Null systems in the non-minimal case

Abstract

In this paper, it is shown that if a dynamical system is null and distal, then it is equicontinuous. It turns out that a null system with closed proximal relation is mean equicontinuous. As a direct application, it follows that a null dynamical system with dense minimal points is also mean equicontinuous. Meanwhile, a distal system with trivial $\text{Ind}_{fip}$-pairs, and a non-trivial regionally proximal relation of order $\infty$ is constructed.