An asymptotic equipartition property for measures on model spaces
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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 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.
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Additional Information
- Tim Austin
- Affiliation: Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, New York 10012
- Address at time of publication: Department of Mathematics, University of California, Los Angeles, Los Angeles, California 90095
- Email: tim@math.ucla.edu
- Received by editor(s): May 5, 2017
- Received by editor(s) in revised form: May 30, 2017
- Published electronically: October 11, 2018
- Additional Notes: This research was supported partly by the Simons Collaboration on Algorithms and Geometry.
- © Copyright 2018 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 371 (2019), 1379-1402
- MSC (2010): Primary 37A35; Secondary 37A50, 28D15, 28D20, 94A17
- DOI: https://doi.org/10.1090/tran/7294
- MathSciNet review: 3885183