Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-11-26T23:23:16.420Z Has data issue: false hasContentIssue false
  • English
  • Français

A characterization of the Morse minimal set up to topological conjugacy

Published online by Cambridge University Press:  01 October 2008

ETHAN M. COVEN ,
MICHAEL KEANE  and
MICHELLE LEMASURIER
ETHAN M. COVEN
Affiliation:
Department of Mathematics, Wesleyan University, Middletown, CT 06459, USA (email: [email protected], [email protected])
MICHAEL KEANE
Affiliation:
Department of Mathematics, Wesleyan University, Middletown, CT 06459, USA (email: [email protected], [email protected])
MICHELLE LEMASURIER
Affiliation:
Department of Mathematics, Hamilton College, Clinton, NY 13323, USA (email: [email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

We establish necessary and sufficient conditions for a dynamical system to be topologically conjugate to the Morse minimal set, the shift orbit closure of the Morse sequence. Conditions for topological conjugacy to the closely related Toeplitz minimal set are also derived.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 28 , Issue 5 , October 2008 , pp. 1443 - 1451
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]Allouche, J.-P. and Shallit, J.. The ubiquitous Prouhet–Thue–Morse sequence. Sequences and their Applications (Singapore 1998) (Springer Series in Discrete Mathematics and Theoretical Computer Science). Springer, London, 1999, pp. 1–16.Google Scholar
[2]Blanchard, F., Durand, F. and Maass, A.. Constant-length substitutions and countable scrambled sets. Nonlinearity 17 (2004), 817–833.CrossRefGoogle Scholar
[3]Gottschalk, W. H.. Substitution minimal sets. Trans. Amer. Math. Soc. 109 (1963), 467–491.CrossRefGoogle Scholar
[4]Gottschalk, W. H. and Hedlund, G. A.. Topological Dynamics (American Mathematical Society Colloquium Publications, 36). American Mathematical Society, Providence, RI, 1955.CrossRefGoogle Scholar
[5]Gottschalk, W. H. and Hedlund, G. A.. A characterization of the Morse minimal set. Proc. Amer. Math. Soc. 15 (1964), 70–74.Google Scholar
[6]Hedlund, G. A.. Remarks on the work of Axel Thue on sequences. Nordisk Math. Tidskr. 15 (1967), 148–150.Google Scholar
[7]Jacobs, K. and Keane, M. S.. 0–1 sequences of Toeplitz type. Z. Wahrs. Verw. Gebiete 13 (1969), 123–131.CrossRefGoogle Scholar
[8]Morse, M.. A solution to the problem of infinite play in chess. Bull. Amer. Math. Soc. 44 (1938), 632 (Abstract 360).Google Scholar
[9]Morse, M. and Hedlund, G. A.. Unending chess, symbolic dynamics and a problem in semigroups. Duke Math. J. 11 (1944), 1–7.Google Scholar
[10]Thue, A.. Über die gegenseitige Lage gleicher Teile gewisser Zeichenreihen. Christiana Vidensk. Selsk. Skr. (7) (1906), 1–22.Google Scholar
[11]Thue, A.. Über unendlichen Zeichenreihen. Christiana Vidensk. Selsk. Skr. (1) (1912), 1–67.Google Scholar
[12]Thue, A.. Selected Mathematical Papers. Universitetsforlaget, Oslo, 1977.Google Scholar