Density of Critical Factorizations

Abstract. We investigate the density of critical factorizations of infinte sequences of words. The density of critical factorizations of a word is the ratio between the number of positions that permit a critical factorization, and the number of all positions of a word. We give a short proof of the Critical Factorization Theorem and show that the maximal number of noncritical positions of a word between two critical ones is less than the period of that word. Therefore, we consider only words of index one, that is words where the shortest period is larger than one half of their total length, in this paper. On one hand, we consider words with the lowest possible number of critical points and show, as an example, that every Fibonacci word longer than five has exactly one critical factorization and every palindrome has at least two critical factorizations. On the other hand, sequences of words with a high density of critical points are considered. We show how to construct an infinite sequen...

• Type: Journal
• Year: 2002
• Where: ITA
• Authors: Tero Harju, Dirk Nowotka
• Keywords: Communications, Critical Factorization Theorem, Critical Factorizations, Critical Points, ITA 2002