a1 Department of Mathematics, University of Toronto, Toronto, Canada
a2 Department of Mathematics, University of Maryland, College Park, Maryland, USA
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.
(Received January 08 1985)
(Revised December 04 1985)
(Revised October 13 1986)