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We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

MiS Preprint
12/2016

A generalization of classical action of Hamiltonian diffeomorphisms to Hamiltonian homeomorphisms

Jian Wang

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.

Received:
Feb 1, 2016
Published:
Feb 10, 2016
MSC Codes:
37E30, 37E45, 37J10

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Preprint
2016 Repository Open Access
Jian Wang

A generalization of classical action of Hamiltonian diffeomorphisms to Hamiltonian homeomorphisms