Approximately transitive dynamical systems and simple spectrum

TL;DR

This paper constructs examples of measure-preserving actions of certain Abelian groups that have simple spectral properties but lack approximate transitivity, challenging assumptions about their relationship.

Contribution

It provides explicit examples of actions with simple spectrum without approximate transitivity, expanding understanding of spectral properties in dynamical systems.

Findings

Existence of actions with simple spectrum and no approximate transitivity

Counterexamples to the presumed link between simplicity of spectrum and transitivity

Insights into the spectral theory of Abelian group actions

Abstract

For some countable discrete torsion Abelian groups we give examples of their finite measure-preserving actions which have simple spectrum and no approximate transitivity property.