Joint ergodicity of actions of an Abelian group

Abstract

Let be a countable abelian group and let be measure preserving -actions on a probability space. We prove that joint ergodicity of implies total joint ergodicity if each is totally ergodic. We also show that if , and the actions commute, then total joint ergodicity of follows from joint ergodicity. This can be seen as a generalization of Berend’s result for commuting -actions.