Combinatorial and arithmetical properties of infinite words associated with non-simple quadratic Parry numbers

Balková, Lubomíra; Pelantová, Edita; Turek, Ondřej

Balková, Lubomíra ; Pelantová, Edita ; Turek, Ondřej

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 41 (2007), p. 307-328 / Harvested from Numdam

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Résumé

We study some arithmetical and combinatorial properties of $β$ -integers for $β$ being the larger root of the equation $x^{2} = m x - n, m, n \in ℕ, m \geq n + 2 \geq 3$ . We determine with the accuracy of $\pm$ 1 the maximal number of $β$ -fractional positions, which may arise as a result of addition of two $β$ -integers. For the infinite word $u_{β}$ coding distances between the consecutive $β$ -integers, we determine precisely also the balance. The word $u_{β}$ is the only fixed point of the morphism $A$ $\to$ $A^{m - 1} B$ and $B$ $\to$ $A^{m - n - 1} B$ . In the case $n = 1$ , the corresponding infinite word $u_{β}$ is sturmian, and, therefore, $1$ -balanced. On the simplest non-sturmian example with $n$ $\geq$ 2, we illustrate how closely the balance and the arithmetical properties of $β$ -integers are related.

Publié le : 2007-01-01

MR 2354360 | Zbl pre05211862 | 2 citations dans Numdam

DOI : https://doi.org/10.1051/ita:2007025
Classification: 68R15, 11A63

@article{ITA_2007__41_3_307_0,
     author = {Balkov\'a, Lubom\'\i ra and Pelantov\'a, Edita and Turek, Ond\v rej},
     title = {Combinatorial and arithmetical properties of infinite words associated with non-simple quadratic Parry numbers},
     journal = {RAIRO - Theoretical Informatics and Applications - Informatique Th\'eorique et Applications},
     volume = {41},
     year = {2007},
     pages = {307-328},
     doi = {10.1051/ita:2007025},
     mrnumber = {2354360},
     zbl = {pre05211862},
     language = {en},
     url = {http://dml.mathdoc.fr/item/ITA_2007__41_3_307_0}
}

Balková, Lubomíra; Pelantová, Edita; Turek, Ondřej. Combinatorial and arithmetical properties of infinite words associated with non-simple quadratic Parry numbers. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 41 (2007) pp. 307-328. doi : 10.1051/ita:2007025. http://gdmltest.u-ga.fr/item/ITA_2007__41_3_307_0/

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