Introduction

An infinite word u may be partially understood by studying its finite subwords. Among the types of natural questions that arise are:

How many distinct subwords of u of length n are there, and what is the growth rate of this quantity as n tends to infinity?

Does every subword of u occur infinitely often in u, and if so, how big are the gaps between successive occurrences?

We start with the first question, which refers to a measure of complexity for infinite words, called subword complexity. This measure is of particular interest because automatic sequences and, more generally, morphic sequences have relatively low subword complexity, while the typical “random” sequence has high subword complexity.

Definition 10.1.1 Let w = a0a1a2 … be an infinite word over a finite alphabet Σ. We define Subw(n) to be the set of all subwords of length n of w. We define Sub(w) to be the set of all finite subwords of w. Finally, we define pw(n), the subword complexity function of w, to be the function counting the number of distinct length-n subwords of w.

Example 10.1.2 If w is an ultimately periodic word, then it is easy to see (Theorem 10.2.6 below) that pw(n) = O(1).