Languages under substitutions and balanced

= zu2(i + k) for all i E I,. 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 Ixl is defined as The usual notation for a finite word of length n is x = xi ... zn where all xi E {a, &}. 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 e). 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 iv, notation x C 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 = y 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 13(w, n) and we write P(w, n) := The mapping P(w, n) : N ~ N is known as the complexity function of w. A factor x C w has multiple right extension (m.r.e.) in w if xa, xb C w. We also say that x