# Complexity of infinite words associated with beta-expansions

Christiane Frougny; Zuzana Masáková; Edita Pelantová

RAIRO - Theoretical Informatics and Applications (2010)

- Volume: 38, Issue: 2, page 163-185
- ISSN: 0988-3754

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topFrougny, Christiane, Masáková, Zuzana, and Pelantová, Edita. "Complexity of infinite words associated with beta-expansions." RAIRO - Theoretical Informatics and Applications 38.2 (2010): 163-185. <http://eudml.org/doc/92737>.

@article{Frougny2010,

abstract = {
We study the complexity of the infinite word uβ associated with the
Rényi expansion of 1 in an irrational base β > 1.
When β is the golden ratio, this is the well known Fibonacci word,
which is Sturmian, and of complexity C(n) = n + 1.
For β such that
dβ(1) = t1t2...tm is finite we provide a simple description of
the structure of special factors of the word uβ. When tm=1
we show that
C(n) = (m - 1)n + 1. In the cases when t1 = t2 = ... tm-1or
t1 > max\{t2,...,tm-1\} we show that the first difference
of the complexity function C(n + 1) - C(n ) takes value in
\{m - 1,m\} for every n, and consequently we determine the
complexity of uβ. We show that
uβ is an Arnoux-Rauzy sequence if and only if
dβ(1) = tt...t1. On the example of
β = 1 + 2cos(2π/7), solution of X3 = 2X2 + X - 1, we illustrate
that the structure of special factors is more complicated for
dβ(1) infinite eventually periodic.
The complexity for this word is equal to 2n+1.
},

author = {Frougny, Christiane, Masáková, Zuzana, Pelantová, Edita},

journal = {RAIRO - Theoretical Informatics and Applications},

keywords = {Beta-expansions; complexity of infinite words.},

language = {eng},

month = {3},

number = {2},

pages = {163-185},

publisher = {EDP Sciences},

title = {Complexity of infinite words associated with beta-expansions},

url = {http://eudml.org/doc/92737},

volume = {38},

year = {2010},

}

TY - JOUR

AU - Frougny, Christiane

AU - Masáková, Zuzana

AU - Pelantová, Edita

TI - Complexity of infinite words associated with beta-expansions

JO - RAIRO - Theoretical Informatics and Applications

DA - 2010/3//

PB - EDP Sciences

VL - 38

IS - 2

SP - 163

EP - 185

AB -
We study the complexity of the infinite word uβ associated with the
Rényi expansion of 1 in an irrational base β > 1.
When β is the golden ratio, this is the well known Fibonacci word,
which is Sturmian, and of complexity C(n) = n + 1.
For β such that
dβ(1) = t1t2...tm is finite we provide a simple description of
the structure of special factors of the word uβ. When tm=1
we show that
C(n) = (m - 1)n + 1. In the cases when t1 = t2 = ... tm-1or
t1 > max{t2,...,tm-1} we show that the first difference
of the complexity function C(n + 1) - C(n ) takes value in
{m - 1,m} for every n, and consequently we determine the
complexity of uβ. We show that
uβ is an Arnoux-Rauzy sequence if and only if
dβ(1) = tt...t1. On the example of
β = 1 + 2cos(2π/7), solution of X3 = 2X2 + X - 1, we illustrate
that the structure of special factors is more complicated for
dβ(1) infinite eventually periodic.
The complexity for this word is equal to 2n+1.

LA - eng

KW - Beta-expansions; complexity of infinite words.

UR - http://eudml.org/doc/92737

ER -

## References

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## Citations in EuDML Documents

top- Ondřej Turek, Balance properties of the fixed point of the substitution associated to quadratic simple Pisot numbers
- Christiane Frougny, Zuzana Masáková, Edita Pelantová, Corrigendum: Complexity of infinite words associated with beta-expansions
- Christiane Frougny, Zuzana Masáková, Edita Pelantová, Corrigendum : “Complexity of infinite words associated with beta-expansions”

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