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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

##### MSC:
 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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