7 - Infinite words with uniform frequencies, and invariant measures

Summary

A fruitful use of word combinatorics is its contribution to the study of dynamical systems, through the symbolic dynamical systems defined in Section 1.6. Indeed, the study of most dynamical systems in the topological and the measure-theoretic categories can be reduced, by appropriate coding techniques, to the study of a suitable symbolic system X_x; and the topological properties of a symbolic system X_x (equipped with the product topology on A^ℕ) can be translated into combinatorial properties of the infinite word x.

In this chapter, we shall study the first two combinatorial properties of infinite words which are significant (and indeed, primordial) for symbolic dynamical systems. The first one is the well-known uniform recurrence which translates the dynamical property of minimality, that is the fact that the topological system cannot be split into smaller systems. The second one is the fact that the topological system has one invariant probability measure; this is called unique ergodicity, a somewhat unhappy expression as it suggests a close association with the classical (i.e., measure-theoretic) ergodic theory, though in fact it is a purely topological notion. Thus, for symbolic systems, unique ergodicity translates into the existence of frequencies for every finite factor of the infinite word x, and the limit defining these frequencies is a uniform one; thus we propose to say that the infinite word x has uniform frequencies. Similarly, the set of invariant measures depends only on the topological structure, and combinatorial properties of the word x will give informations on its structure.