A classification of intrinsic ergodicity for recognisable random substitution systems

TL;DR

This paper classifies measures of maximal entropy in random substitution dynamical systems, providing criteria for intrinsic ergodicity and methods to compute topological entropy.

Contribution

It introduces a classification scheme for measures of maximal entropy based on symmetry invariance and links intrinsic ergodicity to the ergodicity of an associated Markov chain.

Findings

Measures of maximal entropy are classified by symmetry invariance.

Unique measure of maximal entropy corresponds to ergodic Markov chain.

Provides explicit conditions and methods for computing topological entropy.

Abstract

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.