Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A classification of intrinsic ergodicity for recognisable random substitution systems

Philipp Gohlke, Andrew Mitchell

Abstract

We study a class of dynamical systems generated by random substitutions, which contains both intrinsically ergodic systems and instances with several measures of maximal entropy. In this class, we show that the measures of maximal entropy are classified by invariance under an appropriate symmetry relation. All measures of maximal entropy are fully supported and they are generally not Gibbs measures. We prove that there is a unique measure of maximal entropy if and only if an associated Markov chain is ergodic in inverse time. This Markov chain has finitely many states and all transition matrices are explicitly computable. Thereby, we obtain several sufficient conditions for intrinsic ergodicity that are easy to verify. A practical way to compute the topological entropy in terms of inflation words is extended from previous work to a more general geometric setting.

A classification of intrinsic ergodicity for recognisable random substitution systems

Abstract

We study a class of dynamical systems generated by random substitutions, which contains both intrinsically ergodic systems and instances with several measures of maximal entropy. In this class, we show that the measures of maximal entropy are classified by invariance under an appropriate symmetry relation. All measures of maximal entropy are fully supported and they are generally not Gibbs measures. We prove that there is a unique measure of maximal entropy if and only if an associated Markov chain is ergodic in inverse time. This Markov chain has finitely many states and all transition matrices are explicitly computable. Thereby, we obtain several sufficient conditions for intrinsic ergodicity that are easy to verify. A practical way to compute the topological entropy in terms of inflation words is extended from previous work to a more general geometric setting.

Paper Structure

This paper contains 24 sections, 38 theorems, 187 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Symbolic notation
  4. Random substitutions
  5. Probabilistic aspects
  6. Measures along (random) words
  7. Geometric hull
  8. Shuffle group and uniformity measures
  9. The geometric substitution matrix
  10. Induced transformation
  11. Ergodicity of (inverse-time) Markov chains
  12. Conditional entropy
  13. Main results
  14. Topological entropy of the geometric hull
  15. Geometric inflation word entropy
  16. ...and 9 more sections

Key Result

Lemma 2.9

A random substitution $\vartheta$ is primitive if and only if for some (equivalently all) non-degenerate $\mathbf{P}$ the matrix $M(\mathbf{P})$ is primitive and its PF eigenvalue satisfies $\lambda > 1$.

Figures (2)

  • Figure 1: Implication diagram for some conditions on primitive random substitutions.
  • Figure 2: $\vartheta \colon a \mapsto \{ abb\}, \, b \mapsto \{a,bb\}$, geometrically compatible with $\lambda = 2$ and $L = (2,1)$.

Theorems & Definitions (101)

  • Definition 2.1
  • Example 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Remark 2.6
  • Definition 2.7
  • Definition 2.8
  • Lemma 2.9
  • Definition 2.10
  • ...and 91 more