Isomorphic extensions and applications

TL;DR

This paper characterizes isomorphic extensions between minimal topological dynamical systems, especially equicontinuous ones, in terms of mean equicontinuity, and demonstrates that such extensions need not be almost one-to-one.

Contribution

It provides a new characterization of isomorphic extensions using mean equicontinuity and answers open questions about their properties.

Findings

Isomorphic extensions can be characterized by mean equicontinuity.

An isomorphic extension need not be almost one-to-one.

The results apply to minimal and equicontinuous systems.

Abstract

If $\pi:(X,T)\to(Z,S)$ is a topological factor map between uniquely ergodic topological dynamical systems, then $(X,T)$ is called an isomorphic extension of $(Z,S)$ if $\pi$ is also a measure-theoretic isomorphism. We consider the case when the systems are minimal and we pay special attention to equicontinuous $(Z,S)$. We first establish a characterization of this type of isomorphic extensions in terms of mean equicontinuity, and then show that an isomorphic extension need not be almost one-to-one, answering questions of Li, Tu and Ye.