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Complexity of infinite words associated with beta-expansions. (English) Zbl 1104.11013

Summary: 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 \(\mathcal C(n)=n+1\). 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 \(\mathcal C(n)=(m-1)n+1\). In the cases when \(t_1=t_2=\cdots=t_{m-1}\) or \(t_1>\max\{t_2,\dots,t_{m-1}\}\) we show that the first difference of the complexity function \(\mathcal C(n+1)-\mathcal C(n)\) 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 \(d_\beta(1)=t\,t\cdots\,t\,1\). 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 \(2n+1\).
In the corrigendum we add a sufficient condition for validity of Proposition 4.10. This condition is not a necessary one, it is nevertheless convenient, since anyway most of the statements in the paper use it.

MSC:

11B85 Automata sequences
11A63 Radix representation; digital problems
68R15 Combinatorics on words
37B10 Symbolic dynamics
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