Ergodic Theory and Dynamical Systems

Research Article

On ergodic actions whose self-joinings are graphs

A. del Juncoa1 and D. Rudolpha2

a1 Department of Mathematics, University of Toronto, Toronto, Canada

a2 Department of Mathematics, University of Maryland, College Park, Maryland, USA

Abstract

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)