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The structure of hyperfinite Borel equivalence relations. (English) Zbl 0803.28009
The paper studies equivalence relations induced by the orbits of a single Borel automorphism on a standard Borel space. They are called hyperfinite. It is shown that any two such equivalence relations which are not smooth (i.e., do not admit Borel selectors) are Borel embeddable into each other. This uses work of Effros and Weiss. The result is applied to derive a classification of hyperfinite Borel equivalence relations under two different notions of equivalence. It is shown that the cardinality of the set of invariant ergodic measures is a complete invariant for Borel isomorphisms of aperiodic nonsmooth such equivalence relations. A number of canonical examples is discussed.

MSC:
28D05 Measure-preserving transformations
03E15 Descriptive set theory
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