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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

An asymptotic equipartition property for measures on model spaces


Author: Tim Austin
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
Published electronically: October 11, 2018
MathSciNet review: 3885183
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Abstract | References | Similar Articles | Additional Information

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

DOI: https://doi.org/10.1090/tran/7294
Keywords: Sofic entropy, measures on model spaces, asymptotic equipartition property, co-induced dynamical systems
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.
Article copyright: © Copyright 2018 American Mathematical Society