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

Turek, Ondřej

doi:10.1051/ita:2007009

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 $β$ -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.

MR | 3 citations dans Numdam

DOI : 10.1051/ita:2007009

Classification : 68R15
Keywords: balance property, substitution invariant, Parry number

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     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},
     year = {2007},
     publisher = {EDP Sciences},
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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

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