Palindromic complexity of infinite words associated with simple Parry numbers

Petr Ambrož; Zuzana Masáková; Edita Pelantová; Christiane Frougny

•  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
•  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.

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