TY - UNPB

T1 - Measure theoretic entropy of random substitution subshifts

AU - Gohlke, Philipp

AU - Mitchell, Andrew

AU - Rust, Dan

AU - Samuel, Tony

PY - 2021/5/24

Y1 - 2021/5/24

N2 - Subshifts of deterministic substitutions are ubiquitous objects in dynamical systems. Two of their most appealing 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 substitution often have positive topological entropy, and support uncountably many ergodic measures, known as frequency measures. Here, we develop 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.

AB - Subshifts of deterministic substitutions are ubiquitous objects in dynamical systems. Two of their most appealing 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 substitution often have positive topological entropy, and support uncountably many ergodic measures, known as frequency measures. Here, we develop 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.

UR - https://arxiv.org/abs/2105.11224

M3 - Preprint

BT - Measure theoretic entropy of random substitution subshifts

ER -