Overlap - Free Symmetric D 0 L words † Anna

It was rediscovered several times, can be constructed in many alternative ways and occurs in various fields of mathematics (see the survey [1]). The set of all overlap-free words was studied e. g. by E. D. Fife [8] who described all binary overlapfree infinite words and P. Séébold [13] who proved that the Thue-Morse word is essentially the only binary overlap-free word which is a fixed point of a morphism. Nowadays the theory of overlap-free words is a part of a more general theory of pattern avoidance [5]. J.-P. Allouche and J. Shallit [2] asked if the initial Thue’s construction of an overlap-free word could be generalized and found a whole family of overlap-free infinite words built by a similar principle. This paper contains a further generalization of that result; its main theorem was conjectured by J. Shallit [14]. Let us give all the necessary definitions and state the main theorem. Consider a finite alphabet Σ = Σq = {0,1, . . . ,q−1}. For an integer i, let i denote the residue of i modulo q. A morphism φ : Σq→ Σq is called symmetric if for all i ∈ Σq the equality holds φ(i) = t1 + i t2 + i . . . tm + i, †Supported in part by INTAS (grant 97-1001) and RFBR (grant 01-01-06018).