RAIRO - Theoretical Informatics and Applications

Research Article

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

Ondřej Turek

Department of Mathematics, FNSPE, Czech Technical University, Trojanova 13, 120 00 Praha 2, Czech Republic; oturek@centrum.cz


In this paper we will deal with the balance properties of the infinite binary words associated to β-integers when β 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 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 $\varphi(A)=A^pB$, $\varphi(B)=A^q$ is not m-balanced for any m. We exhibit an infinite sequence of pairs of words with the unbalance property.

(Received May 1 2004)

(Accepted June 8 2005)

(Online publication July 18 2007)

Key Words:

  • Balance property;
  • substitution invariant;
  • Parry number

Mathematics Subject Classification:

  • 68R15