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