## 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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### References:

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