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 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 (n)=n+1. For β such that d β (1)=t 1 t 2 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 (n)=(m-1)n+1. In the cases when t 1 =t 2 ==t m-1 or t 1 >max{t 2 ,,t m-1 } we show that the first difference of the complexity function (n+1)-(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)=ttt1. On the example of β=1+2cos(2π/7), solution of X 3 =2X 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 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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