The group of Hamiltonian homeomorphisms and $C^0$ symplectic topology
| dc.creator | Oh, Yong-Geun | |
| dc.creator | Müller, Stefan | |
| dc.date | 2004-02-13 | |
| dc.date | 2006-05-17 | |
| dc.date.accessioned | 2026-07-07T09:19:24Z | |
| dc.date.available | 2026-07-07T09:19:24Z | |
| dc.description | 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} $Hameo(M,ω)$ and the space $Hameo^w(M,ω)$ of {\it Hamiltonian homeomorphisms} such that $$ Ham(M,ω) \subsetneq Hameo(M,ω) \subset Hameo^w(M,ω) \subset Sympeo(M,ω) $$ 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 $C^{1,1}$-functions. We prove that $Hameo(M,ω)$ is path connected and so contained in the identity component $Sympeo_0(M,ω)$ 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 $M \neq S^2$. For the case of $S^2$, we conjecture that $Hameo(S^2,ω)$ is still a proper subgroup of $Homeo^ω_0(S^2) = Sympeo_0(S^2,ω)$. | |
| dc.description | 48 pages ; Many erroneous definitions and details of proofs are corrected. An additional author is added. But all the main theorems stated in the introduction of the previous version remain to hold | |
| dc.identifier | https://arxiv.org/abs/math/0402210 | |
| dc.identifier | http://arxiv.org/abs/math/0402210 | |
| dc.identifier | J. Symp. Geom. 5 (2007), 167-220 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/154375 | |
| dc.subject | Symplectic Geometry | |
| dc.subject | Dynamical Systems | |
| dc.subject | 53D05, 53D35 | |
| dc.title | The group of Hamiltonian homeomorphisms and $C^0$ symplectic topology | |
| dc.type | text |