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
Mots clés : 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},
     publisher = {EDP-Sciences},
     volume = {41},
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     mrnumber = {2350639},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/ita:2007009/}
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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. http://www.numdam.org/articles/10.1051/ita:2007009/

Bibliographie
Cité par

[1] B. Adamczewski, Balances for fixed points of primitive substitutions. Theoret. Comput. Sci. 273 (2002) 197-224. | Zbl

[2] F. Bassino, Beta-expansions for cubic Pisot numbers, in LATIN'02, Springer. Lect. notes Comput. Sci. 2286 (2002) 141-152.

[3] V. Berthé and R. Tijdeman, Balance properties of multi-dimensional words. Theoret. Comput. Sci. 60 (1938) 815-866. | Zbl

[4] E.M. Coven and G.A. Hedlund, Sequences with minimal block growth. Math. Systems Theory 7 (1973) 138-153. | Zbl

[5] Ch. Frougny and B. Solomyak, Finite beta-expansions. Ergod. Theor. Dyn. Syst. 12 (1992) 713-723. | Zbl

[6] Ch. Frougny, J.P. Gazeau and J. Krejcar, Additive and multiplicative properties of point-sets based on beta-integers. Theoret. Comput. Sci. 303 (2003) 491-516. | Zbl

[7] Ch. Frougny, E. Pelantová and Z. Masáková, Complexity of infinite words associated with beta-expansions. RAIRO-Inf. Theor. Appl. 38 (2004) 163-185. | Numdam | Zbl

[8] M. Lothaire, Algebraic combinatorics on words. Cambridge University Press (2002). | MR | Zbl

[9] M. Morse and G.A. Hedlund, Symbolic dynamics. Amer. J. Math. 60 (1938) 815-866. | JFM

[10] M. Morse and G.A. Hedlund, Symbolic dynamics II. Sturmian Trajectories. Amer. J. Math. 62 (1940) 1-42. | JFM

[11] O. Turek, Complexity and balances of the infinite word of $β$ -integers for $β = 1 + \sqrt{3}$ , in Proc. of WORDS'03, Turku (2003) 138-148. | Zbl

[12] L. Vuillon, Balanced words. Bull. Belg. Math. Soc. Simon Stevin 10 (2003) 787-805. | Zbl

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