We study density of rational languages under shift invariant probability measures on spaces of two-sided infinite words, which generalizes the classical notion of density studied in formal languages and automata theory. The density for a language is defined as the limit in average (if it exists) of the probability that a word of a given length belongs to the language. We establish the existence of densities for all rational languages under all shift invariant measures. We also give explicit formulas under certain conditions, in particular when the language is aperiodic. Our approach combines tools and ideas from semigroup theory and ergodic theory.

Inspired by a series of papers initiated in 2015 by Berthé et al., we introduce a new condition called suffix-connectedness. We show that the groups generated by the return sets of a uniformly recurrent suffix-connected language lie in a single conjugacy class of subgroups of the free group. Moreover, the rank of the subgroups in this conjugacy class only depends on the number of connected components in the extension graph of the empty word. We also show how to explicitly compute a representative of this conjugacy class using the first order Rauzy graph. Finally, we provide an example of suffix-connected, uniformly recurrent language that contains infinitely many disconnected words.