Palindromic complexity of infinite words associated with simple Parry numbers

A simple Parry number is a real number β > 1 such that the Rényi expansion of 1 is finite, of the form dβ(1) = t1 · · · tm. We study the palindromic structure of infinite aperiodic words uβ that are the fixed point of a substitution associated with a simple Parry number β. It is shown that the word uβ contains infinitely many palindromes if and only if t1 = t2 = · · · = tm−1 ≥ tm. Numbers β satisfying this condition are the so-called confluent Pisot numbers. If tm = 1 then uβ 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 uβ. We then give a complete description of the set of palindromes, its structure and properties.