Periodicity and the golden ratio

Abstract

We prove a periodicity theorem on words that has strong analogies with the Critical Factorization theorem. The Critical Factorization theorem states, roughly speaking, a connection between local and global periods of a word; the local period at any position in the word is there defined as the shortest repetition (a square) “centered” in that position. We here take into account a different notion of local period by considering, for any position in the word, the shortest repetition “immediately to the left” from that position. In this case a repetition which is a square does not suffices and the golden ratio ϑ (more precisely its square ϑ 2 = 2.618 …) surprisingly appears as a threshold for establishing a connection between local and global periods of the word. We further show that the number ϑ 2 is tight for this result. Two applications are then derived. In the firts we give a characterization of ultimately periodic infinite words. The second application concerns the topological perfectness of some families of infinite words.