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

Part of: Ergodic theory

Published online by Cambridge University Press:  30 October 2020

ANDREI ALPEEV  and
BRANDON SEWARD
ANDREI ALPEEV
Affiliation:
Chebyshev Lab at St. Petersburg State University, 14th Line 29B, Vasilyevsky Island, St. Petersburg199178, Russia (e-mail: a.alpeev@spbu.ru)
BRANDON SEWARD
Affiliation:
Courant Institute of Mathematical Sciences, New York University, 251 Mercer St, New York, NY 10012, USA (e-mail: b.m.seward@gmail.com)
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Abstract

We continue the study of Rokhlin entropy, an isomorphism invariant for probability-measure-preserving (p.m.p.) actions of countablegroups introduced in Part I [B. Seward. Krieger’s finite generator theorem for actions of countable groups I. Invent. Math. 215(1) (2019), 265–310]. In this paper we prove a non-ergodic finite generator theorem and use it to establish sub-additivity and semicontinuity properties of Rokhlin entropy. We also obtain formulas for Rokhlin entropy in terms of ergodic decompositions and inverse limits. Finally, we clarify the relationship between Rokhlin entropy, sofic entropy, and classical Kolmogorov–Sinai entropy. In particular, using Rokhlin entropy we give a new proof of the fact that ergodic actions with positive sofic entropy have finite stabilizers.

Keywords

entropygenerating partitiongeneratornon-ergodicnon-amenable

MSC classification

Primary: 37A35: Entropy and other invariants, isomorphism, classification
Secondary: 37A15: General groups of measure-preserving transformations
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 41 , Issue 10 , October 2021 , pp. 2881 - 2917
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Copyright
© The Author(s), 2020. Published by Cambridge University Press

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