Combinatorics, Automata and Number Theory: Infinite words with uniform frequencies, and invariant measures

For infinite words, we study the properties of uniform recurrence, which translates the dynamical property of minimality, and of uniform frequencies, which corresponds to unique ergodicity; more generally, we look at the set of invariant measures of the associated dynamical system. We present some achievements of word combinatorics, initiated by M. Boshernitzan, which allow us to deduce information on these invariant measures from simple combinatorial properties of the words. Then we review some known examples of words with uniform frequencies, and give important examples which do not have uniform frequencies. We finish by hinting how these basic notions have given birth to very deep problems and high achievements in dynamical systems.

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