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

RAIRO - Theoretical Informatics and Applications (2007)

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

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topTurek, 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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- M. Lothaire, Algebraic combinatorics on words. Cambridge University Press (2002).
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- M. Morse and G.A. Hedlund, Symbolic dynamics II. Sturmian Trajectories. Amer. J. Math.62 (1940) 1–42.
- O. Turek, Complexity and balances of the infinite word of $\beta $-integers for $\beta =1+\sqrt{3}$, in Proc. of WORDS'03, Turku (2003) 138–148.
- L. Vuillon, Balanced words. Bull. Belg. Math. Soc. Simon Stevin10 (2003) 787–805.

## Citations in EuDML Documents

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

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