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Rank and symbolic complexity. (English) Zbl 0858.68051
Summary: We investigate the relation between the complexity function of a sequence, that is the number $$p(n)$$ of its factors of length $$n$$, and the rank of the associated dynamical system, that is the number of Rokhlin towers required to approximate it. We prove that if the rank is one, then $$\liminf_{n\to+\infty}p(n)/n^2\leq{1\over 2}$$, but give examples with $$\limsup_{n\to+\infty}p(n)/G(n)=1$$ for any prescribed function $$G$$ with $$G(n) = o(a^n)$$ for every $$a>I$$. We give exact computations for examples of the ‘staircase’ type, which are strongly mixing systems with quadratic complexity. Conversely, for minimal sequences, if $$p(n)<an + b$$ for some $$a\geq1$$, the rank is at most $$2[a]$$, with bounded strings of spacers, and the system is generated by a finite number of substitutions.

##### MSC:
 68Q45 Formal languages and automata
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##### References:
 [1] Arnoux, Bull. Soc. Math. 119 pp 199– (1991) [2] Queffelec, Lecture Notes in Mathematics 1294 (1987) [3] Ornstein, Mem. Am. Math. Soc. 262 pp none– (1982) [4] Ornstein, Proc. Sixth Berkeley Symp. on Mathematical Statistics and Probability pp 347– (1970) [5] DOI: 10.2307/2371264 · Zbl 0019.33502 · doi:10.2307/2371264 [6] Chacon, Proc. Fifth Berkeley Symp. on Mathematical Statistics and Probability pp 335– (1965) [7] Ferenczi, Bull. Soc. Math. 123 pp 271– (1995) [8] DOI: 10.2307/2371449 · Zbl 0024.41702 · doi:10.2307/2371449 [9] del Junco, Canadian J. Math. 24 pp 836– (1976) · Zbl 0312.47003 · doi:10.4153/CJM-1976-080-3 [10] DOI: 10.1007/BF01706087 · Zbl 0253.02029 · doi:10.1007/BF01706087 [11] Kalikow, Ergod. Th. & Dynam. Sys. 4 pp 237– (1984)
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