An anti-classification theorem for ergodic measure preserving transformations
Finding ways to classify measure preserving transformations up to isomorphism is an important subject in ergodic theory. So far two classes have been classified, the Bernoulli shifts (with the entropy) and the transformations with discrete spectrum ( by countable subgroups of the torus). \textit{J. Feldman} [Proc. Am. Math. Soc. 46, 383--394 (1974; Zbl 0292.28008)] observed that the entropy does not classify K-automorphisms up to isomorphism. His observation was based on a result of \textit{D. S. Ornstein} and \textit{P. C. Shields} [Adv. Math. 10, 63--88 (1973; Zbl 0251.28004)]. Here, the authors show that the group of all invertible measure preservings of [0,1] is turbulent in the sense of \textit{G. Hjorth} [Mathematical Surveys and Monographs, 75, Providence, RI: American Mathematical Society (AMS) (2000; Zbl 0942.03056)]. A consequence of this is the following anti-classification result: it is not possible to associate a countable group with each member of the group of all invertible measure preserving transformations on [0,1] and to expect the isomorphim of the corresponding groups to characterize the isomorphism between the transformations.
scientific article; zbMATH DE number 2102877