The characteristic factors in dynamical systems

It is an ideal situation if we can reduce a problem P stated for general ergodic or minimal systems to the same problem in their "simple factors". We will explain how one did so for the problem on the convergence of multiple ergodic averages in ergodic theory. Moreover, we will present a recent work by Glasner-Huang -Shao-Weiss-Ye on the similar problem in the topological setup.
On the way to do so, we will address the parallels between topological dynamics and ergodic theory, and their applications to combinatoric number theory.