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

Turek, Ondřej

Turek, Ondřej

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 41 (2007), p. 123-135 / Harvested from Numdam

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Résumé

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 $ϕ (A) = A^{p} B$ , $ϕ (B) = A^{q}$ for $p \in ℕ$ , $q \in ℕ$ , $p \geq q$ , where $β = \frac{p + \sqrt{p^{2} + 4 q}}{2}$ . We will 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 [B. Adamczewski, Theoret. Comput. Sci. 273 (2002) 197-224] that the fixed point of the substitution $ϕ (A) = A^{p} B$ , $ϕ (B) = A^{q}$ is not $m$ -balanced for any $m$ . We exhibit an infinite sequence of pairs of words with the unbalance property.

Publié le : 2007-01-01

MR 2350639 | Zbl pre05235503 | 3 citations dans Numdam

DOI : https://doi.org/10.1051/ita:2007009
Classification: 68R15

@article{ITA_2007__41_2_123_0,
     author = {Turek, Ond\v rej},
     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},
     volume = {41},
     year = {2007},
     pages = {123-135},
     doi = {10.1051/ita:2007009},
     mrnumber = {2350639},
     zbl = {pre05235503},
     language = {en},
     url = {http://dml.mathdoc.fr/item/ITA_2007__41_2_123_0}
}

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) pp. 123-135. doi : 10.1051/ita:2007009. http://gdmltest.u-ga.fr/item/ITA_2007__41_2_123_0/

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