On the autonomous norm on the group of Hamiltonian diffeomorphisms of the torus

Michael Brandenbursky, Jarek Kedra, Egor Shelukhin

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Abstract

We prove that the autonomous norm on the group of Hamiltonian diffeomorphisms of the two-dimensional torus is unbounded. We provide examples of Hamiltonian diffeomorphisms with arbitrarily large autonomous norm. For the proofs we construct explicit quasimorphisms on Ham(T2), some of them are C0-continuous and vanish on all autonomous diffeomorphisms, and some of them are Calabi.
Original languageEnglish
Article number1750042
Number of pages27
JournalCommunications in Contemporary Mathematics
Volume20
Issue number2
Early online date24 May 2017
DOIs
Publication statusPublished - 2 Mar 2018

Bibliographical note

We would like to thank Luis Haug for fruitful discussions and Pierre Py for comments on the first version of the paper. We thank the CRM and ISM who supported the visit of Kędra in Montreal. Kędra also thanks the Max Planck Institute for Mathematics for supporting his visit in Bonn.
Part of this work has been done during Brandenbursky’s stay at IHES and CRM. We wish to express his gratitude to both institutes. Brandenbursky was supported by CRM-ISM fellowship and NSF grant No. 1002477.
This work has been done during Shelukhin’s stay in CRM, ICJ Lyon 1, Institut Mittag Leffler, and IAS. He thanks these institutions for their warm hospitality. He was partially supported by CRM-ISM fellowship, ERC Grant RealUMan, Mittag Leffler fellowship, and NSF grant No. DMS-1128155.

Keywords

  • braid groups
  • autonomous norm
  • quasi-morphisms
  • Hamiltonian diffeomorphisms

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