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Inventiones mathematicae

, Volume 215, Issue 1, pp 265–310 | Cite as

Krieger’s finite generator theorem for actions of countable groups I

  • Brandon SewardEmail author
Article

Abstract

For an ergodic p.m.p. action \(G \curvearrowright (X, \mu )\) of a countable group G, we define the Rokhlin entropy \(h^{\mathrm {Rok}}_G(X, \mu )\) to be the infimum of the Shannon entropies of countable generating partitions. It is known that for free ergodic actions of amenable groups this notion coincides with classical Kolmogorov–Sinai entropy. It is thus natural to view Rokhlin entropy as a close analogue to classical entropy. Under this analogy we prove that Krieger’s finite generator theorem holds for all countably infinite groups. Specifically, if \(h^{\mathrm {Rok}}_G(X, \mu ) < \log (k)\) then there exists a generating partition consisting of k sets.

Mathematics Subject Classification

37A15 37A35 

Notes

Acknowledgements

This research was supported by the National Science Foundation Graduate Student Research Fellowship under Grant No. DGE 0718128. The author thanks his advisor, Ralf Spatzier, for numerous helpful discussions, Tim Austin for many suggestions to improve the paper, and Miklos Abért and Benjy Weiss for encouraging the author to coin a name for the new invariant studied here. Part of this work was completed while the author attended the Arbeitsgemeinschaft: Sofic Entropy workshop at the Mathematisches Forschungsinstitut Oberwolfach in Germany. The author thanks the MFO for their hospitality and travel support.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA

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