A Classification of Intrinsic Ergodicity for Recognisable Random Substitution Systems

Area of Science:

• Dynamical Systems and Ergodic Theory
• Stochastic Processes
• Information Theory

Background:

• Random substitution dynamical systems present complex behavior, including intrinsic ergodicity and multiple measures of maximal entropy.
• Understanding the properties and classification of these measures is crucial for characterizing the system's long-term behavior.

Purpose of the Study:

• To classify measures of maximal entropy in random substitution dynamical systems.
• To establish criteria for the existence of a unique measure of maximal entropy.
• To develop practical methods for verifying intrinsic ergodicity and computing topological entropy.

Main Methods:

• Analysis of symmetry properties of measures of maximal entropy.
• Investigation of the ergodicity of associated inverse-time Markov chains with computable transition matrices.
• Extension of methods for computing topological entropy using inflation words in a generalized geometric context.

Main Results:

• Measures of maximal entropy are classified by invariance under specific symmetry relations.
• All measures of maximal entropy are fully supported and typically not Gibbs measures.
• A unique measure of maximal entropy exists if and only if the associated Markov chain is ergodic in inverse time.
• Several easily verifiable sufficient conditions for intrinsic ergodicity are derived.
• A method for computing topological entropy is generalized to a broader geometric setting.

Conclusions:

• Symmetry plays a key role in classifying measures of maximal entropy in these systems.
• The ergodicity of the inverse-time Markov chain provides a definitive test for the uniqueness of the maximal entropy measure.
• The study offers practical tools for analyzing random substitution dynamical systems and their entropic properties.