Applications of orbit equivalence to actions of discrete amenable groups

Details Applications of orbit equivalence to actions of discrete amenable groups Applications of orbit equivalence to actions of discrete amenable groups

2003-04-28

arxiv.org/abs/math/0304456v1

Proceedings of the ICM, Beijing 2002, vol. 3, 339--348

Mathematics

Dynamical Systems

Scientific paper

Since the work of Ornstein and Weiss in 1987 (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 developements have been built. First, a full generalization of the various equivalence theories, including Ornstein's isomorphism theorem itself, exists. Second, one can use the characterization of discrete amenable actions as those which are orbit equivalent to a action of \Z to lift theorems from actions of \Z to those of arbitrary amenable groups.

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