Palindromic complexity of infinite words associated with simple Parry numbers

Abstract

A simple Parry number is a real number \beta>1 such that the R\'enyi expansion of 1 is finite, of the form d_\beta(1)=t_1...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= ... =t_{m-1} \geq 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(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_\beta. We then give a complete description of the set of palindromes, its structure and properties.