# Null systems in the non-minimal case

## TL;DR

This paper explores properties of null dynamical systems, demonstrating conditions under which they are equicontinuous or mean equicontinuous, and constructs examples with specific proximal relations.

## Contribution

It establishes new links between nullness, distalness, and equicontinuity, and constructs examples illustrating these properties in dynamical systems.

## Key findings

- Null and distal systems are equicontinuous.
- Null systems with closed proximal relations are mean equicontinuous.
- Constructed examples with specific proximal relations.

## 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.