Chapter 8 - Periodicity

Summary

Introduction

Periodicity is an important property of words that has applications in various domains. The first significant results on periodicity are the theorem of Fine and Wilf and the critical factorization theorem. These two results refer to two kinds of phenomena concerning periodicity: the theorem of Fine and Wilf considers the simultaneous occurrence of different periods in one finite word, whereas the critical factorization theorem relates local and global periodicity of words. Starting from these basic results the study of periodicity has grown along both directions. This chapter contains a systematic and self-contained exposition of this theory, including very recent results.

In Section 8.1 we analyze the structure of the set of periods of one finite word. This section includes a proof of the theorem of Fine and Wilf and also a generalization of this result to words having three periods. We next give the characterization of Guibas and Odlyzko concerning those sets of integers that yield the periods that can simultaneously occur in a single finite word. Another property is further investigated (similar to the one stated by the theorem of Fine and Wilf) in which the occurrence of two periods in a word of a certain length forces the word to have a shorter period only in a prefix (or suffix) of the word. The golden ratio appears in such a result as an extremal value of a parameter involved in the property. This section also contains some results concerning the squares that can appear as factors in a word. This is a prelude to the next section, since squares describe a special kind of local periodicity.