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Balance properties of the fixed point of the substitution associated to quadratic simple Pisot numbers. (English) Zbl 1146.68410

Summary: We deal with the balance properties of the infinite binary words associated to \(\beta\)-integers when \(\beta\) is a quadratic simple Pisot number. Those words are the fixed points of the morphisms of the type \(\varphi(A)=A^pB\), \(\varphi(B)=A^q\) for \(p\in\mathbb N\), \(q\in\mathbb N\), \(p\geq q\), where \(\beta=\frac{p+\sqrt{p^2+4q}}{2}\). We prove that such word is \(t\)-balanced with \(t=1+[(p-1)/(p+1-q)]\). Finally, in the case that \(p<q\) it is known that the fixed point of the substitution \(\varphi(A)=A^pB\), \(\varphi(B)=A^q\) is not \(m\)-balanced for any \(m\). We exhibit an infinite sequence of pairs of words with the unbalance property.

MSC:

68R15 Combinatorics on words
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References:

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