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ADDITIVITY PROPERTIES OF SOFIC ENTROPY AND MEASURES ON MODEL SPACES

Part of: Special processes Equilibrium statistical mechanics Ergodic theory

Published online by Cambridge University Press:  31 August 2016

TIM AUSTIN
TIM AUSTIN*
Affiliation:
Courant Institute, New York University, New York, NY 10012, USA; [email protected]

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

References

Bekka, B., de la Harpe, P. and Valette, A., Kazhdan’s Property (T), New Mathematical Monographs, 11 (Cambridge University Press, Cambridge, 2008).CrossRefGoogle Scholar
Benjamini, I. and Schramm, O., ‘Recurrence of distributional limits of finite planar graphs’, Electron. J. Probab. 6(23) (2001), 13 pp (electronic).Google Scholar
Bowen, L., ‘The ergodic theory of free group actions: entropy and the f-invariant’, Groups Geom. Dyn. 4(3) (2010), 419–432.CrossRefGoogle Scholar
Bowen, L., ‘Measure conjugacy invariants for actions of countable sofic groups’, J. Amer. Math. Soc. 23(1) (2010), 217–245.CrossRefGoogle Scholar
Bowen, L., ‘Non-abelian free group actions: Markov processes, the Abramov–Rohlin formula and Yuzvinskii’s formula’, Ergodic Theory Dynam. Systems 30(6) (2010), 1629–1663.CrossRefGoogle Scholar
Bowen, L., ‘Entropy for expansive algebraic actions of residually finite groups’, Ergodic Theory Dynam. Systems 31(3) (2011), 703–718.Google Scholar
Bowen, L., ‘Sofic entropy and amenable groups’, Ergodic Theory Dynam. Systems 32(2) (2012), 427–466.CrossRefGoogle Scholar
Bowen, L. P., ‘A measure-conjugacy invariant for free group actions’, Ann. of Math. (2) 171(2) (2010), 1387–1400.CrossRefGoogle Scholar
Dooley, A. H. and Zhang, G., ‘Co-induction in dynamical systems’, Ergodic Theory Dynam. Systems 32(3) (2012), 919–940.CrossRefGoogle Scholar
Goldie, C. M. and Pinch, R. G. E., Communication Theory, London Mathematical Society Student Texts, 20 (Cambridge University Press, Cambridge, 1991).Google Scholar
Gromov, M., ‘Endomorphisms of symbolic algebraic varieties’, J. Eur. Math. Soc. (JEMS) 1(2) (1999), 109–197.CrossRefGoogle Scholar
Hayes, B., ‘Polish models and sofic entropy’, J. Inst. Math. Jussieu. Available on CJO 2016 doi:10.1017/S1474748015000468.CrossRefGoogle Scholar
Kechris, A. S., Global Aspects of Ergodic Group Actions, Mathematical Surveys and Monographs, 160 (American Mathematical Society, Providence, RI, 2010).CrossRefGoogle Scholar
Kerr, D. and Li, H., ‘Bernoulli actions and infinite entropy’, Groups Geom. Dyn. 5(3) (2011), 663–672.CrossRefGoogle Scholar
Kerr, D. and Li, H., ‘Entropy and the variational principle for actions of sofic groups’, Invent. Math. 186(3) (2011), 501–558.Google Scholar
Kerr, D. and Li, H., ‘Soficity, amenability, and dynamical entropy’, Amer. J. Math. 135(3) (2013), 721–761.Google Scholar
Li, H., ‘Compact group automorphisms, addition formulas and Fuglede–Kadison determinants’, Ann. of Math. (2) 176(1) (2012), 303–347.CrossRefGoogle Scholar
Lubotzky, A., Discrete Groups, Expanding Graphs and Invariant Measures, Modern Birkhäuser Classics (Birkhäuser Verlag, Basel, 2010), with an appendix by Jonathan D. Rogawski, reprint of the 1994 edition.Google Scholar
Mézard, M. and Montanari, A., Information, Physics, and Computation, Oxford Graduate Texts (Oxford University Press, Oxford, 2009).Google Scholar
Montanari, A., Mossel, E. and Sly, A., ‘The weak limit of Ising models on locally tree-like graphs’, Probab. Theory Related Fields 152(1–2) (2012), 31–51.CrossRefGoogle Scholar
Mossel, E., Neeman, J. and Sly, A., ‘Reconstruction and estimation in the planted partition model’, Probab. Theory Related Fields 162(3–4) (2015), 431–461.CrossRefGoogle Scholar
Petersen, K. E., Ergodic Theory, (Cambridge University Press, Cambridge, 1983).Google Scholar
Seward, B., Krieger’s finite generator theorem for actions of countable groups. I. Preprint, arXiv:1405.3604.Google Scholar
Weiss, B., ‘Sofic groups and dynamical systems’, Sankhyā Ser. A 62(3) (2000), 350–359. Ergodic theory and harmonic analysis (Mumbai, 1999).Google Scholar