Definitions and Notation

This paper consists of three parts. In the first part we prove a general theorem on the image of a language K under a substitution, in the second we apply this to the special case when K is the language of balanced words and in the third part we deal with recurrent Z-words of minimal block growth. Definitions and notation In this paper a word is a mapping w : I → {a, b} where I is a subinterval of Z. We identify words which are shifts of eachother: hence we identify w1 : I1 → {a, b}, w2 : I2 → {a, b} if there exists an integer k with I1+k = I2 and w1(i) = w2(i + k) for all i ∈ I1. If I = Z we call w a Z-word or a bi-infinite word. If I is finite we call x finite and its length |x| is defined as |I|. The usual notation for a finite word of length n is x = x1 · · ·xn where all xi ∈ {a, b}. We write {a, b}∗ for the collection of finite words and {a, b}+ for the nonempty finite words. (The empty word will be denoted throughout by ). The concatenation xy of two words x, y is defined by writing x in front of y, which is only defined under the obvious restrictions. A finite word x is called a factor of w, notation x ⊂ w, if w = yxz for some words y, z. It is called a left-factor (prefix) of w if w = xy for some y and a right-factor (suffix) if w = yx for some y. The factor-set of w will be denoted by F (w). An n-factor of w is a factor of w of length n. The collection of n-factors is defined by B(w, n) and we write P (w, n) := |B(w, n)|. The mapping P (w, n) : N→ N is known as the complexity function of w. A factor x ⊂ w has multiple right extension (m.r.e.) in w if xa, xb ⊂ w. We also say that x Manuscrit reçu le 31 janvier 2002.