A rewriting of Fife's theorem about overlap-free words

Abstract

The purpose of this expository paper is to present a self-contained proof of a famous theorem of Fife that gives a full description of the set of infinite overlap-free words over a binary alphabet. Fife's characterization consists in a parameterization of these infinite words by a set of infinite words over a ternary alphabet. The result is that the latter is a regular set. The proof is by the explicit construction of the minimal automaton, obtained by the method of left quotients.