.v5.n2.a2
Publisher Statement