Periodicity and the golden ratio

✍ Scribed by Filippo Mignosi; Antonio Restivo; Sergio Salemi

• Book ID: 104326475
• Publisher: Elsevier Science
• Year: 1998
• Tongue: English
• Weight: 983 KB
• Volume: 204
• Category: Article
• ISSN: 0304-3975
• DOI: 10.1016/s0304-3975(98)00037-1

✦ Synopsis

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 rutio cp (more precisely its square cpz = 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 cpz 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.