Balance properties of the fixed point of the substitution associated to quadratic simple Pisot numbers
RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 41 (2007) no. 2, pp. 123-135.

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 $p\in ℕ$, $q\in ℕ$, $p\ge q$, where $\beta =\frac{p+\sqrt{{p}^{2}+4q}}{2}$. We will prove that such word is $t$-balanced with $t=1+\left[\left(p-1\right)/\left(p+1-q\right)\right]$. Finally, in the case that $p 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.

DOI : https://doi.org/10.1051/ita:2007009
Classification : 68R15
Mots clés : balance property, substitution invariant, Parry number
@article{ITA_2007__41_2_123_0,
author = {Turek, Ond\v{r}ej},
title = {Balance properties of the fixed point of the substitution associated to quadratic simple Pisot numbers},
journal = {RAIRO - Theoretical Informatics and Applications - Informatique Th\'eorique et Applications},
pages = {123--135},
publisher = {EDP-Sciences},
volume = {41},
number = {2},
year = {2007},
doi = {10.1051/ita:2007009},
mrnumber = {2350639},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/ita:2007009/}
}
Turek, Ondřej. Balance properties of the fixed point of the substitution associated to quadratic simple Pisot numbers. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 41 (2007) no. 2, pp. 123-135. doi : 10.1051/ita:2007009. http://archive.numdam.org/articles/10.1051/ita:2007009/

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