RAIRO-Theor. Inf. Appl.
Volume 41, Number 3, July-September 2007
|Page(s)||329 - 349|
|Published online||25 September 2007|
On substitution invariant Sturmian words: an application of Rauzy fractals
LIRMM 161 rue Ada F-34392 Montpellier cedex 5, France;
2 Department of Information and System Engineering, Faculty of Science Engineering, 1-13-27 Kasuga, Bunkyo-ku, Tokyo 112-8851, Japan
3 Department of Information and System Engineering, Kanazawa University, Kanazawa, Japan
4 Department of Mathematics, Tsinghua University, Beijing, China
Accepted: 4 January 2007
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.
Mathematics Subject Classification: 11J70 / 37B10 / 68R15
Key words: Sturmian words / Rauzy fractals / invertible substitutions / automorphisms of the free monoid / tilings
© EDP Sciences, 2007
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