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Erschienen in:
2018 | OriginalPaper | Buchkapitel
Introduction
verfasst von : Thierry Giordano, David Kerr, N. Christopher Phillips, Andrew Toms
Erschienen in: Crossed Products of C*-Algebras, Topological Dynamics, and Classification
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In 1959, Dye [19] introduced the notion of orbit equivalence and proved that any two ergodic finite measure preserving transformations on a Lebesgue space are orbit equivalent. In [20], he had also conjectured that an arbitrary ergodic action of a discrete amenable group is orbit equivalent to a ℤ–action. This conjecture was proved in Ornstein–Weiss [73]. The most general case was proved in Connes– Feldman–Weiss [13] by establishing that an amenable nonsingular countable equivalence relation $$\mathcal{R}$$ can be generated by a single transformation, or equivalently, is hyperfinite, i.e., $$\mathcal{R}$$ is up to a null set, a countable increasing union of finite equivalence relations.