# Complexity of infinite words associated with beta-expansions

• [1] Université Paris 7 LIAFA, UMR 7089 CNRS 2 place Jussieu 75251 Paris Cedex 05 (France) and Université Paris 8
• Volume: 38, Issue: 2, page 163-185
• ISSN: 0988-3754

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## Abstract

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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 $2n+1$.

## How to cite

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Frougny, Christiane, Masáková, Zuzana, and Pelantová, Edita. "Complexity of infinite words associated with beta-expansions." RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications 38.2 (2004): 163-185. <http://eudml.org/doc/244788>.

@article{Frougny2004,
abstract = {We study the complexity of the infinite word $u_\beta$ associated with the Rényi expansion of $1$ in an irrational base $\beta &gt;1$. When $\beta$ is the golden ratio, this is the well known Fibonacci word, which is sturmian, and of complexity $\mathbb \{C\}(n)=n+1$. For $\beta$ such that $d_\beta (1)=t_1t_2\cdots t_\{m\}$ 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 $\mathbb \{C\}(n)=(m-1)n+1$. In the cases when $t_1=t_2=\cdots =t_\{m-1\}$ or $t_1&gt;\max \lbrace t_2,\dots ,t_\{m-1\}\rbrace$ we show that the first difference of the complexity function $\mathbb \{C\}(n+1)-\mathbb \{C\}(n)$ takes value in $\lbrace m-1,m\rbrace$ 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 $d_\beta (1)=t\,t\cdots \,t\,1$. On the example of $\beta =1+2\cos (2\pi /7)$, solution of $X^3=2X^2+X-1$, we illustrate that the structure of special factors is more complicated for $d_\beta (1)$ infinite eventually periodic. The complexity for this word is equal to $2n+1$.},
affiliation = {Université Paris 7 LIAFA, UMR 7089 CNRS 2 place Jussieu 75251 Paris Cedex 05 (France) and Université Paris 8},
author = {Frougny, Christiane, Masáková, Zuzana, Pelantová, Edita},
journal = {RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications},
keywords = {beta-expansions; complexity of infinite words},
language = {eng},
number = {2},
pages = {163-185},
publisher = {EDP-Sciences},
title = {Complexity of infinite words associated with beta-expansions},
url = {http://eudml.org/doc/244788},
volume = {38},
year = {2004},
}

TY - JOUR
AU - Frougny, Christiane
AU - Masáková, Zuzana
AU - Pelantová, Edita
TI - Complexity of infinite words associated with beta-expansions
JO - RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications
PY - 2004
PB - EDP-Sciences
VL - 38
IS - 2
SP - 163
EP - 185
AB - We study the complexity of the infinite word $u_\beta$ associated with the Rényi expansion of $1$ in an irrational base $\beta &gt;1$. When $\beta$ is the golden ratio, this is the well known Fibonacci word, which is sturmian, and of complexity $\mathbb {C}(n)=n+1$. For $\beta$ such that $d_\beta (1)=t_1t_2\cdots t_{m}$ 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 $\mathbb {C}(n)=(m-1)n+1$. In the cases when $t_1=t_2=\cdots =t_{m-1}$ or $t_1&gt;\max \lbrace t_2,\dots ,t_{m-1}\rbrace$ we show that the first difference of the complexity function $\mathbb {C}(n+1)-\mathbb {C}(n)$ takes value in $\lbrace m-1,m\rbrace$ 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 $d_\beta (1)=t\,t\cdots \,t\,1$. On the example of $\beta =1+2\cos (2\pi /7)$, solution of $X^3=2X^2+X-1$, we illustrate that the structure of special factors is more complicated for $d_\beta (1)$ infinite eventually periodic. The complexity for this word is equal to $2n+1$.
LA - eng
KW - beta-expansions; complexity of infinite words
UR - http://eudml.org/doc/244788
ER -

## References

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1. [1] J.-P. Allouche, Sur la complexité des suites infinies. Bull. Belg. Math. Soc. Simon Stevin 1 (1994) 133-143. Zbl0803.68094MR1318964
2. [2] P. Arnoux et G. Rauzy, Représentation géométrique de suites de complexité $2n+1$. Bull. Soc. Math. France 119 (1991) 199-215. Zbl0789.28011MR1116845
3. [3] J. Berstel, Recent results on extensions of Sturmian words. J. Algebra Comput. 12 (2003) 371-385. Zbl1007.68141MR1902372
4. [4] A. Bertrand, Développements en base de Pisot et répartition modulo 1. C. R. Acad. Sci. Paris 285A (1977) 419-421. Zbl0362.10040MR447134
5. [5] A. Bertrand-Mathis, Comment écrire les nombres entiers dans une base qui n’est pas entière. Acta Math. Acad. Sci. Hungar. 54 (1989) 237-241. Zbl0695.10005
6. [6] J. Cassaigne, Complexité et facteurs spéciaux. Bull. Belg. Math. Soc. Simon Stevin 4 (1997) 67-88. Zbl0921.68065MR1440670
7. [7] J. Cassaigne, S. Ferenczi and L. Zamboni, Imbalances in Arnoux-Rauzy sequences. Ann. Inst. Fourier 50 (2000) 1265-1276. Zbl1004.37008MR1799745
8. [8] S. Fabre, Substitutions et $\beta$-systèmes de numération. Theoret. Comput. Sci. 137 (1995) 219-236. Zbl0872.11017MR1311222
9. [9] Ch. Frougny, J.-P. Gazeau and R. Krejcar, Additive and multiplicative properties of point sets based on beta-integers. Theoret. Comput. Sci. 303 (2003) 491-516. Zbl1036.11034MR1990778
10. [10] M. Lothaire, Algebraic combinatorics on words. Cambridge University Press (2002). Zbl1001.68093MR1905123
11. [11] W. Parry, On the $\beta$-expansions of real numbers. Acta Math. Acad. Sci. Hungar. 11 (1960) 401-416. Zbl0099.28103MR142719
12. [12] J. Patera, Statistics of substitution sequences. On-line computer program, available at http://kmlinux.fjfi.cvut.cz/~patera/SubstWords.cgi
13. [13] A. Rényi, Representations for real numbers and their ergodic properties. Acta Math. Acad. Sci. Hungar. 8 (1957) 477-493. Zbl0079.08901MR97374
14. [14] K. Schmidt, On periodic expansions of Pisot numbers and Salem numbers. Bull. London Math. Soc. 12 (1980) 269-278. Zbl0494.10040MR576976
15. [15] W.P. Thurston, Groups, tilings, and finite state automata. Geometry supercomputer project research report GCG1, University of Minnesota (1989).
16. [16] O. Turek, Complexity and balances of the infinite word of $\beta$-integers for $\beta =1+\sqrt{3}$, in Proc. of Words’03, Turku. TUCS Publication 27 (2003) 138-148. Zbl1040.68090

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