Let $(M,\omega)$ be a symplectic manifold and let $\operatorname{Ham}(M,\omega)$ be the group of compactly supported Hamiltonian diffeomorphisms equipped with the Hofer metric. I would like to collect properties of the Hofer topology (i.e. the topology on $\operatorname{Ham}(M,\omega)$ induced by the Hofer metric). I am also interested in open problems and examples. For example:
- $\operatorname{Ham}(M,\omega)$ is connected and separable.
- A Hofer open set is also $C^{\infty}$-open and the converse is not true.
- The completion of $\operatorname{Ham}(M,\omega)$ is a Polish group.
- Let $f\in \operatorname{Ham}(M,\omega)$ be an element of infinite order. What are possible topologies on the infinite cyclic subgroup generated by $f$ induced by the Hofer topology?
One property/question/example per answer, please.
