Overlap-free symmetric D0L words

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 Fife [8] who described all binary overlap-free infinite words and 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 this 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 = f0;1; : : : ;q 1g. For an integer i, let i denote the residue of i modulo q. A morphism φ : Σ q ! Σ q is called symmetric if for all i 2 Σq the equality holds φ(i) = t1+ i t2+ i : : : tm+ i; where t1t2 : : : tm is an arbitrary word (equal to φ(0)). Clearly, if t1 = 0, then φ has a fixed point, i. e., a (right) infinite word w = w(φ) satisfying w = φ(w):