The action of Hamiltonian homeomorphisms on surfaces and its applications
Jian Wang(IMPA)

In symplectic geometry, the action function is a classical object defined on the set of contractible fixed points of the time-one map of a Hamiltonian isotopy. Under a weaker boundedness condition (WB for short), we can generalise the classical action function to the case of Hamiltonian homeomorphisms on surfaces. Through studying the properties of the generalised action function, we can generalise several classical results from the smooth world to the C0 world, e.g., the C0-Schwarz’s theorem (that is, the existence of two actions of a non-trivial Hamiltonian homeomorphism), the existence of three actions of a non-trivial Hamiltonian homeomorphism under the WB and a natural topological hypothesis (which is a strengthening of the C0-Arnold Conjecture on surfaces), etc. Mainly in collaboration with Frédéric Le Roux (work in progress).