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Density of group languages in shift spaces

Valérie Berthé, Herman Goulet-Ouellet, Carl-Fredrik Nyberg-Brodda, Dominique Perrin, Karl Petersen

TL;DR

The article develops a unified ergodic-theoretic framework to study the density of group languages in shift spaces. By translating densities into ergodic sums on skew products G ⋊_φ X, it proves existence of δ_μ(L) for all group languages and derives computable formulas, with a particularly clean version δ_μ(L) = |K|/|G| when ν×μ is ergodic on the skew product. The work then specializes to shifts of finite type and minimal shifts, introducing φ-irreducibility and modular cobounding maps to characterize ergodicity and minimality; it provides a density formula in terms of minimal cobounding maps and explores how return words, bifix codes, and coboundaries control the structure of minimal skew products. The results yield concrete equidistribution statements for classical systems (Fibonacci, Sturmian, dendric shifts) and show that, under natural irreducibility hypotheses, densities are often universal across group languages. These insights connect automata-theoretic language density with ergodic theory, offering practical tools for computing pattern frequencies in symbolic dynamical systems.

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.

Density of group languages in shift spaces

TL;DR

The article develops a unified ergodic-theoretic framework to study the density of group languages in shift spaces. By translating densities into ergodic sums on skew products G ⋊_φ X, it proves existence of δ_μ(L) for all group languages and derives computable formulas, with a particularly clean version δ_μ(L) = |K|/|G| when ν×μ is ergodic on the skew product. The work then specializes to shifts of finite type and minimal shifts, introducing φ-irreducibility and modular cobounding maps to characterize ergodicity and minimality; it provides a density formula in terms of minimal cobounding maps and explores how return words, bifix codes, and coboundaries control the structure of minimal skew products. The results yield concrete equidistribution statements for classical systems (Fibonacci, Sturmian, dendric shifts) and show that, under natural irreducibility hypotheses, densities are often universal across group languages. These insights connect automata-theoretic language density with ergodic theory, offering practical tools for computing pattern frequencies in symbolic dynamical systems.

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.
Paper Structure (25 sections, 48 theorems, 120 equations, 11 figures)

This paper contains 25 sections, 48 theorems, 120 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Symbolic dynamics
  3. Group languages and skew products
  4. First definitions
  5. A formula for the density
  6. An example in the Fibonacci shift
  7. The case of Bernoulli measures
  8. Densities in shifts of finite type
  9. Shifts of finite type and phi-irreducibility
  10. Markov measures
  11. Strong irreducibility
  12. A characterization of minimality via return words
  13. More on the theory of codes
  14. Proof of the characterization of minimality
  15. Relation with average length
  16. ...and 10 more sections

Key Result

Theorem A

Let $X$ be a shift space on a finite alphabet $A$ with an ergodic measure $\mu$ and let $\varphi\colon A^*\to G$ be a morphism onto a finite group $G$. For every group language $L=\varphi^{-1}(K)$, where $K\subseteq G$, the density $\delta_\mu(L)$ exists and is given by where $\overline\mu$ is any ergodic measure on the skew product $G \rtimes_{\varphi} X$ that projects to $\mu$.

Figures (11)

  • Figure 1: The invariant probability measure on the Thue-Morse shift.
  • Figure 2: The finite shift $X$ generated by $x=(abc)^\infty$ and its skew product with $\mathbb{Z}/2\mathbb{Z}$ from \ref{['eg:periodicskewprod-1']}.
  • Figure 3: The invariant probability measure on the Fibonacci shift ($\lambda =$ the golden ratio). Circled nodes represent elements from the language $L = \{w\in \{a,b\}^* \mid |w|_a \equiv 0 \mod 2\}$.
  • Figure 4: An automaton recognizing the language $L=\{aa,ab,b\}^*$ with the state $1$ being the initial state and the only final state.
  • Figure 5: The SFT and skew product from \ref{['eg:golden-mean-1']}. The arrows represent the two respective dynamics, namely the shift map and the skew product map.
  • ...and 6 more figures

Theorems & Definitions (122)

  • Theorem A
  • Theorem B
  • Theorem C
  • Example 2.1
  • Lemma 3.1
  • proof
  • Example 3.2
  • Theorem 3.3: \ref{['t:first-main']}
  • Lemma 3.4
  • proof
  • ...and 112 more