Since the work of Ornstein and Weiss in 1987 (Entropy and isomorphism theorems for actions of amenable groups, J. Analyse Math., 48 (1987)) it has been understood that the natural category for classical ergodic theory would be probability measure preserving actions of discrete amenable groups. A conclusion of this work is that all such actions on nonatomic Lebesgue probability spaces were orbit equivalent. From this foundation two broad developments have been built. First, a full generalization of the various equivalence theories, including Ornstein's isomorphism theorem itself, exists. Fixing the amenable group G and an action of it, one can define a metric-like notion on the full-group of the action, called a size. A size breaks the orbit equivalence class of a single action into subsets, those reachable by a Cauchy sequence (in the size) of full group perturbations. These subsets are the equivalence classes associated with the size. Each size possesses a distinguished "most random" set of classes, the "Bernoulli" classes of the relation. An Ornstein-type theoreme can be obtained. Many naturally occurring equivalence relations can be described in this way. perhaps most interesting, entropy itself can be so described. Second, one can use the characterization of discrete amenable actions as those which are orbit equivalent to a action of Z to lift theoremes from actions of Z to those of arbitrary amenable groups. The most interesting of these are first, that actions of completely positive entropy (called K-systems for Z actions) are mixing of all orders (proven jointly with B. Wiess) and that such actions have countable Haar spectrum (proven by Golodets and Dooley). As all ergodic actions are orbit equivalent, only ergodicity is preserved by orbit equivalences in general, but by considering orbit equivalences restricted to be measurable with respect to a sub-σ algebra, many properties relative to that algebra are preserved. This provides the tool for this method to succeed.
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