We study the complexity of the infinite word associated with the Rényi expansion of in an irrational base . When is the golden ratio, this is the well known Fibonacci word, which is sturmian, and of complexity . For such that is finite we provide a simple description of the structure of special factors of the word . When we show that . In the cases when or we show that the first difference of the complexity function takes value in for every , and consequently we determine the complexity of . We show that is an Arnoux-Rauzy sequence if and only if . On the example of , solution of , we illustrate that the structure of special factors is more complicated for infinite eventually periodic. The complexity for this word is equal to .
@article{ITA_2004__38_2_163_0, author = {Frougny, Christiane and Mas\'akov\'a, Zuzana and Pelantov\'a, Edita}, title = {Complexity of infinite words associated with beta-expansions}, journal = {RAIRO - Theoretical Informatics and Applications - Informatique Th\'eorique et Applications}, volume = {38}, year = {2004}, pages = {163-185}, doi = {10.1051/ita:2004009}, mrnumber = {2060775}, zbl = {1104.11013}, language = {en}, url = {http://dml.mathdoc.fr/item/ITA_2004__38_2_163_0} }
Frougny, Christiane; Masáková, Zuzana; Pelantová, Edita. Complexity of infinite words associated with beta-expansions. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 38 (2004) pp. 163-185. doi : 10.1051/ita:2004009. http://gdmltest.u-ga.fr/item/ITA_2004__38_2_163_0/
[1] Sur la complexité des suites infinies. Bull. Belg. Math. Soc. Simon Stevin 1 (1994) 133-143. | MR 1318964 | Zbl 0803.68094
,[2] Représentation géométrique de suites de complexité . Bull. Soc. Math. France 119 (1991) 199-215. | Numdam | MR 1116845 | Zbl 0789.28011
et ,[3] Recent results on extensions of Sturmian words. J. Algebra Comput. 12 (2003) 371-385. | MR 1902372 | Zbl 1007.68141
,[4] Développements en base de Pisot et répartition modulo 1. C. R. Acad. Sci. Paris 285A (1977) 419-421. | MR 447134 | Zbl 0362.10040
,[5] Comment écrire les nombres entiers dans une base qui n'est pas entière. Acta Math. Acad. Sci. Hungar. 54 (1989) 237-241. | Zbl 0695.10005
,[6] Complexité et facteurs spéciaux. Bull. Belg. Math. Soc. Simon Stevin 4 (1997) 67-88. | MR 1440670 | Zbl 0921.68065
,[7] Imbalances in Arnoux-Rauzy sequences. Ann. Inst. Fourier 50 (2000) 1265-1276. | Numdam | MR 1799745 | Zbl 1004.37008
, and ,[8] Substitutions et -systèmes de numération. Theoret. Comput. Sci. 137 (1995) 219-236. | MR 1311222 | Zbl 0872.11017
,[9] Additive and multiplicative properties of point sets based on beta-integers. Theoret. Comput. Sci. 303 (2003) 491-516. | MR 1990778 | Zbl 1036.11034
, and ,[10] Algebraic combinatorics on words. Cambridge University Press (2002). | MR 1905123 | Zbl 1001.68093
,[11] On the -expansions of real numbers. Acta Math. Acad. Sci. Hungar. 11 (1960) 401-416. | MR 142719 | Zbl 0099.28103
,[12] Statistics of substitution sequences. On-line computer program, available at http://kmlinux.fjfi.cvut.cz/~patera/SubstWords.cgi
,[13] Representations for real numbers and their ergodic properties. Acta Math. Acad. Sci. Hungar. 8 (1957) 477-493. | MR 97374 | Zbl 0079.08901
,[14] On periodic expansions of Pisot numbers and Salem numbers. Bull. London Math. Soc. 12 (1980) 269-278. | MR 576976 | Zbl 0494.10040
,[15] Groups, tilings, and finite state automata. Geometry supercomputer project research report GCG1, University of Minnesota (1989).
,[16] Complexity and balances of the infinite word of -integers for , in Proc. of Words'03, Turku. TUCS Publication 27 (2003) 138-148. | Zbl 1040.68090
,