A note on differentiable palindromes

Abstract

We give a characterization of the palindromes in a class of infinite words over Σ={1,2} related to the Kolakoski word K. This characterization, based on the left palindromic closure of all prefixes of K, is obtained by using a bijection between the class of right infinite words over Σ and a class of words over the same alphabet, and reveals the first link between the existence of some palindromes and the recurrence of K. Indeed, the existence of arbitrarily long palindromes implies the recurrence of K, and a stronger assumption implies the closure of the set of its factors by permutation of the letters in Σ.