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On substitution invariant Sturmian words: an application of Rauzy fractals

Published online by Cambridge University Press:  25 September 2007

Valérie Berthé ,
Hiromi Ei ,
Shunji Ito  and
Hui Rao
Valérie Berthé
Affiliation:
LIRMM 161 rue Ada F-34392 Montpellier cedex 5, France; berthe@lirmm.fr
Hiromi Ei
Affiliation:
Department of Information and System Engineering, Faculty of Science Engineering, 1-13-27 Kasuga, Bunkyo-ku, Tokyo 112-8851, Japan
Shunji Ito
Affiliation:
Department of Information and System Engineering, Kanazawa University, Kanazawa, Japan
Hui Rao
Affiliation:
Department of Mathematics, Tsinghua University, Beijing, China
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Abstract

Sturmian words are infinite words that have exactly n+1 factors of length n for every positive integer n. A Sturmian word sα,p is also defined as a coding over a two-letter alphabet of the orbit of point ρ under the action of the irrational rotation Rα : x → x + α (mod 1). A substitution fixes a Sturmian word if and only if it is invertible. The main object of the present paper is to investigate Rauzy fractals associated with two-letter invertible substitutions. As an application, we give an alternative geometric proof of Yasutomi's characterization of all pairs (α,p) such that sα,p is a fixed point of some non-trivial substitution.

Keywords

Sturmian wordsRauzy fractalsinvertible substitutionsautomorphisms of the free monoidtilings
Type
Research Article
Information
RAIRO - Theoretical Informatics and Applications , Volume 41 , Issue 3 , July 2007 , pp. 329 - 349
Copyright
© EDP Sciences, 2007

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