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Computability of $\mathcal{G}$-Beroulli Measures and Measures of Maximal Entropy on Coded Shift Spaces

Tamara Kucherenko, Marco López, Christian Wolf

TL;DR

This work studies the computability of $\mathcal{G}$-Bernoulli measures and measures of maximal entropy on coded shifts, focusing on when the unique MME is computable from oracles for the generating set and language. It develops a computability framework linking entropy data to computability via the Vere-Jones parameter $\kappa$, proving that the unique MME $\mu_{max}$ is computable when $h_{con}(X)>h_{res}(X)$ and $\kappa$ is $\mathcal{G}$-computable, with explicit criteria such as $h_{top}(X)>r(\mathcal{G})$ guaranteeing $\kappa$ computability. The paper applies these results to prominent classes (e.g., $\beta$-shifts, $S$-gap and generalized gap shifts, and the Dyck shift), establishing computability of their MMEs under oracle access to $\mathcal{G}$ and, when available, $\mathcal{L}(X)$; it also shows that $\kappa$ need not be computable in general and that even a unique MME can be noncomputable in certain constructions. A key contribution is showing computability of both the measure and the entropy in several natural coded-shift families, alongside precise limitations that delineate the boundary between computable and noncomputable MMEs in this setting.

Abstract

In this paper, we investigate the computability of $\mathcal{G}$-Bernoulli measures, with a particular focus on measures of maximal entropy (MMEs) on coded shift spaces. Coded shifts are natural generalizations of sofic shifts and are defined as the closure of all bi-infinite concatenations of words (generators) drawn from a countable generating set $\mathcal{G}$. We begin by establishing a computability criterion for $\mathcal{G}$-Bernoulli measures which are invariant measures given by assigning probability weights to the generators. We then apply this criterion to the setting in which the concatenation entropy exceeds the residual entropy, showing that in this case the unique measure of maximal entropy $μ_{\rm max}$ on $X$ is computable, provided the Vere--Jones parameter $κ$ of $\mathcal{G}$ is computable, based on having oracle access to the generators and the language of $X$. As a consequence, the unique MME is computable for several well-known classes of shift spaces, including $S$-gap shifts, multiple-gap shifts, and $β$-shifts. Moreover, the two ergodic MMEs of the Dyck shift are also computable. Finally, we examine the opposite situation, where the residual entropy exceeds the concatenation entropy and the MME is known to be non-unique in general. We show that even when $μ_{\rm max}$ is unique and the parameter $κ$ is computable, the measure $μ_{\rm max}$ may still fail to be computable.

Computability of $\mathcal{G}$-Beroulli Measures and Measures of Maximal Entropy on Coded Shift Spaces

TL;DR

This work studies the computability of -Bernoulli measures and measures of maximal entropy on coded shifts, focusing on when the unique MME is computable from oracles for the generating set and language. It develops a computability framework linking entropy data to computability via the Vere-Jones parameter , proving that the unique MME is computable when and is -computable, with explicit criteria such as guaranteeing computability. The paper applies these results to prominent classes (e.g., -shifts, -gap and generalized gap shifts, and the Dyck shift), establishing computability of their MMEs under oracle access to and, when available, ; it also shows that need not be computable in general and that even a unique MME can be noncomputable in certain constructions. A key contribution is showing computability of both the measure and the entropy in several natural coded-shift families, alongside precise limitations that delineate the boundary between computable and noncomputable MMEs in this setting.

Abstract

In this paper, we investigate the computability of -Bernoulli measures, with a particular focus on measures of maximal entropy (MMEs) on coded shift spaces. Coded shifts are natural generalizations of sofic shifts and are defined as the closure of all bi-infinite concatenations of words (generators) drawn from a countable generating set . We begin by establishing a computability criterion for -Bernoulli measures which are invariant measures given by assigning probability weights to the generators. We then apply this criterion to the setting in which the concatenation entropy exceeds the residual entropy, showing that in this case the unique measure of maximal entropy on is computable, provided the Vere--Jones parameter of is computable, based on having oracle access to the generators and the language of . As a consequence, the unique MME is computable for several well-known classes of shift spaces, including -gap shifts, multiple-gap shifts, and -shifts. Moreover, the two ergodic MMEs of the Dyck shift are also computable. Finally, we examine the opposite situation, where the residual entropy exceeds the concatenation entropy and the MME is known to be non-unique in general. We show that even when is unique and the parameter is computable, the measure may still fail to be computable.
Paper Structure (19 sections, 17 theorems, 76 equations)

This paper contains 19 sections, 17 theorems, 76 equations.

Table of Contents

  1. Introduction
  2. Entropy and Measures of Maximal Entropy
  3. Coded shifts
  4. Computability
  5. Statement of the results
  6. Background and Definitions
  7. Basics from computability theory
  8. Coded shifts
  9. Computability from oracles of $\mathcal{L}(X)$ and $\mathcal{G}$
  10. Proofs of Computability results for $\mathcal{G}$-Bernoulli measures and the MME
  11. Proofs of Theorem \ref{['thmmain']} and Corollary \ref{['cormain11']}
  12. Proof of Theorem \ref{['thmmain2']}
  13. Proof of Theorem \ref{['thmmain3']}
  14. Proof of Theorem \ref{['thmnoncomput']}
  15. Computability of the Vere-Jones parameter
  16. ...and 4 more sections

Key Result

Theorem A

Let $X = X(\mathcal{G})$ be a coded shift with generating set $\mathcal{G}$ such that $\mathcal{G}$ uniquely represents $X_{\rm con}(\mathcal{G})$. Let $\mu$ be a $\mathcal{G}$-Bernoulli measure given by positive real numbers $p_g$ with normalizing constant $c<\infty$, see equation measure. Suppose

Theorems & Definitions (41)

  • Theorem A
  • Corollary 1.1
  • Theorem B
  • Theorem C
  • Corollary 1.2
  • Corollary 1.3
  • Corollary 1.4
  • Theorem D
  • Definition 2.1
  • Definition 2.2
  • ...and 31 more