DOL Schemes and Recurrent Words

Let X be a finite alphabet, X* the free monoid generated by X and X+ = X* {1}, where 1 denotes the empty word. Elements of X* are called words and subsets of X* are called languages. A DOL scheme is a pair D = (X,h) where X is a finite alphabet and h is an endomorphism of X*; a DOL system is a triple G = (X,h,ω) where (X,h) is a DOL scheme and ω is a word of X* called the axiom of G (see [4], [5]). The language L(G) generated by G is defined by L(G) = {hn(ω) | n ≥ 0}. A word u is called alive if hn(u) ≠ 1 for n ≥ 1.