I will present some ongoing work about densities of regular languages in minimal shift spaces. We will focus on the density of group languages, i.e. languages recognized by morphisms onto finite groups. Working within the skew product of the shift space and the recognizing group, a simple formula is derived for the density, which holds under the condition that the minimal components of the skew product are ergodic. In the process, we give a description of these minimal components and relate them with subgroups generated by return words. We also give some sufficient conditions under which the skew product is ergodic.

This is an ongoing collaboration with Valérie Berthé, Carl-Fredrik Nyberg-Brodda, Dominique Perrin and Karl Petersen.