A classification of intrinsic ergodicity for recognisable random substitution systems

We study a class of dynamical systems generated by random substitutions, which contains both intrinsically ergodic systems and instances with several measures of maximal entropy. In this class, we show that the measures of maximal entropy are classified by invariance under an appropriate symmetry relation. All measures of maximal entropy are fully supported and they are generally not Gibbs measures. We prove that there is a unique measure of maximal entropy if and only if an associated Markov chain is ergodic in inverse time. This Markov chain has finitely many states and all transition matrices are explicitly computable. Thereby, we obtain several sufficient conditions for intrinsic ergodicity that are easy to verify. A practical way to compute the topological entropy in terms of inflation words is extended from previous work to a more general geometric setting.

💡 Research Summary

This paper investigates intrinsic ergodicity – the uniqueness of the measure of maximal entropy (MME) – for a broad class of symbolic dynamical systems generated by random substitutions. The authors focus on primitive random substitutions that satisfy two structural assumptions: geometric compatibility and recognisability. Geometric compatibility guarantees that every marginal (deterministic) substitution associated with the random rule shares a common left eigenvector and expansion factor λ, allowing the substitution to be interpreted as a geometric inflation rule. Recognisability ensures that each bi‑infinite configuration in the subshift X_θ can be uniquely decomposed into a concatenation of inflation words drawn from the iterates θⁿ(a), a property essential for defining a natural symmetry group.

The symmetry group, called the shuffle group, consists of local exchanges of words that belong to the same level‑n image θⁿ(a) for a fixed letter a. A probability measure invariant under all such exchanges is termed a uniformity measure. The first main result shows that, under the two structural assumptions, the set of MMEs on the suspension flow Y_θ coincides exactly with the set of uniformity measures. Consequently every MME has full topological support, but, unlike many classical subshifts, these measures need not satisfy a Gibbs property nor the specification property.

To decide whether the uniformity measure is unique, the authors construct a finite‑state Markov chain Q_θ = (Qⁿ)_{n∈ℕ}. The entries of Qⁿ are explicit functions of the combinatorial data #θⁿ(a) and the abelianisation vectors of the substitution. This chain describes the backward evolution of inflation words under the substitution. The second main theorem proves that the uniformity measure is unique if and only if the backward‑time Markov process is ergodic (i.e., irreducible and aperiodic). This condition can be checked with standard tools such as the Perron–Frobenius theorem or classical criteria for Markov chain recurrence. The paper supplies several easily verifiable sufficient conditions (e.g., primitive transition matrix, period one) and presents an explicit counterexample where the backward chain fails to be ergodic, leading to multiple MMEs for both X_θ and Y_θ.

A further contribution is the identification of the topological entropy of the suspension flow with the growth rate of the number of inflation words, called the inflation‑word entropy. The authors prove that  h_top(Y_θ) = lim_{n→∞} (1/n) log #θⁿ(a) for any letter a, without requiring recognisability. This extends earlier results for deterministic substitutions and for random substitutions of constant length, and provides a practical method for computing entropy simply by counting words.

The paper is organized as follows. Section 2 introduces the symbolic framework, random substitution definitions, geometric compatibility, recognisability, and the probabilistic structure (choice of probability vectors on each θ(a)). Section 3 states the main results. Section 4 establishes the equality between topological entropy and inflation‑word entropy, illustrating why the suspension flow Y_θ is the appropriate setting. Section 5 analyses the structural consequences of recognisability, leading to a description of the subshift’s language. Section 6 develops transfer operators that encode the action of θ on measures. Section 7 defines inverse‑limit measures (generalising frequency measures) and studies their uniqueness via the Markov chain Q_θ. Section 8 proves the classification of intrinsic ergodicity, presents the counterexample, and discusses the implications for the shuffle group. The final sections contain examples and remarks on possible extensions.

In summary, the authors achieve a complete classification of intrinsic ergodicity for recognisable, geometrically compatible random substitution systems by (i) characterising MMEs as uniformity measures invariant under the shuffle group, (ii) linking uniqueness to the ergodicity of an explicitly computable backward‑time Markov chain, and (iii) providing a simple word‑counting formula for topological entropy. These results bridge combinatorial substitution theory, probabilistic Markov dynamics, and thermodynamic formalism, and they open the way for systematic analysis of entropy and phase transitions in random tiling models, quasicrystals with defects, and other statistically self‑similar structures.