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

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

Published online by Cambridge University Press:  18 March 2021

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

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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 systems53D35: Global theory of symplectic and contact manifolds

Information

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

Footnotes

To the memory of Anatole Katok

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