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ADDITIVITY PROPERTIES OF SOFIC ENTROPY AND MEASURES ON MODEL SPACES

Part of: Ergodic theory Special processes Equilibrium statistical mechanics

Published online by Cambridge University Press:  31 August 2016

TIM AUSTIN
TIM AUSTIN*
Affiliation:
Courant Institute, New York University, New York, NY 10012, USA; tim@cims.nyu.edu

Abstract

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Sofic entropy is an invariant for probability-preserving actions of sofic groups. It was introduced a few years ago by Lewis Bowen, and shown to extend the classical Kolmogorov–Sinai entropy from the setting of amenable groups. Some parts of Kolmogorov–Sinai entropy theory generalize to sofic entropy, but in other respects this new invariant behaves less regularly. This paper explores conditions under which sofic entropy is additive for Cartesian products of systems. It is always subadditive, but the reverse inequality can fail. We define a new entropy notion in terms of probability distributions on the spaces of good models of an action. Using this, we prove a general lower bound for the sofic entropy of a Cartesian product in terms of separate quantities for the two factor systems involved. We also prove that this lower bound is optimal in a certain sense, and use it to derive some sufficient conditions for the strict additivity of sofic entropy itself. Various other properties of this new entropy notion are also developed.

MSC classification

Primary: 37A35: Entropy and other invariants, isomorphism, classification
Secondary: 37A50: Relations with probability theory and stochastic processes 60K35: Interacting random processes; statistical mechanics type models; percolation theory 82B20: Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs
Type
Research Article
Information
Forum of Mathematics, Sigma , Volume 4 , 2016 , e25
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author 2016

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