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Pseudorotations of the $2$-disc and Reeb flows on the $3$-sphere

Part of: Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems Symplectic geometry, contact geometry

Published online by Cambridge University Press:  18 March 2021

PETER ALBERS Open the ORCID record for PETER ALBERS [Opens in a new window] ,
HANSJÖRG GEIGES Open the ORCID record for HANSJÖRG GEIGES [Opens in a new window]  and
KAI ZEHMISCH Open the ORCID record for KAI ZEHMISCH [Opens in a new window]
PETER ALBERS
Affiliation:
Mathematisches Institut, Universität Heidelberg, Im Neuenheimer Feld 205, 69120Heidelberg, Germany (e-mail: peter.albers@uni-heidelberg.de)
HANSJÖRG GEIGES*
Affiliation:
Mathematisches Institut, Universität zu Köln, Weyertal 86–90, 50931Köln, Germany
KAI ZEHMISCH
Affiliation:
Fakultät für Mathematik, Ruhr-Universität Bochum, Universitätsstraße 150, 44780Bochum, Germany (e-mail: kai.zehmisch@rub.de)
*
e-mail: geiges@math.uni-koeln.de
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Abstract

We use Lerman’s contact cut construction to find a sufficient condition for Hamiltonian diffeomorphisms of compact surfaces to embed into a closed $3$ -manifold as Poincaré return maps on a global surface of section for a Reeb flow. In particular, we show that the irrational pseudorotations of the $2$ -disc constructed by Fayad and Katok embed into the Reeb flow of a dynamically convex contact form on the $3$ -sphere.

Keywords

area-preserving diffeomorphisms of the disccontact cutglobal surface of sectionopen book decompositionpseudorotation

MSC classification

Primary: 37J05: General theory, relations with symplectic geometry and topology
Secondary: 37J55: Contact systems 53D35: Global theory of symplectic and contact manifolds
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 42 , Issue 2: Anatole Katok Memorial Issue Part 1: Special Issue of Ergodic Theory and Dynamical Systems , February 2022 , pp. 402 - 436
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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Footnotes

To the memory of Anatole Katok

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