A characterization of overlap-free morphisms

Abstract

We prove that a morphism h over a two-letter alphabet { a, b} is overlap-free, i.e., maps overlap-free words into overlap-free words, iff the word h( abbabaab) is overlap-free. As a consequence, we obtain a simple proof of the fact that the only infinite overlap-free words that can be obtained by iterating a morphism are the Thue–Morse sequence and its opposite.