A generalization of classical action of Hamiltonian diffeomorphisms to Hamiltonian homeomorphisms on fixed points

We define boundedness properties on the contractible fixed points set of the time-one map of an identity isotopy on a closed oriented surface with genus $g\geq1$. In symplectic geometry, a classical object is the notion of action function, defined on the set of contractible fixed points of the time-one map of a Hamiltonian isotopy. We give a dynamical interpretation of this function that permits us to generalize it in the case of a homeomorphism isotopic to identity that preserves a Borel finite measure of rotation vector zero, provided that a boundedness condition is satisfied. We give some properties of the generalized action. In particular, we generalize a result of Schwarz [Pacific J. Math.,2000] about the action function being non-constant which has been proved by using Floer homology. As applications, we generalize some results of Polterovich [Invent. Math.,2002] about the symplectic and Hamiltonian diffeomorphisms groups on closed oriented surfaces being distortion free, which permits us to give an alternative proof of the $C^1$-version of the Zimmer conjecture on closed oriented surfaces.