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

Ondřej Turek

RAIRO - Theoretical Informatics and Applications (2007)

  • Volume: 41, Issue: 2, page 123-135
  • ISSN: 0988-3754

Abstract

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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 ∈ ℕ , q ∈ ℕ , p ≥ q , where β = p + 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.

How to cite

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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 41.2 (2007): 123-135. <http://eudml.org/doc/250032>.

@article{Turek2007,
abstract = { 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 $\varphi(A)=A^pB$, $\varphi(B)=A^q$ for $p\in\mathbb N$, $q\in\mathbb N$, $p\geq q$, where $\beta=\frac\{p+\sqrt\{p^2+4q\}\}\{2\}$. We will prove that such word is t-balanced with $t=1+\left[(p-1)/(p+1-q)\right]$. 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 $\varphi(A)=A^pB$, $\varphi(B)=A^q$ is not m-balanced for any m. We exhibit an infinite sequence of pairs of words with the unbalance property. },
author = {Turek, Ondřej},
journal = {RAIRO - Theoretical Informatics and Applications},
keywords = {Balance property; substitution invariant; Parry number; infinite binary words},
language = {eng},
month = {7},
number = {2},
pages = {123-135},
publisher = {EDP Sciences},
title = {Balance properties of the fixed point of the substitution associated to quadratic simple Pisot numbers},
url = {http://eudml.org/doc/250032},
volume = {41},
year = {2007},
}

TY - JOUR
AU - Turek, Ondřej
TI - Balance properties of the fixed point of the substitution associated to quadratic simple Pisot numbers
JO - RAIRO - Theoretical Informatics and Applications
DA - 2007/7//
PB - EDP Sciences
VL - 41
IS - 2
SP - 123
EP - 135
AB - 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 $\varphi(A)=A^pB$, $\varphi(B)=A^q$ for $p\in\mathbb N$, $q\in\mathbb N$, $p\geq q$, where $\beta=\frac{p+\sqrt{p^2+4q}}{2}$. We will prove that such word is t-balanced with $t=1+\left[(p-1)/(p+1-q)\right]$. 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 $\varphi(A)=A^pB$, $\varphi(B)=A^q$ is not m-balanced for any m. We exhibit an infinite sequence of pairs of words with the unbalance property.
LA - eng
KW - Balance property; substitution invariant; Parry number; infinite binary words
UR - http://eudml.org/doc/250032
ER -

References

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  1. B. Adamczewski, Balances for fixed points of primitive substitutions. Theoret. Comput. Sci.273 (2002) 197–224.  
  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.  
  4. E.M. Coven and G.A. Hedlund, Sequences with minimal block growth. Math. Systems Theory7 (1973) 138–153.  
  5. Ch. Frougny and B. Solomyak, Finite beta-expansions. Ergod. Theor. Dyn. Syst.12 (1992) 713–723.  
  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.  
  7. Ch. Frougny, E. Pelantová and Z. Masáková, Complexity of infinite words associated with beta-expansions. RAIRO-Inf. Theor. Appl.38 (2004) 163–185.  
  8. M. Lothaire, Algebraic combinatorics on words. Cambridge University Press (2002).  
  9. M. Morse and G.A. Hedlund, Symbolic dynamics. Amer. J. Math.60 (1938) 815–866.  
  10. M. Morse and G.A. Hedlund, Symbolic dynamics II. Sturmian Trajectories. Amer. J. Math.62 (1940) 1–42.  
  11. O. Turek, Complexity and balances of the infinite word of β -integers for β = 1 + 3 , in Proc. of WORDS'03, Turku (2003) 138–148.  
  12. L. Vuillon, Balanced words. Bull. Belg. Math. Soc. Simon Stevin10 (2003) 787–805.  

Citations in EuDML Documents

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  1. L'ubomíra Balková, Zuzana Masáková, Palindromic complexity of infinite words associated with non-simple Parry numbers
  2. L'ubomíra Balková, Zuzana Masáková, Palindromic complexity of infinite words associated with non-simple Parry numbers
  3. Lubomíra Balková, Edita Pelantová, Ondřej Turek, Combinatorial and arithmetical properties of infinite words associated with non-simple quadratic Parry numbers
  4. Ondřej Turek, Balances and Abelian Complexity of a Certain Class of Infinite Ternary Words
  5. Z. Masáková, T. Vávra, Integers in number systems with positive and negative quadratic Pisot base
  6. L'ubomíra Balková, Edita Pelantová, Štěpán Starosta, Sturmian jungle (or garden?) on multiliteral alphabets
  7. L'ubomíra Balková, Edita Pelantová, Štěpán Starosta, Sturmian jungle (or garden?) on multiliteral alphabets

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