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

TL;DR

This paper explores the computability of measures of maximal entropy on coded shift spaces, providing criteria and demonstrating computability for various classes, while also identifying cases where the measure is non-computable.

Contribution

It establishes a computability criterion for $ ext{G}$-Bernoulli measures and applies it to show the computability of MMEs on several classes of coded shifts, including $S$-gap, multiple-gap, and $eta$-shifts.

Findings

Unique MME is computable when concatenation entropy exceeds residual entropy and $ ext{Vere--Jones}$ parameter is computable.

MMEs on Dyck shifts are computable.

In cases where residual entropy exceeds concatenation entropy, the MME may not be computable even if it is unique and the parameter is computable.

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 $\mu_{\rm max}$ on $X$ is computable, provided the Vere--Jones parameter $\kappa$ 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 $\beta$-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 $\mu_{\rm max}$ is unique and the parameter $\kappa$ is computable, the measure $\mu_{\rm max}$ may still fail to be computable.