Q-Factorization of suffixes of two-way infinite extensions of irrational characteristic words

Let α be an irrational number between 0 and 1 with continued fraction expansion [0;a1+1,a2,a3,…], where an≥1 (n≥1). Define a sequence of numbers {qn}n≥−1 by q−1=1, q0=1, qn=anqn−1+qn−2 (n≥1). For each integer k≥−1, we consider the kth-order Q-factorization of each suffix H of a two-way infinite extension X of the characteristic word of α of the form: H=ukuk+1uk+2⋯, where the length of the factor ui is qi (i≥k). We show that in such a factorization of the simple Sturmian word H, either all ui are singular words, or there exists a nonnegative integer q such that H begins with q singular words, and uk+q,uk+q+1,uk+q+2,… are α-words. Moreover, the number q and the labels of these α-words are uniquely determined by the Q-representation of a nonnegative integer obtained from the position of H in X. These results lead, naturally, to a new method for generating suffixes of X.