On the group of strong symplectic homeomorphisms

We generalize the hamiltonian topology on hamiltonian isotopies to an intrinsic symplectic topology on the space of symplectic isotopies. We use it to define the group $SSympeo(M,omega)$ of strong symplectic homeomorphisms, which generalizes the group $Hameo(M,omega)$ of hamiltonian homeomorphisms introduced by Oh and Muller. The group $SSympeo(M,omega)$ is arcwise connected, is contained in the identity component of $Sympeo(M,omega)$; it contains $Hameo(M,omega)$ as a normal subgroup and coincides with it when $M$ is simply connected. Finally its commutator subgroup $[SSympeo(M,omega),SSympeo(M,omega)]$ is contained in $Hameo(M,omega)$.

A Hofer-like metric on the group of symplectic diffeomorphisms

Using a Hodge decomposition of symplectic isotopies on a compact symplectic manifold $(M,omega)$, we construct a norm on the identity component in the group of all symplectic diffeomorphisms of $(M,omega)$ whose restriction to the group $Ham(M,omega)$ of hamiltonian diffeomorphisms is bounded from above by the Hofer norm. Moreover, $Ham(M,omega)$ is closed in $Symp(M,omega)$ equipped with the topology induced by the extended norm. We give an application to the $C^0$ symplectic topology. We also discuss extensions of Ohs spectral distance.