On the k-freeness of morphisms on free monoids

Abstract

Let an integer k≧2 be fixed. A word is called k-repetition free, or shortly k-free, if it does not contain any non-empty subword of the form R^k. A morphism h: X* → Y* is called k-free if the word h(w) is k-free for every k-free word w in X*. We investigate the general structure of k-free morphisms and give outlines for the proof of the following result: if a non-trivial morphism h: X* → Y*, where card(X)≧2 and card(Y)≧2, is k-free for some integer k≧2, then, except a certain possibility concerning one infrequent situation in the case k=3, h is a primitive ps-code. Moreover, an effective characterization is provided for all k-free morphisms h: X* → Y* in the case k≧q_h+1, and for a wide class of morphisms in the case 2≦k≦q_h, where q_h=max{|h(a)| | a∈X}.