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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)
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Abstract

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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.

Information

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

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