On substitution invariant sturmian words : an application of Rauzy fractals

Berthé, Valérie; Ei, Hiromi; Ito, Shunji; Rao, Hui

doi:10.1051/ita:2007026

Berthé, Valérie ; Ei, Hiromi ¹ ; Ito, Shunji ² ; Rao, Hui ³

¹ Chuo University Kasuga, Bunkyo-ku Department of Information and System Engineering Tokyo (Japan)
² Kanazawa University Department of Mathematical Kanazawa (Japan)
³ Tsinghua University Department of Mathematics Beijing (China)

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 41 (2007) no. 3, pp. 329-349.

Voir la notice de l'article dans Numdam

Résumé

sturmian words are infinite words that have exactly $n + 1$ factors of length $n$ for every positive integer $n$ . A sturmian word $s_{α, ρ}$ is also defined as a coding over a two-letter alphabet of the orbit of point $ρ$ under the action of the irrational rotation $R_{α} : x \mapsto x + α$ (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 $(α, ρ)$ such that $s_{α, ρ}$ is a fixed point of some non-trivial substitution.

MR Zbl

DOI : 10.1051/ita:2007026

Classification : 11J70, 37B10, 68R15
Mots-clés : sturmian words, Rauzy fractals, invertible substitutions, automorphisms of the free monoid, tilings

Affiliations des auteurs :

Berthé, Valérie ; Ei, Hiromi ¹ ; Ito, Shunji ² ; Rao, Hui ³

¹ Chuo University Kasuga, Bunkyo-ku Department of Information and System Engineering Tokyo (Japan)
² Kanazawa University Department of Mathematical Kanazawa (Japan)
³ Tsinghua University Department of Mathematics Beijing (China)

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     title = {On substitution invariant sturmian words : an application of {Rauzy} fractals},
     journal = {RAIRO - Theoretical Informatics and Applications - Informatique Th\'eorique et Applications},
     pages = {329--349},
     publisher = {EDP-Sciences},
     volume = {41},
     number = {3},
     year = {2007},
     doi = {10.1051/ita:2007026},
     zbl = {1140.11014},
     mrnumber = {2354361},
     language = {en},
     url = {https://geodesic-test.mathdoc.fr/articles/10.1051/ita:2007026/}
}

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Berthé, Valérie; Ei, Hiromi; Ito, Shunji; Rao, Hui. On substitution invariant sturmian words : an application of Rauzy fractals. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 41 (2007) no. 3, pp. 329-349. doi : 10.1051/ita:2007026. https://geodesic-test.mathdoc.fr/articles/10.1051/ita:2007026/

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