Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-27T08:12:48.574Z Has data issue: false hasContentIssue false
  • English
  • Français

Complexity of infinite words associated withbeta-expansions

Published online by Cambridge University Press:  15 June 2004

Christiane Frougny ,
Zuzana Masáková  and
Edita Pelantová
Christiane Frougny
Affiliation:
LIAFA, CNRS UMR 7089, 2 place Jussieu, 75251 Paris Cedex 05, France; [email protected]. Université Paris 8.
Zuzana Masáková
Affiliation:
Department of Mathematics, FNSPE, Czech Technical University, Trojanova 13, 120 00 Praha 2, Czech Republic; [email protected].,[email protected].
Edita Pelantová
Affiliation:
Department of Mathematics, FNSPE, Czech Technical University, Trojanova 13, 120 00 Praha 2, Czech Republic; [email protected].,[email protected].
Get access

Abstract

We study the complexity of the infinite word uβ associated with theRényi 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 thatdβ(1) = t1t2...tm is finite we provide a simple description ofthe structure of special factors of the word uβ . When tm =1we show thatC(n) = (m - 1)n + 1. In the cases when t1 = t2 = ... tm-1 ort1 > max{t2,...,tm-1 } we show that the first differenceof the complexity function C(n + 1) - C(n ) takes value in{m - 1,m} for every n, and consequently we determine thecomplexity of uβ . We show thatuβ is an Arnoux-Rauzy sequence if and only ifdβ(1) = tt...t1. On the example ofβ = 1 + 2cos(2π/7), solution of X3 = 2X2 + X - 1, we illustratethat the structure of special factors is more complicated fordβ (1) infinite eventually periodic.The complexity for this word is equal to 2n+1.

Keywords

Beta-expansionscomplexity of infinite words.
Type
Research Article
Information
RAIRO - Theoretical Informatics and Applications , Volume 38 , Issue 2 , April 2004 , pp. 163 - 185
Copyright
© EDP Sciences, 2004

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Allouche, J.-P., Sur la complexité des suites infinies. Bull. Belg. Math. Soc. Simon Stevin 1 (1994) 133-143.
Arnoux, P. et Rauzy, G., Représentation géométrique de suites de complexité 2n + 1. Bull. Soc. Math. France 119 (1991) 199-215. CrossRef
Berstel, J., Recent results on extensions of Sturmian words. J. Algebra Comput. 12 (2003) 371-385. CrossRef
Bertrand, A., Développements en base de Pisot et répartition modulo 1. C. R. Acad. Sci. Paris 285A (1977) 419-421.
Bertrand-Mathis, A., Comment écrire les nombres entiers dans une base qui n'est pas entière. Acta Math. Acad. Sci. Hungar. 54 (1989) 237-241. CrossRef
Cassaigne, J., Complexité et facteurs spéciaux. Bull. Belg. Math. Soc. Simon Stevin 4 (1997) 67-88.
Cassaigne, J., Ferenczi, S. and Zamboni, L., Imbalances in Arnoux-Rauzy sequences. Ann. Inst. Fourier 50 (2000) 1265-1276. CrossRef
Fabre, S., Substitutions et β-systèmes de numération. Theoret. Comput. Sci. 137 (1995) 219-236. CrossRef
Ch. Frougny, J.-P. Gazeau, R. Krejcar, Additive and multiplicative properties of point sets based on beta-integers. Theoret. Comput. Sci. 303 (2003) 491-516. CrossRef
M. Lothaire, Algebraic combinatorics on words. Cambridge University Press (2002).
Parry, W., On the β-expansions of real numbers. Acta Math. Acad. Sci. Hungar. 11 (1960) 401-416. CrossRef
J. Patera, Statistics of substitution sequences. On-line computer program, available at http://kmlinux.fjfi.cvut.cz/~patera/SubstWords.cgi
Rényi, A., Representations for real numbers and their ergodic properties. Acta Math. Acad. Sci. Hungar. 8 (1957) 477-493. CrossRef
Schmidt, K., On periodic expansions of Pisot numbers and Salem numbers. Bull. London Math. Soc. 12 (1980) 269-278. CrossRef
W.P. Thurston, Groups, tilings, and finite state automata. Geometry supercomputer project research report GCG1, University of Minnesota (1989).
O. Turek, Complexity and balances of the infinite word of β-integers for β = 1 + √3, in Proc. of Words'03, Turku. TUCS Publication 27 (2003) 138-148.