An asymptotic equipartition property for measures on model spaces

Author
Austin, Tim
Year 2017
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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 in the sofic setting of the Shannon--McMillan theorem. 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.

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Details

Title
An asymptotic equipartition property for measures on model spaces
Author
Austin, Tim
Year
2017
Type
Research Article
Language
eng
Comment
30 pages
History
2017-01-30 00:00:00
Categories
Probability · Dynamical Systems · Information Theory
This is Version 1 of this record. We added this version on February 1, 2017. This version is based on an original data import from arXiv.org e-Print archive.