The group of Hamiltonian homeomorphisms and $C^0$ symplectic topology

Details The group of Hamiltonian homeomorphisms and $C^0$ symplectic topology The group of Hamiltonian homeomorphisms and $C^0$ symplectic topology

2004-02-13

arxiv.org/abs/math/0402210v4

J. Symp. Geom. 5 (2007), 167-220

Mathematics

Symplectic Geometry

48 pages ; Many erroneous definitions and details of proofs are corrected. An additional author is added. But all the main the

Scientific paper

The main purpose of this paper is to carry out some of the foundational study of $C^0$-Hamiltonian geometry and $C^0$-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} $Hameo(M,\omega)$ and the space $Hameo^w(M,\omega)$ of {\it Hamiltonian homeomorphisms} such that $$ Ham(M,\omega) \subsetneq Hameo(M,\omega) \subset Hameo^w(M,\omega) \subset Sympeo(M,\omega) $$ where $Sympeo(M,\omega)$ is the group of symplectic homeomorphisms. We prove that $Hameo(M,\omega)$ is a {\it normal subgroup} of $Sympeo(M,\omega)$ and contains all the time-one maps of Hamiltonian vector fields of $C^{1,1}$-functions. We prove that $Hameo(M,\omega)$ is path connected and so contained in the identity component $Sympeo_0(M,\omega)$ of $Sympeo(M,\omega)$. In the case of an orientable surface, we prove that the {\it mass flow} of any element from $Hameo(M,\omega)$ vanishes, which in turn implies that $Hameo(M,\omega)$ is strictly smaller than the identity component of the group of area preserving homeomorphisms when $M \neq S^2$. For the case of $S^2$, we conjecture that $Hameo(S^2,\omega)$ is still a proper subgroup of $Homeo^\omega_0(S^2) = Sympeo_0(S^2,\omega)$.

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