On simplicity concepts for ergodic actions

Abstract

A weakly mixing measure preserving action of a locally compact second countable group on a standard probability space is called 2-fold near simple if every ergodic joining of it with itself is either product measure or is supported on a ‘convex combination’ of graphs. A similar definition can be given for near simplicity of higher order. This generalizes the Veech-del Junco-Rudolph notion of simplicity. Our main results include the following. An analogue of Veech theorem on factors holds for the 2-fold near simple actions. A weakly mixing group extension of an action with near MSJ is near simple. The action of a normal co-compact subgroup is near simple if and only if the whole action is near simple. The subset of all 2-fold near simple transformations (i.e., ℤ-actions) is meager in the group of measure preserving transformations endowed with the weak topology. Via the (C, F)-construction, we produce a near simple quasi-simple transformation which is disjoint from any simple map.