Chapter 1 - Finite and Infinite Words

Summary

Introduction

The aim of this chapter is to provide an introduction to several concepts used elsewhere in the book. It fixes the general notation on words used elsewhere. It also introduces more specialized notions of general interest. For instance, the notion of a uniformly recurrent word used in several other chapters is introduced here.

We start with the notation concerning finite and infinite words. We also describe the Cantor space topology on the space of infinite words.

We provide a basic introduction to the theory of automata. It covers the determinization algorithm, part of Kleene's theorem, syntactic monoids and basic facts about transducers. These concepts are illustrated on the classical combinatorial examples of the de Bruijn graph, and the Morse-Hedlund theorem.

We also consider the relationship with generating series, as a useful tool for the enumeration of words.

We introduce some basic concepts of symbolic dynamical systems, in relation with automata. We prove the equivalence between the notions of minimality and uniform recurrence. Entropy is considered, and we show how to compute it for a sofic system.

We also present a more specialized subject, namely unavoidable sets. This notion is easy to define but leads to interesting and significant results. In this sense, the last section of this chapter is a foretaste of the rest of the book.