Density of group languages under ergodic probability measures

I will present recent work about densities of group languages under ergodic probability measures. Group languages are the rational languages recognized by morphisms onto finite groups, or equivalently recognized by automata where letters act as permutations of the states. We consider the density of such languages under probability measures defined on spaces of infinite words. In this setting, the density quantifies how frequently words from the language are found in infinite words sampled at random using the measure. I will explain how we can prove that group languages have well-defined densities with respect to any ergodic probability measure, and how the proof of this fact leads to equidistribution results in some important cases.

This is joint work with Valérie Berthé, Carl-Fredrik Nyberg-Brodda, Dominique Perrin and Karl Petersen