In symplectic geometry, the action function is a classical object defined on the set of contractible fixed points of the time-one map of a Hamiltonian isotopy. On closed aspherical surfaces, we give a dynamical interpretation of this function, which permits us to generalize it to the case of a diffeomorphism that is isotopic to identity and 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. We generalize the classical action function to any Hamiltonian homeomorphism, provided that the proposed boundedness condition is satisfied. We prove that the generalized action function only depends on the time-one map but not on the isotopy. Finally, we define the action spectrum and show that it is invariant under conjugation by an orientation and measure preserving homeomorphism.
Annales Henri Poincaré – Springer Journals
Published: Jun 13, 2017
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