On $\theta$-episturmian words

Abstract

In this paper we study a class of infinite words on a finite alphabet $A$ whose factors are closed under the image of an involutory antimorphism $\theta$ of the free monoid $A^*$. We show that given a recurrent infinite word $\omega \in A^{\mathbb N}$, if there exists a positive integer $K$ such that for each $n\geq 1$ the word $\omega$ has

1. $\mathrm{card} A+(n-1)K$ distinct factors of length $n$, and
2. a unique right and a unique left special factor of length $n$,

then there exists an involutory antimorphism $\theta$ of the free monoid $A^*$ preserving the set of factors of $\omega$.