Measure Theoretic Entropy of Random Substitution Subshifts

• Published in: Annales Henri Poincaré Vol. 24; no. 1; pp. 277 - 323
• Main Authors: Gohlke, P., Mitchell, A., Rust, D., Samuel, T.
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 2023; Springer Nature B.V
• Subjects: Classical and Quantum Gravitation; Dynamical Systems and Ergodic Theory; Elementary Particles; Entropy; Ergodic processes; Markov analysis; Markov processes; Mathematical and Computational Physics; Mathematical Methods in Physics; Original Paper; Physics; Physics and Astronomy; Quantum Field Theory; Quantum Physics; Relativity Theory; Specifications; Substitutes; Theoretical; Topology; 37A50; 37A25; 37B10; 52C23
• ISSN: 1424-0637; 1424-0661
• DOI: 10.1007/s00023-022-01212-x

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.