Sharp Characterizations of Squarefree Morphisms

Squarefree morphisms, i.e. morphisms that preserve squarefreeness of words, have been a subject of interest since the works of Thue [8]. They are a powerful tool to prove properties on squarefree words and are used in particular to prove the existence of infinite sequences without repetition. The study of squarefree words is connected with cluestions in algebra and game theory, and recentiy several results have been published on that topic [2-71. In this note we prove the following theorem: “A semigroup morphism defined on a three letter alphabet is squarefree iff the images of the squarefree words of length 5 are squarefree”. The first effective characterization of squarefree morphisms has been given by Berstel[2] who found a bound (on the length of words to check) depending on the morphism. We improve his result in the general case (not only for three letters) and obtain the above theorem as a consequence. On a related subject, i.e. morphisms defined on a three letter alphabet and which generate infinite squarefree sequences by iteration, Karhumaki [6] gave a bound independent from the morphism. We also show that the bound 5 in our theorem is optimal and that on a greater alphabet such a result does not exist.