An asymptotic equipartition property for measures on model spaces

Abstract

Let $G$ be a sofic group, and let $\Sigma =({\sigma }_n{)}_{n\ge 1}$ 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.

30 pages. [v2:] Slight editing, no substantial changes [v3 & v4:] Some small corrections following referee's report, no substantial changes