The D-representation of nonnegative integers and the Fibonacci factorization of suffixes of infinite Fibonacci words

For each suffix X of a two-way infinite Fibonacci word, we consider the factorization X=ukuk+1uk+2⋯, where k is a positive integer, and the length of the factor ui is the ith Fibonacci number (i≥k). It is called the Fibonacci factorization of X of order k. We show that in such a factorization, either all ui are singular words, or there exists a positive integer l≥k such that ul,ul+1,ul+2,… are the Fibonacci words along an infinite path in the tree of Fibonacci words and the rest of the uis are singular words. The labels of such infinite paths are determined by the D-representation of nonnegative integers.