# An asymptotic equipartition property for measures on model spaces

## TL;DR

This paper establishes that sequences with the asymptotic equipartition property suffice for analyzing sofic entropy in $G$-systems, extending Shannon--McMillan type results to the sofic group setting and providing new entropy formulas.

## Contribution

It proves that sequences with the asymptotic equipartition property can be used to study sofic entropy, generalizing classical information theory results to the sofic group context.

## Key findings

- Sequences with the asymptotic equipartition property are sufficient for sofic entropy analysis.
- A new formula for the sofic entropy of co-induced systems is derived.
- The result extends Shannon--McMillan theorem to the setting of sofic groups.

## Abstract

Let $G$ be a sofic group, and let $\Sigma = (\sigma_n)_{n\geq 1}$ be a sofic approximation to it. For a probability-preserving $G$-system, a variant of the sofic entropy relative to $\Sigma$ has recently been defined in terms of sequences of measures on its model spaces that `converge' to the system in a certain sense. Here we prove that, in order to study this notion, one may restrict attention to those sequences that have the asymptotic equipartition property. This may be seen as a relative of the Shannon--McMillan theorem in the sofic setting. We also give some first applications of this result, including a new formula for the sofic entropy of a $(G\times H)$-system obtained by co-induction from a $G$-system, where $H$ is any other infinite sofic group.