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

Michael Brandenbursky; Jarek Kedra; Egor Shelukhin

doi:10.1142/S0219199717500420

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

Michael Brandenbursky, Jarek Kedra, Egor Shelukhin

Mathematical Science

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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 language	English
Article number	1750042
Number of pages	27
Journal	Communications in Contemporary Mathematics
Volume	20
Issue number	2
Early online date	24 May 2017
DOIs	https://doi.org/10.1142/S0219199717500420
Publication status	Published - 2 Mar 2018

Keywords

braid groups
autonomous norm
quasi-morphisms
Hamiltonian diffeomorphisms

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10.1142/S0219199717500420Licence: Unspecified

Accepted Manuscript
Electronic version of an article published as Communications in Contemporary Mathematics [Volume 20, Issue , 2018, 1750042 ] DOI: 10.1142/S0219199717500420 © copyright World Scientific Publishing Company. http://www.worldscientific.com/worldscinet/ccm
Accepted author manuscript, 286 KBLicence: Unspecified

https://arxiv.org/abs/1602.03287Licence: Unspecified

Jarek Kedra
- School of Natural & Computing Sciences, Mathematical Science - Personal Chair
Person: Academic

Cite this

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title = "On the autonomous norm on the group of Hamiltonian diffeomorphisms of the torus",

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.",

keywords = "braid groups, autonomous norm, quasi-morphisms, Hamiltonian diffeomorphisms",

author = "Michael Brandenbursky and Jarek Kedra and Egor Shelukhin",

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 K{\c e}dra in Montreal. K{\c e}dra also thanks the Max Planck Institute for Mathematics for supporting his visit in Bonn. Part of this work has been done during Brandenbursky{\textquoteright}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{\textquoteright}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.",

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T1 - On the autonomous norm on the group of Hamiltonian diffeomorphisms of the torus

AU - Brandenbursky, Michael

AU - Kedra, Jarek

AU - Shelukhin, Egor

N1 - 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.

PY - 2018/3/2

Y1 - 2018/3/2

N2 - 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.

AB - 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.

KW - braid groups

KW - autonomous norm

KW - quasi-morphisms

KW - Hamiltonian diffeomorphisms

U2 - 10.1142/S0219199717500420

DO - 10.1142/S0219199717500420

M3 - Article

VL - 20

JO - Communications in Contemporary Mathematics

JF - Communications in Contemporary Mathematics

SN - 0219-1997

IS - 2

M1 - 1750042

ER -

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

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Jarek Kedra

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