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Sofic entropy and amenable groups

  • LEWIS BOWEN (a1)

Abstract

In previous work, the author introduced a measure-conjugacy invariant for sofic group actions called sofic entropy. Here, it is proven that the sofic entropy of an amenable group action equals its classical entropy. The proof uses a new measure-conjugacy invariant called upper-sofic entropy and a theorem of Rudolph and Weiss for the entropy of orbit-equivalent actions relative to the orbit change σ-algebra.

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[Bo10a]Bowen, L.. Measure conjugacy invariants for actions of countable sofic groups. J. Amer. Math. Soc. 23 (2010), 217245.
[Bo10b]Bowen, L.. Entropy for expansive algebraic actions of residually finite groups. Ergod. Th. & Dynam. Sys. to appear, arXiv:0909.4770.
[CFW81]Connes, A., Feldman, J. and Weiss, B.. An amenable equivalence relation is generated by a single transformation. Ergod. Th. & Dynam. Sys. 1(4) (1982), 431450.
[Dy59]Dye, H. A.. On groups of measure preserving transformation. I. Amer. J. Math. 81 (1959), 119159.
[Dy63]Dye, H. A.. On groups of measure preserving transformations. II. Amer. J. Math. 85 (1963), 551576.
[ES10]Elek, G. and Szabó, E.. Sofic representations of amenable groups. Preprint, arXiv:1010.3424.
[Gl03]Glasner, E.. Ergodic Theory via Joinings (Mathematical Surveys and Monographs, 101). American Mathematical Society, Providence, RI, 2003.
[Gro99]Gromov, M.. Endomorphisms of symbolic algebraic varieties. J. Eur. Math. Soc. 1(2) (1999), 109197.
[KL1]Kerr, D. and Li, H.. Entropy and the variational principle for actions of sofic groups. Preprint, arXiv:1005.0399.
[KL2]Kerr, D. and Li, H.. Soficity, amenability and dynamical entropy. Preprint, arXiv:1008.1429.
[Ma40]Mal’cev, A. I.. On faithful representations of infinite groups of matrices. Mat. Sb. 8 (1940), 405422, Amer. Math. Soc. Transl. (2) 45 (1965), 1–18.
[Ol85]Moulin Ollagnier, J.. Ergodic Theory and Statistical Mechanics (Lecture Notes in Mathematics, 1115). Springer, Berlin, 1985.
[OW80]Ornstein, D. and Weiss, B.. Ergodic theory of amenable group actions. I. The Rohlin lemma. Bull. Amer. Math. Soc. (N.S.) 2(1) (1980), 161164.
[OW87]Ornstein, D. and Weiss, B.. Entropy and isomorphism theorems for actions of amenable groups. J. Anal. Math. 48 (1987), 1141.
[Pe08]Pestov, V.. Hyperlinear and sofic groups: a brief guide. Bull. Symbolic Logic 14(4) (2008), 449480.
[Ro88]Rosenthal, A.. Finite uniform generators for ergodic, finite entropy, free actions of amenable groups. Probab. Theory Related Fields 77(2) (1988), 147166.
[RW00]Rudolph, D. J. and Weiss, B.. Entropy and mixing for amenable group actions. Ann. of Math. (2) 151(3) (2000), 11191150.
[We00]Weiss, B.. Sofic groups and dynamical systems. Ergodic Theory and Harmonic Analysis, Mumbai, 1999. Sankhyā Ser. A 62(3) (2000), 350–359.
[WZ92]Ward, T. and Zhang, Q.. The Abramov–Rohlin entropy addition formula for amenable group actions. Monatsh. Math. 114(3–4) (1992), 317329.

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Sofic entropy and amenable groups

  • LEWIS BOWEN (a1)

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