# Palindromic complexity of infinite words associated with simple Parry numbers

Petr Ambrož[1]; Zuzana Masáková[2]; Edita Pelantová[2]; Christiane Frougny[3]

• [1] Czech Technical University Doppler Institute for Mathematical Physics and Applied Mathematics Department of Mathematics, FNSPE Trojanova 13, 120 00 Praha 2 (Czech Republic)
• [2] Doppler Institute for Mathematical Physics and Applied Mathematics and Department of Mathematics, FNSPE, Czech Technical University, Trojanova 13, 120 00 Praha 2 Czech Republic
• [3] Université Paris 7 LIAFA, UMR 7089 CNRS 2 place Jussieu 75251 Paris Cedex 05 (France) and Université Paris 8
• Volume: 56, Issue: 7, page 2131-2160
• ISSN: 0373-0956

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## Abstract

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A simple Parry number is a real number $\beta >1$ such that the Rényi expansion of $1$ is finite, of the form ${d}_{\beta }\left(1\right)={t}_{1}\cdots {t}_{m}$. We study the palindromic structure of infinite aperiodic words ${u}_{\beta }$ that are the fixed point of a substitution associated with a simple Parry number $\beta$. It is shown that the word ${u}_{\beta }$ contains infinitely many palindromes if and only if ${t}_{1}={t}_{2}=\cdots ={t}_{m-1}\ge {t}_{m}$. Numbers $\beta$ satisfying this condition are the so-called confluent Pisot numbers. If ${t}_{m}=1$ then ${u}_{\beta }$ is an Arnoux-Rauzy word. We show that if $\beta$ is a confluent Pisot number then $𝒫\left(n+1\right)+𝒫\left(n\right)=𝒞\left(n+1\right)-𝒞\left(n\right)+2$, where $𝒫\left(n\right)$ is the number of palindromes and $𝒞\left(n\right)$ is the number of factors of length $n$ in ${u}_{\beta }$. We then give a complete description of the set of palindromes, its structure and properties.

## How to cite

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Ambrož, Petr, et al. "Palindromic complexity of infinite words associated with simple Parry numbers." Annales de l’institut Fourier 56.7 (2006): 2131-2160. <http://eudml.org/doc/10200>.

@article{Ambrož2006,
abstract = {A simple Parry number is a real number $\beta &gt;1$ such that the Rényi expansion of $1$ is finite, of the form $d_\beta (1)=t_1 \cdots t_m$. We study the palindromic structure of infinite aperiodic words $u_\beta$ that are the fixed point of a substitution associated with a simple Parry number $\beta$. It is shown that the word $u_\beta$ contains infinitely many palindromes if and only if $t_1=t_2= \cdots =t_\{m-1\}\ge t_m$. Numbers $\beta$ satisfying this condition are the so-called confluent Pisot numbers. If $t_m=1$ then $u_\beta$ is an Arnoux-Rauzy word. We show that if $\beta$ is a confluent Pisot number then $\{\mathcal\{P\}\}(n+1)+ \{\mathcal\{P\}\}(n) = \{\mathcal\{C\}\}(n+1) - \{\mathcal\{C\}\}(n)+2$, where $\{\mathcal\{P\}\}(n)$ is the number of palindromes and $\{\mathcal\{C\}\}(n)$ is the number of factors of length $n$ in $u_\beta$. We then give a complete description of the set of palindromes, its structure and properties.},
affiliation = {Czech Technical University Doppler Institute for Mathematical Physics and Applied Mathematics Department of Mathematics, FNSPE Trojanova 13, 120 00 Praha 2 (Czech Republic); Doppler Institute for Mathematical Physics and Applied Mathematics and Department of Mathematics, FNSPE, Czech Technical University, Trojanova 13, 120 00 Praha 2 Czech Republic; Doppler Institute for Mathematical Physics and Applied Mathematics and Department of Mathematics, FNSPE, Czech Technical University, Trojanova 13, 120 00 Praha 2 Czech Republic; Université Paris 7 LIAFA, UMR 7089 CNRS 2 place Jussieu 75251 Paris Cedex 05 (France) and Université Paris 8},
author = {Ambrož, Petr, Masáková, Zuzana, Pelantová, Edita, Frougny, Christiane},
journal = {Annales de l’institut Fourier},
keywords = {beta-expansions; palindromic complexity},
language = {eng},
number = {7},
pages = {2131-2160},
publisher = {Association des Annales de l’institut Fourier},
title = {Palindromic complexity of infinite words associated with simple Parry numbers},
url = {http://eudml.org/doc/10200},
volume = {56},
year = {2006},
}

TY - JOUR
AU - Ambrož, Petr
AU - Masáková, Zuzana
AU - Pelantová, Edita
AU - Frougny, Christiane
TI - Palindromic complexity of infinite words associated with simple Parry numbers
JO - Annales de l’institut Fourier
PY - 2006
PB - Association des Annales de l’institut Fourier
VL - 56
IS - 7
SP - 2131
EP - 2160
AB - A simple Parry number is a real number $\beta &gt;1$ such that the Rényi expansion of $1$ is finite, of the form $d_\beta (1)=t_1 \cdots t_m$. We study the palindromic structure of infinite aperiodic words $u_\beta$ that are the fixed point of a substitution associated with a simple Parry number $\beta$. It is shown that the word $u_\beta$ contains infinitely many palindromes if and only if $t_1=t_2= \cdots =t_{m-1}\ge t_m$. Numbers $\beta$ satisfying this condition are the so-called confluent Pisot numbers. If $t_m=1$ then $u_\beta$ is an Arnoux-Rauzy word. We show that if $\beta$ is a confluent Pisot number then ${\mathcal{P}}(n+1)+ {\mathcal{P}}(n) = {\mathcal{C}}(n+1) - {\mathcal{C}}(n)+2$, where ${\mathcal{P}}(n)$ is the number of palindromes and ${\mathcal{C}}(n)$ is the number of factors of length $n$ in $u_\beta$. We then give a complete description of the set of palindromes, its structure and properties.
LA - eng
KW - beta-expansions; palindromic complexity
UR - http://eudml.org/doc/10200
ER -

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