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There is no minimal action of ℤ2 on the plane

Published online by Cambridge University Press:  05 April 2011

FRÉDÉRIC LE ROUX
FRÉDÉRIC LE ROUX*
Affiliation:
Laboratoire de Mathématique (CNRS UMR 8628), Université Paris Sud, 91405 Orsay Cedex, France (email: [email protected])
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Abstract

We prove that there is no minimal action of ℤ2 by homeomorphisms on the plane. This may be seen as a generalization of Le Calvez–Yoccoz’s theorem: there exists no minimal homeomorphism of the infinite annulus.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 32 , Issue 1 , February 2012 , pp. 167 - 172
Copyright
Copyright © Cambridge University Press 2011

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References

[BarFra93]Barge, M. and Franks, J.. Recurrent sets for planar homeomorphisms. From Topology to Computation: Proceedings of the Smalefest (Berkeley, CA, 1990). Springer, New York, 1993, pp. 186–195.CrossRefGoogle Scholar
[BegLeR03]Béguin, F. and Le Roux, F.. Ensemble oscillant d’un homéomorphisme de Brouwer, homéomorphismes de Reeb. Bull. Soc. Math. France 131 (2003), 149–210.CrossRefGoogle Scholar
[Ber00]Bernardes, N. C.. On the set of points with a dense orbit. Proc. Amer. Math. Soc. 128(11) (2000), 3421–3423.CrossRefGoogle Scholar
[Bes51]Besicovitch, A. S.. A problem on topological transformations of the plane. II. Proc. Cambridge Philos. Soc. 47 (1951), 38–45.CrossRefGoogle Scholar
[Bon04]Bonino, M.. A Brouwer-like theorem for orientation reversing homeomorphisms of the sphere. Fund. Math. 182(1) (2004), 1–40.CrossRefGoogle Scholar
[Bro12]Brouwer, L. E. J.. Beweis des ebenen Translationssatzes. Math. Ann. 72 (1912), 37–54.CrossRefGoogle Scholar
[ConKol94]Constantin, A. and Kolev, B.. The theorem of Kérékjártó on periodic homeomorphisms of the disk and the sphere. Enseign. Math. (2) 40(3–4) (1994), 193–204.Google Scholar
[Fra92]Franks, J.. A new proof of the Brouwer plane translation theorem. Ergod. Th. & Dynam. Sys. 12 (1992), 217–226.CrossRefGoogle Scholar
[Ham65]Hamstrom, M.-E.. Homotopy properties of the space of homeomorphisms on P 2 and the Klein bottle. Trans. Amer. Math. Soc. 120 (1965), 37–45.Google Scholar
[Gui94]Guillou, L.. Théorème de translation plane de Brouwer et généralisations du théorème de Poincaré–Birkhoff. Topology 33 (1994), 331–351.CrossRefGoogle Scholar
[HomTer53]Homma, T. and Terasaka, H.. On the structure of the plane translation of Brouwer. Osaka J. Math. 5 (1953), 233–266.Google Scholar
[LecYoc97]Le Calvez, P. and Yoccoz, J.-C.. Un théorème d’indice pour les homéomorphismes du plan au voisinage d’un point fixe. Ann. of Math. (2) 146(2) (1997), 241–293.CrossRefGoogle Scholar
[LeR04]Le Roux, F.. Homéomorphismes de surfaces: théorèmes de la fleur de Leau–Fatou et de la variété stable. Astérisque 292 (2004).Google Scholar