On k-repetition free words generated by length uniform morphisms over a binary alphabet

Abstract

Let an integer k ≧ 3 be fixed. A word is called k-repetition free, or shortly k-free, if it does not contain a subword of the form p^k ≠ λ. Let a morphism h: {a,b}* → γ* be length uniform (meaning that |h(a)|=|h(b)|) and h(a) ≠ h(b). Assume that pⁿ, k ≦ n ε ℕ, is a subword of h(w), where w in {a,b}* is k-free. In this case we give an optimal upper bound for the length of pⁿ. Moreover, we give outlines for the proof of the following result: when deciding whether a given morphism h, of the form mentioned above, is k-free, one has only to examine (in an easy way) the words h(w_o), where the length of w_o is ≦ 4 (or, in some special cases, even less). Finally, we characterize sharply k-free DOL and NDOL sequences obtained by using length uniform binary morphisms. For example, in the case k=3 we have the following result: if a length uniform binary endomorphism generates a cube in a DOL sequence, then it does so in three steps.