Hostname: page-component-77c89778f8-7drxs Total loading time: 0 Render date: 2024-07-24T18:53:21.198Z Has data issue: false hasContentIssue false
  • English
  • Français

Free curves and periodic points for torus homeomorphisms

Published online by Cambridge University Press:  15 September 2008

ALEJANDRO KOCSARD  and
ANDRES KOROPECKI
ALEJANDRO KOCSARD
Affiliation:
Instituto Nacional de Matemática Pura e Aplicada, Estrada Dona Castorina, 110, 22460-320, Rio de Janeiro, RJ, Brasil (email: alejo@impa.br)
ANDRES KOROPECKI
Affiliation:
Universidade Federal Fluminense, Instituto de Matemática, Rua Mário Santos Braga S/N, 24020-140 Niterói, RJ, Brasil (email: koro@mat.uff.br)
Get access Rights & Permissions [Opens in a new window]

Abstract

We study the relationship between free curves and periodic points for torus homeomorphisms in the homotopy class of the identity. By free curve we mean a homotopically non-trivial simple closed curve that is disjoint from its image. We prove that every rational point in the rotation set is realized by a periodic point provided that there is no free curve and the rotation set has empty interior. This gives a topological version of a theorem of Franks. Using this result, and inspired by a theorem of Guillou, we prove a version of the Poincaré–Birkhoff theorem for torus homeomorphisms: in the absence of free curves, either there is a fixed point or the rotation set has non-empty interior.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 28 , Issue 6 , December 2008 , pp. 1895 - 1915
Copyright
Copyright © 2008 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.)

References

[1] Béguin, F., Crovisier, S., Le Roux, F. and Patou, A.. Pseudo-rotations of the closed annulus: variation on a theorem of J. Kwapisz. Nonlinearity 17(4) (2004), 1427–1453.CrossRefGoogle Scholar
[2] Bestvina, M. and Handel, M.. An area preserving homeomorphism of 𝕋2 that is fixed point free but does not move any essential simple closed curve off itself. Ergod. Th. & Dynam. Sys. 12 (1992), 673–676.CrossRefGoogle Scholar
[3] Birkhoff, G.. An extension of Poincaré’s last geometric theorem. Acta Math. 47 (1925), 297–311.CrossRefGoogle Scholar
[4] Brouwer, L. E. J.. Beweis des Ebenen Translationssatzes. Math. Ann. 72 (1912), 37–54.CrossRefGoogle Scholar
[5] Fathi, A.. An orbit closing proof of Brouwer’s lemma on translation arcs. Enseign. Math. 33 (1987), 315–322.Google Scholar
[6] Franks, J. and Misiurewicz, M.. Rotation sets for toral flows. Proc. Amer. Math. Soc. 109 (1990), 243–249.CrossRefGoogle Scholar
[7] Franks, J.. Recurrence and fixed points of surface homeomorphisms. Ergod. Th. & Dynam. Sys. 8 (1988), 99–107 (Charles Conley Memorial Issue).Google Scholar
[8] Franks, J.. Realizing rotation vectors for torus homeomorphisms. Trans. Amer. Math. Soc. 311(1) (1989), 107–115.CrossRefGoogle Scholar
[9] Franks, J.. A new proof of the Brouwer plane translation theorem. Ergod. Th. & Dynam. Sys. 12 (1992), 217–226.CrossRefGoogle Scholar
[10] Franks, J.. The rotation set and periodic points for torus homeomorphisms. Dynamical Systems and Chaos. Eds. N. Aoki, K. Shiraiwa and Y. Takahashi. World Scientific, Singapore, 1995, pp. 41–48.Google Scholar
[11] Guillou, L.. Théorème de translation plane de Brouwer et généralisations du théorème de Poincaré–Birkhoff. Topology 33 (1994), 331–351.CrossRefGoogle Scholar
[12] Guillou, L.. Free lines for homeomorphisms of the open annulus. Trans. Amer. Math. Soc. 360(4) (2008), 2191–2204.CrossRefGoogle Scholar
[13] Hardy, G. and Wright, E.. An Introduction to the Theory of Numbers, 5th edn. Oxford Science Publications, Oxford University Press, New York, 1990.Google Scholar
[14] Jonker, L. and Zhang, L.. Torus homeomorphisms whose rotation sets have empty interior. Ergod. Th. & Dynam. Sys. 18 (1998), 1173–1185.CrossRefGoogle Scholar
[15] de Kerékjártó, B.. Vorlesungen über Topologie (I). Springer, Berlin, 1923.CrossRefGoogle Scholar
[16] de Kerékjártó, B.. The plane translation theorem of Brouwer and the last geometric theorem of poincaré. Acta Sci. Math. Szeged 4 (1928–29), 86–102.Google Scholar
[17] Kwapisz, J.. A priori degenracy of one-dimensional rotation sets for periodic point free torus maps. Trans. Amer. Math. Soc. 354(7) (2002), 2865–2895.CrossRefGoogle Scholar
[18] Le Calvez, P.. Propriétés dynamiques de l’anneau et du tore. Astérisque 204 (1991).Google Scholar
[19] Le Calvez, P.. Une version feuilletée équivariante du théorème de translation de Brouwer. Publ. Math. Inst. Hautes Études Sci. 102(1) (2005), 1–98.CrossRefGoogle Scholar
[20] Llibre, J. and Mackay, R. S.. Rotation vectors and entropy for homeomorphisms of the torus isotopic to the identity. Ergod. Th. & Dynam. Sys. 11 (1991), 115–128.CrossRefGoogle Scholar
[21] Misiurewicz, M. and Ziemian, K.. Rotation sets for maps of tori. J. London Math. Soc. 40(2) (1989), 490–506.CrossRefGoogle Scholar
[22] Misiurewicz, M. and Ziemian, K.. Rotation sets and ergodic measures for torus homeomorphisms. Fund. Math. 137 (1991), 44–52.CrossRefGoogle Scholar