A generalization of classical action of Hamiltonian diffeomorphisms to Hamiltonian homeomorphisms
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Submission date: 01. Feb. 2016
MSC-Numbers: 37E30, 37E45, 37J10
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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.