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On ergodic actions whose self-joinings are graphs

Published online by Cambridge University Press:  19 September 2008

A. del Junco  and
D. Rudolph
A. del Junco
Affiliation:
Department of Mathematics, University of Toronto, Toronto, Canada
D. Rudolph
Affiliation:
Department of Mathematics, University of Maryland, College Park, Maryland, USA
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Abstract

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We call an ergodic measure-preserving action of a locally compact group G on a probability space simple if every ergodic joining of it to itself is either product measure or is supported on a graph, and a similar condition holds for multiple self-joinings. This generalizes Rudolph's notion of minimal self-joinings and Veech's property S.

Main results The joinings of a simple action with an arbitrary ergodic action can be explicitly descnbed. A weakly mixing group extension of an action with minimal self-joinings is simple. The action of a closed, normal, co-compact subgroup in a weakly-mixing simple action is again simple. Some corollaries. Two simple actions with no common factors are disjoint. The time-one map of a weakly mixing flow with minimal self-joinings is prime Distinct positive times in a -action with minimal self-joinings are disjoint.

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 7 , Issue 4 , December 1987 , pp. 531 - 557
Copyright
Copyright © Cambridge University Press 1987

References

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