  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.
