Ergodic frequency measures for random substitutions

TL;DR

This paper develops ergodic measures for random substitution subshifts, ensuring the existence of word frequencies and identifying measures of maximal entropy for certain classes, advancing understanding of stochastic symbolic dynamics.

Contribution

It introduces a family of ergodic measures for RS-subshifts and demonstrates their role as frequency measures and measures of maximal entropy in specific cases.

Findings

Word frequencies exist for almost all elements under these measures.

Measures of maximal entropy are identified as frequency measures for certain random substitutions.

The construction applies to primitive random substitutions, broadening the scope of ergodic analysis.

Abstract

We construct a family of ergodic measures on random substitution subshifts (RS-subshifts) associated to a primitive random substitution. In particular, the word frequencies of every finite legal word exist for almost every element of the random substitution subshift with respect to these measures. As an application, we show that for a certain class of random substitutions the measures of maximal entropy are frequency measures.