On discrete semi-flows with universally measurable Ellis semigroup

Abstract

We show that, in the framework of ZFC axioms, any continuous compact discrete semi-flow whose Ellis semigroup consists of universally measurable transformations is tame, and the union of its minimal sets is dense in the minimal attraction center. At the same time, under ZFC, the Ellis semigroups of a broad class of discrete dynamical systems contain transformations that are not universally measurable.