Measure theoretic entropy of random substitution subshifts

TL;DR

This paper develops new methods to compute the entropy of frequency measures in random substitution subshifts, revealing cases with unique maximal entropy measures and expanding understanding of their ergodic properties.

Contribution

It introduces novel techniques for entropy calculation in random substitution subshifts and establishes conditions for unique maximal entropy measures.

Findings

Closed form formulas for entropy of frequency measures.

Existence of frequency measures of maximal entropy in many cases.

Identification of a new class of intrinsically ergodic subshifts without Bowen's property.

Abstract

Subshifts of deterministic substitutions are ubiquitous objects in dynamical systems and aperiodic order (the mathematical theory of quasicrystals). Two of their most striking features are that they have low complexity (zero topological entropy) and are uniquely ergodic. Random substitutions are a generalisation of deterministic substitutions where the substituted image of a letter is determined by a Markov process. In stark contrast to their deterministic counterparts, subshifts of random substitutions often have positive topological entropy, and support uncountably many ergodic measures. The underlying Markov process singles out one of the ergodic measures, called the frequency measure. Here, we develop new techniques for computing and studying the entropy of these frequency measures. As an application of our results, we obtain closed form formulas for the entropy of frequency measures for a wide range of random substitution subshifts and show that in many cases there exists a frequency measure of maximal entropy. Further, for a class of random substitution subshifts, we prove that this measure is the unique measure of maximal entropy. These subshifts do not satisfy Bowen's specification property or the weaker specification property of Climenhaga and Thompson and hence provide an interesting new class of intrinsically ergodic subshifts.