Characterization of Test-sets for Overlap-free Morphisms

We give a characterization of all the sets X such that any morphism h on fa; bg is overlap-free ii for all x in X , h(x) is overlap-free. As a consequence, we observe the particular case X = fbbabaag which improves the previous characterization of Berstel-S e ebold 2]. R esum e Nous donnons une caract erisation de tous les ensembles X tels qu'un morphisme h sur fa; bg est sans chevauchement si et seulement si, pour tout x dans X , h(x) est sans chevauchement. En particulier, on prouve que h est un morphisme sans chevauchement si et seulement si h(bbabaa) est sans chevauchement, ce qui am eliore un r esultat pr ec edemment prouv e par Berstel-S e ebold 2]. De plus, ce r esultat est optimal. 1 Notations and Deenitions Let A be a nite alphabet, and A the free monoid generated by A. We denote by " the empty word. A word u is a subword or factor of a word v if there exist some words x and y (possibly empty) such that v = xuy. We denote by Fact(X) the set of all the subwords of words of X. A word on an alphabet A is overlap-free if it has no subword of the form xuxux for some word u and some non empty word x (or some letter x). Remark that a word u is overlap-free ii its mirror image ~ u is overlap-free. An overlap-free morphism h on A is a morphism such that for any overlap-free word x, h(x) is overlap-free. A morphism on A is erasing if for at least one letter a in A, h(a) = ". Note that an erasing morphism can never be overlap-free. From here, we restrict our attention on binary alphabet A = fa; bg. The basic overlap-free morphisms on A are the identity Id, the morphism E which exchanges a and b, and the morphism such that (a) = ab and (b) = ba. Clearly, the composition of two overlap-free morphisms is overlap-free. Thus for any integer k, k and E k are overlap-free. The converse was proved by Thue 4] (see also 1, 5]): Theorem 1 A morphism h on fa; bg is overlap-free ii h = k or h = E k for some integer k 0. In this note, we will characterize all the test-sets for the overlap-freeness of morphism …