The Structure of Hyperfinite Borel Equivalence Relations

We study the structure of the equivalence relations induced by the orbits of a single Borel automorphism on a standard Borel space. We show that any two such equivalence relations which are not smooth, i.e., do not admit Borel selectors, are Borel embeddable into each other. (This utilizes among oth...

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Bibliographic Details
Published in:Transactions of the American Mathematical Society 1994-01, Vol.341 (1), p.193-225
Main Authors: Dougherty, R., Jackson, S., Kechris, A. S.
Format: Article
Language:English
Subjects:
Ales
Automorphisms
Borel sets
Cardinality
Compressibility
Equivalence relation
Ergodic theory
Mathematical relations
Mathematical theorems
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Summary:We study the structure of the equivalence relations induced by the orbits of a single Borel automorphism on a standard Borel space. We show that any two such equivalence relations which are not smooth, i.e., do not admit Borel selectors, are Borel embeddable into each other. (This utilizes among other things work of Effros and Weiss.) Using this and also results of Dye, Varadarajan, and recent work of Nadkarni, we show that the cardinality of the set of ergodic invariant measures is a complete invariant for Borel isomorphism of aperiodic nonsmooth such equivalence relations. In particular, since the only possible such cardinalities are the finite ones, countable infinity, and the cardinality of the continuum, there are exactly countably infinitely many isomorphism types. Canonical examples of each type are also discussed.
ISSN:0002-9947
1088-6850
DOI:10.1090/S0002-9947-1994-1149121-0