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

TL;DR

This paper extends the classical action function from Hamiltonian diffeomorphisms to Hamiltonian homeomorphisms on surfaces, providing new dynamical insights and applications to group properties and the Zimmer conjecture.

Contribution

It introduces a generalized action function for Hamiltonian homeomorphisms on surfaces, broadening the scope of symplectic invariants beyond smooth settings.

Findings

Generalized action function is well-defined under boundedness conditions.

Proved the non-constancy of the generalized action function, extending Schwarz's result.

Applied the generalization to show groups of Hamiltonian homeomorphisms are distortion free.

Abstract

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.