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Simple noncommutative counterexamples in the joining theory. (English. Russian original) Zbl 1028.28015

Mosc. Univ. Math. Bull. 53, No. 4, 15-17 (1998); translation from Vestn. Mosk. Univ., Ser. I 1998, No. 4, 16-19 (1998).
Summary: Actions preserving a Haar measure and generated by automorphisms of a compact abelian group are considered and joinings of such systems are studied. In many problems of ergodic theory the study of the classes \(MSJ(n)\) of actions with minimal self-joinings of order \(n\) are of great interest. The authors present examples of noncommutative actions that belong to the classes \(MSJ(2)\setminus MSJ(3)\) and \(MSJ!(3)\setminus MSJ!(4)\), where \(MSJ!(n)\) is the class of actions having trivial joinings of order \(n\).

MSC:

28D05 Measure-preserving transformations
37A15 General groups of measure-preserving transformations and dynamical systems
28D15 General groups of measure-preserving transformations
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