Complexity of infinite words associated with beta-expansions
RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Volume 38 (2004) no. 2, pp. 163-185.

We study the complexity of the infinite word ${u}_{\beta }$ associated with the Rényi expansion of $1$ in an irrational base $\beta >1$. When $\beta$ is the golden ratio, this is the well known Fibonacci word, which is sturmian, and of complexity $ℂ\left(n\right)=n+1$. For $\beta$ such that ${d}_{\beta }\left(1\right)={t}_{1}{t}_{2}\cdots {t}_{m}$ is finite we provide a simple description of the structure of special factors of the word ${u}_{\beta }$. When ${t}_{m}=1$ we show that $ℂ\left(n\right)=\left(m-1\right)n+1$. In the cases when ${t}_{1}={t}_{2}=\cdots ={t}_{m-1}$ or ${t}_{1}>max\left\{{t}_{2},\cdots ,{t}_{m-1}\right\}$ we show that the first difference of the complexity function $ℂ\left(n+1\right)-ℂ\left(n\right)$ takes value in $\left\{m-1,m\right\}$ for every $n$, and consequently we determine the complexity of ${u}_{\beta }$. We show that ${u}_{\beta }$ is an Arnoux-Rauzy sequence if and only if ${d}_{\beta }\left(1\right)=t\phantom{\rule{0.166667em}{0ex}}t\cdots \phantom{\rule{0.166667em}{0ex}}t\phantom{\rule{0.166667em}{0ex}}1$. On the example of $\beta =1+2cos\left(2\pi /7\right)$, solution of ${X}^{3}=2{X}^{2}+X-1$, we illustrate that the structure of special factors is more complicated for ${d}_{\beta }\left(1\right)$ infinite eventually periodic. The complexity for this word is equal to $2n+1$.

DOI: 10.1051/ita:2004009
Classification: 11A63,  11A67,  37B10,  68R15
Keywords: beta-expansions, complexity of infinite words
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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Frougny, Christiane; Masáková, Zuzana; Pelantová, Edita. Complexity of infinite words associated with beta-expansions. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Volume 38 (2004) no. 2, pp. 163-185. doi : 10.1051/ita:2004009. http://www.numdam.org/articles/10.1051/ita:2004009/

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