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Entropy and isomorphism theorems for actions of amenable groups. (English) Zbl 0637.28015
Ornstein’s isomorphism theorem states that two Bernoulli transformations are isomorphic if they have the same entropy. If \(G\) is a group, an action of \(G\) is a homomorphism from \(G\) into the group of invertible measure- preserving transformations on some probability space. Letting \(Z\) be the group of integers, every action of \(Z\) comes about from a measure- preserving transformation on the underlying probability space and its iterates. Defining an action of \(Z\) to be Bernoulli if the corresponding measure-preserving transformation is Bernoulli, we can restate the isomorphism theorem to say that any two Bernoulli actions of \(Z\) of the same entropy are isomorphic. The following natural question arose soon after the isomorphism theorem was proved. For what groups \(G\) are Bernoulli actions of \(G\) of the same entropy isomorphic (where the notion of Bernoulli action of \(G\) is defined appropriately)? The authors show that such a result holds for a certain type of amenable group \(G\) that would take us too long to explain here. Along the way, the authors develop a wealth of material concerning the entropy theory associated with actions of groups.
Reviewer: J.C.Kieffer

28D15 General groups of measure-preserving transformations
28D20 Entropy and other invariants
28A05 Classes of sets (Borel fields, \(\sigma\)-rings, etc.), measurable sets, Suslin sets, analytic sets
37A15 General groups of measure-preserving transformations and dynamical systems
37A05 Dynamical aspects of measure-preserving transformations
37A20 Algebraic ergodic theory, cocycles, orbit equivalence, ergodic equivalence relations
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