Monotone periodic orbits for torus homeomorphisms

Parwani, Kamlesh

doi:10.1090/S0002-9939-04-07877-3

Monotone periodic orbits for torus homeomorphisms
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Proc. Amer. Math. Soc. 133 (2005), 1677-1683 Request permission

Abstract:

Let $f$ be a homeomorphism of the torus isotopic to the identity and suppose that there exists a periodic orbit with a non-zero rotation vector $(\frac {p}{q},\frac {r}{q})$. Then $f$ has a topologically monotone periodic orbit with the same rotation vector.

References

Philip Boyland, Rotation sets and monotone periodic orbits for annulus homeomorphisms, Comment. Math. Helv. 67 (1992), no. 2, 203–213. MR 1161281, DOI 10.1007/BF02566496
Philip Boyland, Isotopy stability of dynamics on surfaces, Geometry and topology in dynamics (Winston-Salem, NC, 1998/San Antonio, TX, 1999) Contemp. Math., vol. 246, Amer. Math. Soc., Providence, RI, 1999, pp. 17–45. MR 1732369, DOI 10.1090/conm/246/03772
Philip Boyland, John Guaschi, and Toby Hall, L’ensemble de rotation des homéomorphismes pseudo-Anosov, C. R. Acad. Sci. Paris Sér. I Math. 316 (1993), no. 10, 1077–1080 (French, with English and French summaries). MR 1222976
Andrew J. Casson and Steven A. Bleiler, Automorphisms of surfaces after Nielsen and Thurston, London Mathematical Society Student Texts, vol. 9, Cambridge University Press, Cambridge, 1988. MR 964685, DOI 10.1017/CBO9780511623912
Albert Fathi, An orbit closing proof of Brouwer’s lemma on translation arcs, Enseign. Math. (2) 33 (1987), no. 3-4, 315–322. MR 925994
Travaux de Thurston sur les surfaces, Astérisque, vol. 66, Société Mathématique de France, Paris, 1979 (French). Séminaire Orsay; With an English summary. MR 568308
John Franks, Generalizations of the Poincaré-Birkhoff theorem, Ann. of Math. (2) 128 (1988), no. 1, 139–151. MR 951509, DOI 10.2307/1971464
Michael Handel, The rotation set of a homeomorphism of the annulus is closed, Comm. Math. Phys. 127 (1990), no. 2, 339–349. MR 1037109
A. Katok, Some remarks of Birkhoff and Mather twist map theorems, Ergodic Theory Dynam. Systems 2 (1982), no. 2, 185–194 (1983). MR 693974, DOI 10.1017/s0143385700001504
Patrice Le Calvez, Dynamical properties of diffeomorphisms of the annulus and of the torus, SMF/AMS Texts and Monographs, vol. 4, American Mathematical Society, Providence, RI; Société Mathématique de France, Paris, 2000. Translated from the 1991 French original by Philippe Mazaud. MR 1747730
J. Llibre and R. S. MacKay, Rotation vectors and entropy for homeomorphisms of the torus isotopic to the identity, Ergodic Theory Dynam. Systems 11 (1991), no. 1, 115–128. MR 1101087, DOI 10.1017/S0143385700006040
K. Parwani, Simple braids for surface homeomorphisms, preprint.
K. Parwani, Thesis, Northwestern University, 2003.
William P. Thurston, On the geometry and dynamics of diffeomorphisms of surfaces, Bull. Amer. Math. Soc. (N.S.) 19 (1988), no. 2, 417–431. MR 956596, DOI 10.1090/S0273-0979-1988-15685-6
Heiner Zieschang, Finite groups of mapping classes of surfaces, Lecture Notes in Mathematics, vol. 875, Springer-Verlag, Berlin, 1981. MR 643627, DOI 10.1007/BFb0090465