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Complexity and balances of the infinite word of \(\beta\)-integers for \(\beta=1+\sqrt 3\). (English) Zbl 1040.68090

Harju, Tero (ed.) et al., Proceedings of WORDS’03, the 4th international conference on combinatorics on words, Turku, Finland, September 10–13, 2003. Turku: Turku Centre for Computer Science (ISBN 952-12-1211-X/pbk). TUCS General Publication 27, 138-148 (2003).
Summary: We consider the infinite binary word \(u\) corresponding to the \(\beta\)-integers for \(\beta= 1+\sqrt{3}\), which is the fixed point of the morphism \(\varphi(A)= AAB\), \(\varphi(B)= AA\). Since \(\beta\) is not a quadratic unit, the infinite word \(u\) is not a Sturmian word; therefore it has complexity greater than \(n+1\) and it is not balanced. We describe the structure of left special factors of \(u\) which is more complicated than in the case of Sturmian words. This allows us to determine the complexity of \(u\), in particular, we show that \(\liminf_{k\in\infty}{\mathcal C}(k)/k\simeq 1.18\), \(\limsup_{k\to\infty}{\mathcal C}(k)/k\simeq 1.38\). Moreover, we show that \(u\) is a 2-balanced word.
For the entire collection see [Zbl 1030.00026].

MSC:

68R15 Combinatorics on words
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