

Preprint 12/2016
A generalization of classical action of Hamiltonian diffeomorphisms to Hamiltonian homeomorphisms
Jian Wang
Contact the author: Please use for correspondence this email.
Submission date: 01. Feb. 2016
Pages: 41
Bibtex
MSC-Numbers: 37E30, 37E45, 37J10
Download full preprint: PDF (421 kB)
Abstract:
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. On closed surfaces, we give a
dynamical interpretation of this function that permits us to
generalize it in the case of a diffeomorphism isotopic to identity
that preserves a Borel finite measure of rotation vector zero. We
define a boundedness property on the contractible fixed points set
of the time-one map of an identity isotopy, which includes the case
where the time-one map is a diffeomorphism and the simple case where
the set of contractible fixed points of the time-one map is finite.
We generalize the classical function to any homeomorphism, provided
that the boundedness condition is satisfied. Finally, we define the
action spectrum which is invariant under conjugation by an
orientation and measure preserving homeomorphism.