On the existence of ergodic automorphisms in ergodic ${\mathbb Z} ^d$-actions on compact groups

TL;DR

This paper investigates conditions under which ergodic actions of finitely generated groups of commuting automorphisms on compact groups contain ergodic automorphisms, focusing on the structure of the automorphism group's center.

Contribution

It establishes that ergodicity implies the existence of ergodic automorphisms when the center of the action has DCC, extending previous results by Berend and Schmidt.

Findings

Ergodicity of the group action implies the presence of ergodic automorphisms under certain conditions.

The condition on the center of the action having DCC is not restrictive and applies to various abelian groups.

The results generalize known theorems in the context of ergodic automorphisms on compact groups.

Abstract

Let $K$ be a compact metrizable group and $\Ga$ be a finitely generated group of commuting automorphisms of $K$. We show that ergodicity of $\Ga$ implies $\Ga$ contains ergodic automorphisms if center of the action, $Z(\Ga) = \{\ap \in {\rm Aut}(K) \mid \ap {\rm commutes with elements of \rm} \Ga \}$ has DCC. To explain that the condition on the center of the action is not restrictive, we discuss certain abelian groups which in particular, retrieves Theorems of Berend \cite{Be} and Schmidt \cite{Sc1} proved in this context.