Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-26T00:57:51.693Z Has data issue: false hasContentIssue false
  • English
  • Français

Translation numbers for a class of maps on the dynamical systems arising from quasicrystals in the real line

Published online by Cambridge University Press:  03 August 2009

JOSÉ ALISTE-PRIETO
JOSÉ ALISTE-PRIETO*
Affiliation:
Departmento de Ingeniería Matemática, Universidad de Chile, Av. Blanco Encalada 2120, Santiago, Chile Laboratorie J.-A. Dieudonné, UMR 6621 CNRS, Université de Nice Sophia Antipolis, Parc Valrose, 06108 Nice Cedex 02, France (email: [email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper, we study translation sets for non-decreasing maps of the real line with a pattern-equivariant displacement with respect to a quasicrystal. First, we establish a correspondence between these maps and self maps of the continuous hull associated with the quasicrystal that are homotopic to the identity and preserve orientation. Then, by using first-return times and induced maps, we provide a partial description for the translation set of the latter maps in the case where they have fixed points and obtain the existence of a unique translation number in the case where they do not have fixed points. Finally, we investigate the existence of a semiconjugacy from a fixed-point-free map homotopic to the identity on the hull to the translation given by its translation number. We concentrate on semiconjugacies that are also homotopic to the identity and, under a boundedness condition, we prove a generalization of Poincaré’s theorem, finding a sufficient condition for such a semiconjugacy to exist depending on the translation number of the given map.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 30 , Issue 2 , April 2010 , pp. 565 - 594
Copyright
Copyright © Cambridge University Press 2009

