Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-24T00:10:50.842Z Has data issue: false hasContentIssue false
  • English
  • Français

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

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

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: [email protected])
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: [email protected])
*
e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

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
Check for updates
Copyright
© The Author(s), 2021. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

To the memory of Anatole Katok

References

Abbondandolo, A., Bramham, B., Hryniewicz, U. L. and Salomão, P. A. S.. Sharp systolic inequalities for Reeb flows on the $3$ -sphere. Invent. Math. 211 (2018), 687–778.CrossRefGoogle Scholar
Albers, P., Fish, J. W., Frauenfelder, U., Hofer, H. and van Koert, O.. Global surfaces of section in the planar restricted $3$ -body problem. Arch. Ration. Mech. Anal. 204 (2012), 273–284.CrossRefGoogle Scholar
Albers, P., Frauenfelder, U., van Koert, O. and Paternain, G.. Contact geometry of the restricted three-body problem. Comm. Pure Appl. Math. 65 (2012), 229–263.CrossRefGoogle Scholar
Anosov, D. V. and Katok, A. B.. New examples in smooth ergodic theory. Ergodic diffeomorphisms. Trans. Moscow Math. Soc. 23 (1970), 1–35.Google Scholar
Bourgeois, F., Cieliebak, K. and Ekholm, T.. A note on Reb dynamics on the tight $3$ -sphere. J. Mod. Dyn. 1 (2007), 597–613.10.3934/jmd.2007.1.597CrossRefGoogle Scholar
Bramham, B.. Periodic approximations of irrational pseudo-rotations using pseudo-holomorphic curves. Ann. of Math. (2) 181 (2015), 1033–1086.CrossRefGoogle Scholar
Bramham, B.. Pseudo-rotations with sufficiently Liouvillean rotation number are ${C}^0$ -rigid. Invent. Math. 199 (2015), 561–580.CrossRefGoogle Scholar
Cristofaro-Gardiner, D. and Hutchings, M.. From one Reeb orbit to two. J. Differential Geom. 102 (2016), 25–36.Google Scholar
Cristofaro-Gardiner, D. and Mazzucchelli, M.. The action spectrum characterizes closed contact $3$ -manifolds all of whose Reeb orbits are closed. Comment. Math. Helv. 95 (2020), 461–481.CrossRefGoogle Scholar
Ding, F. and Geiges, H.. Contact structures on principal circle bundles. Bull. Lond. Math. Soc. 44 (2012), 1189–1202.10.1112/blms/bds042CrossRefGoogle Scholar
Dörner, M.. The space of contact forms adapted to an open book. Inaugural Dissertation, Universität zu Köln, 2014.Google Scholar
Fayad, B. and Katok, A.. Constructions in elliptic dynamics. Ergod. Th. & Dynam. Sys. 24 (2004), 1477–1520.CrossRefGoogle Scholar
Geiges, H.. An Introduction to Contact Topology (Cambridge Studies in Advanced Mathematics, 109). Cambridge University Press, Cambridge, 2008.CrossRefGoogle Scholar
Geiges, H.. Controlled Reeb dynamics—three lectures not in Cala Gonone. Complex Manifolds 6 (2019), 118–137.CrossRefGoogle Scholar
Geiges, H. and Zehmisch, K.. Odd-symplectic forms via surgery and minimality in symplectic dynamics. Ergod. Th. & Dynam. Sys. 40 (2020), 699–713.10.1017/etds.2018.60CrossRefGoogle Scholar
Giroux, E.. Géométrie de contact: de la dimension trois vers les dimensions supérieures. Proc. Int. Cong. of Math., Vol. II (Beijing, 2002). Higher Education Press, Beijing, 2002, pp. 405–414.Google Scholar
