On the isomorphism problem for non-ergodic systems with discrete spectrum

TL;DR

This paper introduces a novel approach to the isomorphism problem for non-ergodic systems with discrete spectrum by linking ergodic theory and topological dynamics through topological models, leading to new characterizations.

Contribution

It solves the isomorphism problem for a class of topological systems and derives measure-preserving results from topological models, providing new insights into mean ergodicity.

Findings

Solved the isomorphism problem for certain topological dynamical systems.

Deduced measure-preserving case results from topological models.

Provided a new characterization of mean ergodicity.

Abstract

The article presents a new perspective on the isomorphism problem for non-ergodic measure-preserving dynamical systems with discrete spectrum which is based on the connection between ergodic theory and topological dynamics constituted by topological models. By first solving the isomorphism problem for a certain class of topological dynamical systems, it is shown that the measure-preserving case can in fact be deduced from the topological one via the construction of topological models. As a byproduct, a new characterization of mean ergodicity for topological dynamical systems is obtained.