Balance properties of the fixed point of the substitution associated to quadratic simple Pisot numbers

Turek, Ondřej

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 41 (2007) no. 2, pp. 123-135.

Résumé

In this paper we will 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 $\varphi \left(A\right)=A^{p}B$, $\varphi \left(B\right)=A^q$ for $p\in \mathbb{N}$, $q\in \mathbb{N}$, $p\ge q$, where $\beta =\frac{p+\sqrt{p^2+4q}}{2}$. We will prove that such word is $t$-balanced with $t=1+\left[(p-1)/(p+1-q)\right]$. Finally, in the case that $p<q$ it is known [B. Adamczewski, Theoret. Comput. Sci. 273 (2002) 197-224] that the fixed point of the substitution $\varphi \left(A\right)=A^{p}B$, $\varphi \left(B\right)=A^q$ is not $m$-balanced for any $m$. We exhibit an infinite sequence of pairs of words with the unbalance property.

Détail

DOI : 10.1051/ita:2007009

Classification : 68R15
Mots-clés : balance property, substitution invariant, Parry number