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]Bellissard, J., Benedetti, R. and Gambaudo, J.-M.. Spaces of tilings, finite telescopic approximations and gap-labeling. Commun. Math. Phys. 261(1) (2006), 1–41.CrossRefGoogle Scholar
[2]Benedetti, R. and Gambaudo, J.-M.. On the dynamics of 𝔾-solenoids. Applications to Delone sets. Ergod. Th. & Dynam. Sys. 23(3) (2003), 673–691.CrossRefGoogle Scholar
[3]Cornfeld, I. P., Fomin, S. V. and Sinaĭ, Ya. G.. Ergodic Theory (Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 245). Springer, New York, 1982 (Translated from the Russian by A. B. Sosinskiĭ).CrossRefGoogle Scholar
[4]Clark, A.. The dynamics of maps of solenoids homotopic to the identity. Continuum Theory (Denton, TX, 1999) (Lecture Notes in Pure and Applied Mathematics, 230). Dekker, New York, 2002, pp. 127–136.Google Scholar
[5]Durand, F., Host, B. and Skau, C.. Substitutional dynamical systems, Bratteli diagrams and dimension groups. Ergod. Th. & Dynam. Sys. 19(4) (1999), 953–993.CrossRefGoogle Scholar
[6]Fogg, N. P.. Substitutions in Dynamics, Arithmetics and Combinatorics (Lecture Notes in Mathematics, 1794). Eds. Berthé, V., Ferenczi, S., Mauduit, C. and Siegel, A.. Springer, Berlin, 2002.CrossRefGoogle Scholar
[7]Gambaudo, J.-M., Guiraud, P. and Petite, S.. Minimal configurations for the Frenkel–Kontorova model on a quasicrystal. Commun. Math. Phys. 265(1) (2006), 165–188.CrossRefGoogle Scholar
[8]Glasner, E.. Ergodic Theory via Joinings (Mathematical Surveys and Monographs, 101). American Mathematical Society, Providence, RI, 2003.CrossRefGoogle Scholar
[9]Geller, W. and Misiurewicz, M.. Rotation and entropy. Trans. Amer. Math. Soc. 351(7) (1999), 2927–2948.CrossRefGoogle Scholar
[10]Herman, M.-R.. 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(3) (1983), 453–502.CrossRefGoogle Scholar
[11]Hof, A.. A remark on Schrödinger operators on aperiodic tilings. J. Statist. Phys. 81(3–4) (1995), 851–855.CrossRefGoogle Scholar
[12]Herman, R. H., Putnam, I. F. and Skau, C. F.. Ordered Bratteli diagrams, dimension groups and topological dynamics. Int. J. Math. 3(6) (1992), 827–864.CrossRefGoogle Scholar
[13]Jäger, T. H.. Linearization of conservative toral homeomorphisms. Invent. Math. 176(3) (2009), 601–616.CrossRefGoogle Scholar
[14]Jäger, T. H. and Stark, J.. Towards a classification for quasiperiodically forced circle homeomorphisms. J. London Math. Soc. (2) 73(3) (2006), 727–744.CrossRefGoogle Scholar
[15]Kellendonk, J.. Pattern-equivariant functions and cohomology. J. Phys. A: Math. Gen. 36(21) (2003), 5765–5772.CrossRefGoogle Scholar
[16]Katok, A. and Hasselblatt, B.. Introduction to the Modern Theory of Dynamical Systems (Encyclopedia of Mathematics and its Applications, 54). Cambridge University Press, Cambridge, 1995 (With a supplementary chapter by Katok and Leonardo Mendoza).CrossRefGoogle Scholar
[17]Kellendonk, J. and Putnam, I. F.. Tilings, C*-algebras, and K-theory. Directions in Mathematical Quasicrystals (CRM Monograph Series, 13). American Mathematical Society, Providence, RI, 2000, pp. 177–206.Google Scholar
[18]Kwapisz, J.. Poincaré rotation number for maps of the real line with almost periodic displacement. Nonlinearity 13(5) (2000), 1841–1854.CrossRefGoogle Scholar
[19]Lee, J.-Y., Moody, R. V. and Solomyak, B.. Pure point dynamical and diffraction spectra. Ann. Henri Poincaré 3(5) (2002), 1003–1018.CrossRefGoogle Scholar
[20]Lagarias, J. C. and Pleasants, P. A. B.. Repetitive Delone sets and quasicrystals. Ergod. Th. & Dynam. Sys. 23(3) (2003), 831–867.CrossRefGoogle Scholar
[21]Mañé, R.. Ergodic Theory and Differentiable Dynamics (Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 8). Springer, Berlin, 1987 (Translated from the Portuguese by Silvio Levy).CrossRefGoogle Scholar
[22]Misiurewicz, M. and Ziemian, K.. Rotation sets for maps of tori. J. London Math. Soc. (2) 40(3) (1989), 490–506.CrossRefGoogle Scholar
[23]Putnam, I. F.. The C*-algebras associated with minimal homeomorphisms of the Cantor set. Pacific J. Math. 136(2) (1989), 329–353.CrossRefGoogle Scholar
[24]Arthur Robinson, E. Jr. Symbolic dynamics and tilings of ℝd. Symbolic Dynamics and its Applications (Proceedings of Symposia in Applied Mathematics, 60). American Mathematical Society, Providence, RI, 2004, pp. 81–119.CrossRefGoogle Scholar
[25]Radin, C. and Sadun, L.. Isomorphism of hierarchical structures. Ergod. Th. & Dynam. Sys. 21(4) (2001), 1239–1248.CrossRefGoogle Scholar
[26]Rudolph, D. J.. Markov tilings of Rn and representations of Rn actions. Measure and Measurable Dynamics (Rochester, NY, 1987) (Contemporary Mathematics, 94). American Mathematical Society, Providence, RI, 1989, pp. 271–290.CrossRefGoogle Scholar
[27]Shecthman, D., Blech, I., Gratias, D. and Cahn, J. W.. Metallic phase with long range orientational order and no translational symmetry. Phys. Rev. Lett. 53(20) (1984), 1951–1954.CrossRefGoogle Scholar
[28]Schlottmann, M.. Generalized model sets and dynamical systems. Directions in Mathematical Quasicrystals (CRM Monograph Series, 13). American Mathematical Society, Providence, RI, 2000, pp. 143–159.Google Scholar
[29]Shvetsov, Y.. Rotation of flows on generalized solenoids. PhD Thesis, Montana State University, 2003.Google Scholar
[30]Wang, H.. Proving theorems by pattern recognition II. Bell Syst. Tech. J. 40 (1961), 1–41.CrossRefGoogle Scholar