Palindromic complexity of infinite words associated with simple Parry numbers

Ambrož, Petr; Masáková, Zuzana; Pelantová, Edita; Frougny, Christiane

doi:10.5802/aif.2236

Ambrož, Petr ¹; Masáková, Zuzana ²; Pelantová, Edita ²; Frougny, Christiane 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

Annales de l'Institut Fourier, Volume 56 (2006) no. 7, pp. 2131-2160.

Abstract
Résumé

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

Un nombre de Parry simple est un nombre réel $β > 1$ tel que le développement de Rényi de $1$ est fini, de la forme $d_{β} (1) = t_{1} \dots t_{m}$ . Nous étudions la structure palindromique des mots infinis apériodiques $u_{β}$ qui sont point fixe d’une substitution associée à un nombre de Parry simple $β$ . Nous montrons que le mot $u_{β}$ contient un nombre infini de palindromes si et seulement si $t_{1} = t_{2} = \dots = t_{m - 1} \geq t_{m}$ . Les nombres $β$ satisfaisant cette condition sont connus sous le nom de nombres de Pisot confluents. Si de plus $t_{m} = 1$ alors $u_{β}$ est un mot d’Arnoux-Rauzy. Nous montrons que si $β$ est un nombre de Pisot confluent alors $𝒫 (n + 1) + 𝒫 (n) = 𝒞 (n + 1) - 𝒞 (n) + 2$ , où $𝒫 (n)$ est le nombre de facteurs de longueur $n$ de $u_{β}$ . Nous donnons aussi une description complète de l’ensemble des palindromes, de sa structure et de ses propriétés.

MR Zbl | 5 citations in Numdam

DOI: 10.5802/aif.2236

Classification: 68R15, 11A63
Keywords: beta-expansions, palindromic complexity
Mot clés : beta-développements, complexité palindromique

Author's affiliations:

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

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     author = {Ambro\v{z}, Petr and Mas\'akov\'a, Zuzana and Pelantov\'a, Edita and Frougny, Christiane},
     title = {Palindromic complexity of infinite words associated with simple {Parry} numbers},
     journal = {Annales de l'Institut Fourier},
     pages = {2131--2160},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {56},
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     year = {2006},
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Ambrož, Petr; Masáková, Zuzana; Pelantová, Edita; Frougny, Christiane. Palindromic complexity of infinite words associated with simple Parry numbers. Annales de l'Institut Fourier, Volume 56 (2006) no. 7, pp. 2131-2160. doi : 10.5802/aif.2236. http://www.numdam.org/articles/10.5802/aif.2236/

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