# Complexity of infinite words associated with beta-expansions

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

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## Abstract

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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.

## How to cite

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Frougny, 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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