Distal actions and ergodic actions on compact groups

TL;DR

This paper investigates the relationship between distal and ergodic actions of automorphism groups on compact groups, establishing conditions under which local properties imply global ones and identifying when ergodic automorphisms exist.

Contribution

It proves that distal automorphisms imply the entire group action is distal under certain conditions and shows the existence of ergodic automorphisms in nilpotent groups acting on connected finite-dimensional compact abelian groups.

Findings

Distal automorphisms imply the entire group is distal under specific conditions.

Existence of ergodic automorphisms in nilpotent groups on certain compact groups.

Local to global correspondence for distal actions in particular group settings.

Abstract

Let $K$ be a compact metrizable group and $\Ga$ be a group of automorphisms of $K$. We first show that each $\ap \in \Ga$ is distal on $K$ implies $\Ga$ itself is distal on $K$, a local to global correspondence provided $\Ga$ is a generalized $\ov{FC}$-group or $K$ is a connected finite-dimensional group. We show that $\Ga$ contains an ergodic automorphism when $\Ga$ is nilpotent and ergodic on a connected finite-dimensional compact abelian group $K$.