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Generalizations of the Action Function in Symplectic Geometry

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Abstract

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.

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Acknowledgements

I would like to thank Patrice Le Calvez for many helpful discussions and suggestions. I am grateful to Lucien Guillou and Bassam Fayad for careful reading the manuscript and many useful remarks. Lastly I would like to thank the anonymous referees for pointing out several inaccuracies and for many helpful suggestions which have improved the exposition. I would like to express my deep sorrow over the passing away of Lucien Guillou in 2015.

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Correspondence to Jian Wang.

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Communicated by Dmitry Dolgopyat.

This work was supported by China Postdoctoral Science Foundation (2013T60251); International Postdoctoral Exchange Fellowship Program (20130045); and National Natural Science Foundation of China (11401320).

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Wang, J. Generalizations of the Action Function in Symplectic Geometry. Ann. Henri Poincaré 18, 2945–2993 (2017). https://doi.org/10.1007/s00023-017-0596-8

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  • DOI: https://doi.org/10.1007/s00023-017-0596-8

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