A priori degeneracy of one-dimensional rotation sets for periodic point free torus maps

Kwapisz, Jaroslaw

doi:10.1090/S0002-9947-02-02952-5

A priori degeneracy of one-dimensional rotation sets for periodic point free torus maps
HTML articles powered by AMS MathViewer

by Jaroslaw Kwapisz

Trans. Amer. Math. Soc. 354 (2002), 2865-2895

DOI: https://doi.org/10.1090/S0002-9947-02-02952-5

Published electronically: March 7, 2002

PDF

Abstract:

Diffeomorphisms of the two torus that are isotopic to the identity have rotation sets that are convex compact subsets of the plane. We show that certain line segments (including all rationally sloped segments with no rational points) cannot be realized as a rotation set.

References

Lars V. Ahlfors, Lectures on quasiconformal mappings, Van Nostrand Mathematical Studies, No. 10, D. Van Nostrand Co., Inc., Toronto, Ont.-New York-London, 1966. Manuscript prepared with the assistance of Clifford J. Earle, Jr. MR 0200442
Lluís Alsedà, Jaume Llibre, and MichałMisiurewicz, Combinatorial dynamics and entropy in dimension one, Advanced Series in Nonlinear Dynamics, vol. 5, World Scientific Publishing Co., Inc., River Edge, NJ, 1993. MR 1255515, DOI 10.1142/1980
C. Baesens, J. Guckenheimer, S. Kim, and R. S. MacKay, Three coupled oscillators: mode-locking, global bifurcations and toroidal chaos, Phys. D 49 (1991), no. 3, 387–475. MR 1115870, DOI 10.1016/0167-2789(91)90155-3
G. E. Bredon, Topology and geometry, Springer, New York, 1993.
Patrice Le Calvez, Propriétés dynamiques des difféomorphismes de l’anneau et du tore, Astérisque 204 (1991), 131 (French, with English and French summaries). MR 1183304
John Franks, Realizing rotation vectors for torus homeomorphisms, Trans. Amer. Math. Soc. 311 (1989), no. 1, 107–115. MR 958891, DOI 10.1090/S0002-9947-1989-0958891-1
John Franks, A new proof of the Brouwer plane translation theorem, Ergodic Theory Dynam. Systems 12 (1992), no. 2, 217–226. MR 1176619, DOI 10.1017/S0143385700006702
John Franks, The rotation set and periodic points for torus homeomorphisms, Dynamical systems and chaos, Vol. 1 (Hachioji, 1994) World Sci. Publ., River Edge, NJ, 1995, pp. 41–48. MR 1479903
John Franks, Rotation vectors and fixed points of area preserving surface diffeomorphisms, Trans. Amer. Math. Soc. 348 (1996), no. 7, 2637–2662. MR 1325916, DOI 10.1090/S0002-9947-96-01502-4
John Franks and MichałMisiurewicz, Rotation sets of toral flows, Proc. Amer. Math. Soc. 109 (1990), no. 1, 243–249. MR 1021217, DOI 10.1090/S0002-9939-1990-1021217-5
David Fried, The geometry of cross sections to flows, Topology 21 (1982), no. 4, 353–371. MR 670741, DOI 10.1016/0040-9383(82)90017-9
Michael Handel, Periodic point free homeomorphism of $T^2$, Proc. Amer. Math. Soc. 107 (1989), no. 2, 511–515. MR 965243, DOI 10.1090/S0002-9939-1989-0965243-2
Michael-R. Herman, Une méthode pour minorer les exposants de Lyapounov et quelques exemples montrant le caractère local d’un théorème d’Arnol′d et de Moser sur le tore de dimension $2$, Comment. Math. Helv. 58 (1983), no. 3, 453–502 (French). MR 727713, DOI 10.1007/BF02564647
Michael Herman, Some open problems in dynamical systems, Proceedings of the International Congress of Mathematicians, Vol. II (Berlin, 1998), 1998, pp. 797–808. MR 1648127
Russell A. Johnson, Ergodic theory and linear differential equations, J. Differential Equations 28 (1978), no. 1, 23–34. MR 487484, DOI 10.1016/0022-0396(78)90077-3
R. Johnson and J. Moser, The rotation number for almost periodic potentials, Comm. Math. Phys. 84 (1982), no. 3, 403–438. MR 667409
Leo B. Jonker and Lei Zhang, Torus homeomorphisms whose rotation sets have empty interior, Ergodic Theory Dynam. Systems 18 (1998), no. 5, 1173–1185. MR 1653236, DOI 10.1017/S0143385798117959
Jaroslaw Kwapisz, Every convex polygon with rational vertices is a rotation set, Ergodic Theory Dynam. Systems 12 (1992), no. 2, 333–339. MR 1176627, DOI 10.1017/S0143385700006787
—, Rotation sets and entropy, SUNY at Stony Brook PhD thesis, 1995.
Jaroslaw Kwapisz, A toral diffeomorphism with a nonpolygonal rotation set, Nonlinearity 8 (1995), no. 4, 461–476. MR 1342499
Jaroslaw Kwapisz, Poincaré rotation number for maps of the real line with almost periodic displacement, Nonlinearity 13 (2000), no. 5, 1841–1854. MR 1781820, DOI 10.1088/0951-7715/13/5/320
—, Combinatorics of torus diffeomorphisms, to be published in Erg. Th. & Dyn. Sys., 2001.
—, Renormalizing torus and annulus maps, in progress, 2001.
O. Lehto and K. I. Virtanen, Quasiconformal mappings in the plane, 2nd ed., Die Grundlehren der mathematischen Wissenschaften, Band 126, Springer-Verlag, New York-Heidelberg, 1973. Translated from the German by K. W. Lucas. MR 0344463
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
Ricardo Mañé, Ergodic theory and differentiable dynamics, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 8, Springer-Verlag, Berlin, 1987. Translated from the Portuguese by Silvio Levy. MR 889254, DOI 10.1007/978-3-642-70335-5
Dusa McDuff and Dietmar Salamon, Introduction to symplectic topology, 2nd ed., Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1998. MR 1698616
MichałMisiurewicz and Krystyna Ziemian, Rotation sets for maps of tori, J. London Math. Soc. (2) 40 (1989), no. 3, 490–506. MR 1053617, DOI 10.1112/jlms/s2-40.3.490
MichałMisiurewicz and Krystyna Ziemian, Rotation sets and ergodic measures for torus homeomorphisms, Fund. Math. 137 (1991), no. 1, 45–52. MR 1100607, DOI 10.4064/fm-137-1-45-52
Edwin E. Moise, Geometric topology in dimensions $2$ and $3$, Graduate Texts in Mathematics, Vol. 47, Springer-Verlag, New York-Heidelberg, 1977. MR 0488059
Edward E. Slaminka, A Brouwer translation theorem for free homeomorphisms, Trans. Amer. Math. Soc. 306 (1988), no. 1, 277–291. MR 927691, DOI 10.1090/S0002-9947-1988-0927691-X
V. V. Veremenyuk, The existence of the rotation number of the equation $\dot x=\lambda (t,x)$ with a right-hand side that is periodic in $x$ and almost periodic in $t$, Differentsial′nye Uravneniya 27 (1991), no. 6, 1073–1076, 1102 (Russian). MR 1130294