Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-26T04:20:51.905Z Has data issue: false hasContentIssue false
  • English
  • Français

Aperiodic substitution systems and their Bratteli diagrams

Published online by Cambridge University Press:  01 February 2009

S. BEZUGLYI ,
J. KWIATKOWSKI  and
K. MEDYNETS
S. BEZUGLYI
Affiliation:
Department of Mathematics, Institute for Low Temperature Physics, Kharkov 61103, Ukraine (email: bezuglyi@ilt.kharkov.ua)
J. KWIATKOWSKI
Affiliation:
Faculty of Mathematics and Computer Science, University of Warmia and Mazury, Olsztyn 10561, Poland (email: jkwiat@mat.uni.torun.pl)
K. MEDYNETS
Affiliation:
Department of Mathematics, Institute for Low Temperature Physics, Kharkov 61103, Ukraine (email: medynets@ilt.kharkov.ua)
Get access Rights & Permissions [Opens in a new window]

Abstract

We study aperiodic substitution dynamical systems arising from non-primitive substitutions. We prove that the Vershik homeomorphism φ of a stationary ordered Bratteli diagram is topologically conjugate to an aperiodic substitution system if and only if no restriction of φ to a minimal component is conjugate to an odometer. We also show that every aperiodic substitution system generated by a substitution with nesting property is conjugate to the Vershik map of a stationary ordered Bratteli diagram. It is proved that every aperiodic substitution system is recognizable. The classes of m-primitive substitutions and derivative substitutions associated with them are studied. We discuss also the notion of expansiveness for Cantor dynamical systems of finite rank.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 29 , Issue 1 , February 2009 , pp. 37 - 72
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]Bezuglyi, S., Dooley, A. H. and Medynets, K.. The Rokhlin lemma for homeomorphisms of a Cantor set. Proc. Amer. Math. Soc. 133(10) (2005), 2957–2964.CrossRefGoogle Scholar
[2]Carlsen, T. and Eilers, S.. Augmenting dimension group invariants for substitution dynamics. Ergod. Th. & Dynam. Sys. 24(4) (2004), 1015–1039.CrossRefGoogle Scholar
[3]Carlsen, T. and Eilers, S.. Ordered k-groups associated to substitutional dynamics. J. Funct. Anal. 238(1) (2006), 99–117.CrossRefGoogle Scholar
[4]Durand, F., Host, B. and Skau, B.. Substitutional dynamical systems, Bratteli diagrams and dimension groups. Ergod. Th. & Dynam. Sys. 19(4) (1999), 953–993.CrossRefGoogle Scholar
[5]Downarowicz, T. and Maass, A.. Finite rank Bratteli–Vershik diagrams are expansive. Ergod. Th. & Dynam. Sys. 28(3) (2008), 739–747.CrossRefGoogle Scholar
[6]Durand, F.. A characterization of substitutive sequences using return words. Discrete Math. 179 (1998), 89–101.CrossRefGoogle Scholar
[7]Durand, F.. A theorem of Cobham for non-primitive substitutions. Acta Arith. 104(2) (2002), 225–241.CrossRefGoogle Scholar
[8]Ferenczi, S.. Substitution dynamical systems on infinite alphabets (Substitutions sur un alphabet infini). Ann. Inst. Fourier (Grenoble) 56(7) (2006), 2315–2343.CrossRefGoogle Scholar
[9]Fogg, P. N.. Substitutions in Dynamics, Arithmetics and Combinatorics (Lecture Notes in Mathematics, 1794). Springer, Berlin, 2002.CrossRefGoogle Scholar
[10]Forrest, A. H.. K-groups associated with substitution minimal systems. Israel J. Math. 98 (1997), 101–139.CrossRefGoogle Scholar
[11]Gjerde, R. and Johansen, Ø.. Bratteli–Vershik models for Cantor minimal systems: applications to Toeplitz flows. Ergod. Th. & Dynam. Sys. 20 (2000), 1687–1710.CrossRefGoogle Scholar
[12]Glasner, E.. Ergodic Theory via Joinings (Mathematical Surveys and Monographs, 101). American Mathematical Society, Providence, RI, 2003.CrossRefGoogle Scholar
[13]Giordano, T., Matui, H., Putnam, I. and Skau, C.. Orbit equivalence for Cantor minimal ℤ2-systems. J. Amer. Math. Soc. 21(3) (2008), 863–892.CrossRefGoogle Scholar
[14]Giordano, T., Putnam, I. and Skau, C.. Topological orbit equivalence and C *-crossed products. J. Reine Angew. Math. 469 (1995), 51–111.Google Scholar
[15]Giordano, T., Putnam, I. and Skau, C.. Full groups of Cantor minimal systems. Israel J. Math. 111 (1999), 285–320.CrossRefGoogle Scholar
[16]Giordano, T., Putnam, I. and Skau, C.. Affable equivalence relations and orbit structure of Cantor dynamical systems. Ergod. Th. & Dynam. Sys. 24(2) (2004), 441–475.CrossRefGoogle Scholar
[17]Glasner, E. and Weiss, B.. Weak orbit equivalence of Cantor minimal systems. Internat. J. Math. 6(4) (1995), 569–579.CrossRefGoogle Scholar
[18]Host, B.. Valeurs propres des systèmes dynamiques definis par des substitutions de longueur variable. Ergod. Th. & Dynam. Sys. 6(4) (1986), 529–540.CrossRefGoogle Scholar
[19]Herman, R. H., Putnam, I. and Skau, C.. Ordered Bratteli diagrams, dimension groups, and topological dynamics. Internat. J. Math. 3(6) (1992), 827–864.CrossRefGoogle Scholar
[20]Kurka, P.. Topological and Symbolic Dynamics (Cours Spécialisés (Paris) 11). Société Mathématique de France, Paris, 2003.Google Scholar
[21]Medynets, K.. Cantor aperiodic systems and Bratteli diagrams. C. R. Math. Acad. Sci. Paris 342(1) (2006), 43–46.CrossRefGoogle Scholar
[22]Mossé, B.. Puissances de mots et reconnaissabilité des points fixes d’une substitution. Theoret. Comput. Sci. 99(2) (1992), 327–334.CrossRefGoogle Scholar
[23]Mossé, B.. Reconnaissabilité des substitutions et complexité des suites automatiques. Bull. Soc. Math. France 124(2) (1996), 329–346.CrossRefGoogle Scholar
[24]Queffelec, M.. Substitution Dynamical Systems – Spectral Analysis (Lecture Notes in Mathematics, 1294). Springer, Berlin, 1987.CrossRefGoogle Scholar
[25]Walters, P.. An Introduction to Ergodic Theory (Graduate Texts in Mathematics, 79). Springer, Berlin, 1982.CrossRefGoogle Scholar
[26]Yuasa, H.. On the topological orbit equivalence in a class of substitutional minimal systems. Tokyo J. Math. 25(2) (2002), 221–240.CrossRefGoogle Scholar
[27]Yuasa, H.. Invariant measures for the subshifts arising from non-primitive substitutions. J. Anal. Math. 102 (2007), 143–180.CrossRefGoogle Scholar