An anti-classification theorem for ergodic measure preserving transformations
type: Item
instance of: scholarly article
title: An anti-classification theorem for ergodic measure preserving transformations
zbMATH Open document ID: 1063.37004
DOI: 10.4171/JEMS/10
author: Matthew Foreman
author: Benjamin Weiss
published in: Journal of the European Mathematical Society (JEMS)
publication date: 2004-09-24T00:00:00Z
review text: 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.
reviewed by: Idris Assani
Mathematics Subject Classification ID: 37A35
Mathematics Subject Classification ID: 28D05
Mathematics Subject Classification ID: 37A05
zbMATH DE Number: 2102877
zbMATH Keywords: anti-classification
zbMATH Keywords: entropy
zbMATH Keywords: isomorphism
MaRDI profile type: Publication
cites work: Borel Structures and Invariants for Measurable Transformations
cites work: Q4511096
cites work: Q4379361
cites work: Q3274377
full work available at URL: https://doi.org/10.4171/jems/10
OpenAlex ID: W2074909313
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description: scientific article; zbMATH DE number 2102877