Giroux, E.. Ideal Liouville domains, a cool gadget. J. Symplectic Geom. 18 (2020), 769–790.10.4310/JSG.2020.v18.n3.a5CrossRefGoogle Scholar
Giroux, E. and Massot, P.. On the contact mapping class group of Legendrian circle bundles. Compos. Math. 153 (2017), 294–312.CrossRefGoogle Scholar
Harrison, J. and Pugh, C.. A fixed-point free ergodic flow on the $3$ -sphere. Michigan Math. J. 36 (1989), 261–266.CrossRefGoogle Scholar
Hofer, H., Wysocki, K. and Zehnder, E.. A characterisation of the tight three-sphere. Duke Math. J. 81 (1995), 159–226.CrossRefGoogle Scholar
Hofer, H., Wysocki, K. and Zehnder, E.. The dynamics on three-dimensional strictly convex energy surfaces. Ann. of Math. (2) 148 (1998), 197–289.CrossRefGoogle Scholar
Hofer, H., Wysocki, K. and Zehnder, E.. Finite energy foliations of tight three-spheres and Hamiltonian dynamics. Ann. of Math. (2) 157 (2003), 125–255.CrossRefGoogle Scholar
Hryniewicz, U. L., Licata, J. E. and Salomão, P. A. S.. A dynamical characterization of universally tight lens spaces. Proc. Lond. Math. Soc. (3) 110 (2015), 213–269.CrossRefGoogle Scholar
Hryniewicz, U. L., Momin, A. and Salomão, P. A. S.. A Poincaré–Birkhoff theorem for tight Reeb flows on ${S}^3$ . Invent. Math. 199 (2015), 333–422.CrossRefGoogle Scholar
Hryniewicz, U. L. and Salomão, P. A. S.. On the existence of disk-like global sections for Reeb flows on the tight $3$ -sphere. Duke Math. J. 160 (2011), 415–465.CrossRefGoogle Scholar
Hryniewicz, U. L. and Salomão, P. A. S.. Global surfaces of section for Reeb flows in dimension three and beyond. Proc. Int. Cong. of Math., Vol. II (Rio de Janeiro, 2018). World Scientific, Hackensack, NJ, 2018, pp. 937–964.Google Scholar
Hutchings, M.. Mean action and the Calabi invariant. J. Mod. Dyn. 10 (2016), 511–539.CrossRefGoogle Scholar
Katok, A. B.. Ergodic perturbations of degenerate integrable Hamiltonian systems. Math. USSR Izv. 7 (1973), 535–572.10.1070/IM1973v007n03ABEH001958CrossRefGoogle Scholar
Le Calvez, P. and Yoccoz, J.-C.. Un théorème d’indice pur les homéomorphismes du plan au voisinage d’un point. Ann. of Math. (2) 146 (1997), 241–293.CrossRefGoogle Scholar
Lerman, E.. Contact cuts. Israel J. Math. 124 (2001), 77–92.CrossRefGoogle Scholar
Massot, P., Niederkrüger, K. and Wendl, C.. Weak and strong fillability of higher dimensional contact manifolds. Invent. Math. 192 (2013), 287–373.CrossRefGoogle Scholar
McDuff, D. and Salamon, D.. Introduction to Symplectic Topology (Oxford Graduate Texts in Mathematics, 27), 3rd edn. Oxford University Press, Oxford, 2017.CrossRefGoogle Scholar
Morse, M.. Relations between the critical points of a real function in $n$ independent variables. Trans. Amer. Math. Soc. 27 (1925), 345–396.Google Scholar
Munkres, J.. Differentiable isotopies on the $2$ -sphere. Michigan Math. J. 7 (1960), 193–197.CrossRefGoogle Scholar
Rolfsen, D., Knots and Links (Mathematics Lecture Series, 7). Publish or Perish, Berkeley, CA, 1976.Google Scholar
Schneider, A.. Global surfaces of section for dynamically convex Reeb flows on lens spaces. Trans. Amer. Math. Soc. 373 (2020), 2775–2803.CrossRefGoogle Scholar
Smale, S.. Diffeomorphisms of the $2$ -sphere. Proc. Amer. Math. Soc. 10 (1959), 621–626.CrossRefGoogle Scholar
Wall, C. T. C.. Differential Topology (Cambridge Studies in Advanced Mathematics, 156). Cambridge University Press, Cambridge, 2016.Google Scholar