Complexity of infinite words associated with beta-expansions

Frougny, Christiane; Masáková, Zuzana; Pelantová, Edita

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 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 $ℂ\left(n\right)=(m-1)n+1$. In the cases when $t_1=t_2=\cdots =t_{m-1}$ or $t_1>max\{t_2,\cdots ,t_{m-1}\}$ we show that the first difference of the complexity function $ℂ(n+1)-ℂ\left(n\right)$ takes value in $\{m-1,m\}$ 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 $d_{\beta }\left(1\right)=tt\cdots t1$. On the example of $\beta =1+2cos(2\pi /7)$, solution of $X^3=2X^2+X-1$, we illustrate that the structure of special factors is more complicated for $d_{\beta }\left(1\right)$ infinite eventually periodic. The complexity for this word is equal to $2n+1$.

[1] J.-P. Allouche, Sur la complexité des suites infinies. Bull. Belg. Math. Soc. Simon Stevin 1 (1994) 133-143. | MR | Zbl

[2] P. Arnoux et G. Rauzy, Représentation géométrique de suites de complexité $2n+1$. Bull. Soc. Math. France 119 (1991) 199-215. | Numdam | MR | Zbl

[3] J. Berstel, Recent results on extensions of Sturmian words. J. Algebra Comput. 12 (2003) 371-385. | MR | Zbl

[4] A. Bertrand, Développements en base de Pisot et répartition modulo 1. C. R. Acad. Sci. Paris 285A (1977) 419-421. | MR | Zbl

[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. | Zbl

[6] J. Cassaigne, Complexité et facteurs spéciaux. Bull. Belg. Math. Soc. Simon Stevin 4 (1997) 67-88. | MR | Zbl

[7] J. Cassaigne, S. Ferenczi and L. Zamboni, Imbalances in Arnoux-Rauzy sequences. Ann. Inst. Fourier 50 (2000) 1265-1276. | Numdam | MR | Zbl

[8] S. Fabre, Substitutions et $\beta$-systèmes de numération. Theoret. Comput. Sci. 137 (1995) 219-236. | MR | Zbl

[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. | MR | Zbl

[10] M. Lothaire, Algebraic combinatorics on words. Cambridge University Press (2002). | MR | Zbl

[11] W. Parry, On the $\beta$-expansions of real numbers. Acta Math. Acad. Sci. Hungar. 11 (1960) 401-416. | MR | Zbl

[12] J. Patera, Statistics of substitution sequences. On-line computer program, available at http://kmlinux.fjfi.cvut.cz/~patera/SubstWords.cgi

[13] A. Rényi, Representations for real numbers and their ergodic properties. Acta Math. Acad. Sci. Hungar. 8 (1957) 477-493. | MR | Zbl

[14] K. Schmidt, On periodic expansions of Pisot numbers and Salem numbers. Bull. London Math. Soc. 12 (1980) 269-278. | MR | Zbl

[15] W.P. Thurston, Groups, tilings, and finite state automata. Geometry supercomputer project research report GCG1, University of Minnesota (1989).

[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. | Zbl