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A Classification of Intrinsic Ergodicity for Recognisable Random Substitution Systems.

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Measures of maximal entropy in random substitution dynamical systems are classified by symmetry. A unique measure exists if and only if an associated Markov chain is ergodic in inverse time, providing verifiable conditions for intrinsic ergodicity.

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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.