Complexity of infinite words associated with beta-expansions

Frougny, Christiane; Masáková, Zuzana; Pelantová, Edita

doi:10.1051/ita:2004009

Frougny, Christiane ¹ ; Masáková, Zuzana ; Pelantová, Edita

¹ Université Paris 7 LIAFA, UMR 7089 CNRS 2 place Jussieu 75251 Paris Cedex 05 (France) and Université Paris 8

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 38 (2004) no. 2, pp. 163-185

corrigé par Corrigendum : “Complexity of infinite words associated with beta-expansions”

Résumé

We study the complexity of the infinite word $u_{β}$ associated with the Rényi expansion of $1$ in an irrational base $β > 1$ . When $β$ is the golden ratio, this is the well known Fibonacci word, which is sturmian, and of complexity $ℂ (n) = n + 1$ . For $β$ such that $d_{β} (1) = t_{1} t_{2} \dots t_{m}$ is finite we provide a simple description of the structure of special factors of the word $u_{β}$ . When $t_{m} = 1$ we show that $ℂ (n) = (m - 1) n + 1$ . In the cases when $t_{1} = t_{2} = \dots = t_{m - 1}$ or $t_{1} > max {t_{2}, \dots, t_{m - 1}}$ we show that the first difference of the complexity function $ℂ (n + 1) - ℂ (n)$ takes value in ${m - 1, m}$ for every $n$ , and consequently we determine the complexity of $u_{β}$ . We show that $u_{β}$ is an Arnoux-Rauzy sequence if and only if $d_{β} (1) = t t \dots t 1$ . On the example of $β = 1 + 2 cos (2 π / 7)$ , solution of $X^{3} = 2 X^{2} + X - 1$ , we illustrate that the structure of special factors is more complicated for $d_{β} (1)$ infinite eventually periodic. The complexity for this word is equal to $2 n + 1$ .

MR Zbl | 4 citations dans Numdam

DOI : 10.1051/ita:2004009

Classification : 11A63, 11A67, 37B10, 68R15
Keywords: beta-expansions, complexity of infinite words

Affiliations des auteurs :

Frougny, Christiane ¹ ; Masáková, Zuzana ; Pelantová, Edita

¹ Université Paris 7 LIAFA, UMR 7089 CNRS 2 place Jussieu 75251 Paris Cedex 05 (France) and Université Paris 8

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     author = {Frougny, Christiane and Mas\'akov\'a, Zuzana and Pelantov\'a, Edita},
     title = {Complexity of infinite words associated with beta-expansions},
     journal = {RAIRO - Theoretical Informatics and Applications - Informatique Th\'eorique et Applications},
     pages = {163--185},
     year = {2004},
     publisher = {EDP Sciences},
     volume = {38},
     number = {2},
     doi = {10.1051/ita:2004009},
     mrnumber = {2060775},
     zbl = {1104.11013},
     language = {en},
     url = {https://www.numdam.org/articles/10.1051/ita:2004009/}
}

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Frougny, Christiane; Masáková, Zuzana; Pelantová, Edita. Complexity of infinite words associated with beta-expansions. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 38 (2004) no. 2, pp. 163-185. doi: 10.1051/ita:2004009

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