Density of group languages in shift spaces

TL;DR

This paper investigates the density of group languages within shift spaces, providing formulas and conditions for their existence and computation, especially in minimal and ergodic cases, linking dynamics with algebraic properties.

Contribution

It introduces a method to compute the density of group languages in shift spaces using ergodic measures and characterizes minimal invariant subsets via cocycles and coboundaries.

Findings

Density exists for all group languages under ergodic measures.

A closed formula for density in minimal shifts is derived.

Conditions for ergodic minimal subsets in skew products are provided.

Abstract

The density of a rational language can be understood as the frequency of some "pattern" in the shift space, for example a pattern like "words with an even number of a given letter." We study the density of group languages, i.e. rational languages recognized by morphisms onto finite groups, inside shift spaces. We show that the density with respect to any given ergodic measure on a shift space exists for every group language, because it can be computed by using any ergodic lift of the given measure to some skew product between the shift space and the recognizing group. We then further study densities in shifts of finite type (with a suitable notion of irreducibility), and then in minimal shifts. In the latter case, we obtain a closed formula for the density under the condition that the skew product has minimal closed invariant subsets which are ergodic under the product of the original measure and the uniform probability measure on the group. The formula is derived in part from a characterization of minimal closed invariant subsets for skew products relying on notions of cocycles and coboundaries. In the case where the whole skew product is ergodic under the product measure, then the density is just the cardinality of the subset of the group which defines the language divided by the cardinality of the group. Moreover, we provide sufficient conditions for the skew product to have minimal closed invariant subsets that are ergodic under the product measure. Finally, we investigate the link between minimal closed invariant subsets, return words and bifix codes.