On Continuity of Quasimorphisms for Symplectic Maps

  • Michael Entov
  • Leonid Polterovich
  • Pierre Py
  • Michael Khanevsky
Chapter
First Online:
Part of the Progress in Mathematics book series (PM, volume 296)

Abstract

We discuss C0-continuous homogeneous quasimorphisms on the identity component of the group of compactly supported symplectomorphisms of a symplectic manifold. Such quasimorphisms extend to the C0-closure of this group inside the homeomorphism group. We show that for standard symplectic balls of any dimension, as well as for compact oriented surfaces other than the sphere, the space of such quasimorphisms is infinite-dimensional. In the case of surfaces, we give a user-friendly topological characterization of such quasimorphisms. We also present an application to Hofer’s geometry on the group of Hamiltonian diffeomorphisms of the ball.

Keywords

Symplectomorphism Quasimorphism Calabi homomorphism Hofer metric 
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Notes

Acknowledgments

This text started as an attempt to understand a remark of Dieter Kotschick. We thank him for stimulating discussions and in particular for communicating to us the idea of getting the continuity from the C0-fragmentation, which appeared in a preliminary version of [29]. The authors would like to thank warmly Frédéric Le Roux for his comments on this work and for the thrilling discussions we had during the preparation of this article, Felix Schlenk for critical remarks on the first draft of this paper, as well as Dusa McDuff for a useful discussion. The third author would like to thank Tel-Aviv University for its hospitality during the spring of 2008, when this work began. The second author expresses his deep gratitude to Oleg Viro for generous help and support at the beginning of his research in topology.

Finally, the authors would like to thank warmly the anonymous referee for his careful reading and for finding several inaccuracies in the first version of the text.

Michael Entov was partially supported by the Israel Science Foundation grant # 881/06. Leonid Polterovich was partially supported by the Israel Science Foundation grant # 509/07. Pierre Py was partially supported by the NSF (grant DMS-0905911).

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Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  • Michael Entov
    • 1
  • Leonid Polterovich
    • 2
    • 3
  • Pierre Py
    • 3
  • Michael Khanevsky
    • 2
  1. 1.Department of MathematicsTechnion – Israel Institute of TechnologyHaifaIsrael
  2. 2.School of Mathematical SciencesTel Aviv UniversityTel AvivIsrael
  3. 3.Department of MathematicsUniversity of ChicagoChicagoUSA

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