Metric mean dimension of flows

TL;DR

This paper develops a metric mean dimension theory for continuous flows, providing variational principles to measure their complexity, especially for flows with infinite topological entropy.

Contribution

It introduces the concept of metric mean dimension for flows and establishes variational principles relating it to various entropy measures, including for uniformly Lipschitz flows.

Findings

Established variational principles for metric mean dimension of flows.

Connected metric mean dimension with local and Kolmogorov-Sinai entropy.

Extended the theory to uniformly Lipschitz flows.

Abstract

The present paper aims to investigate the metric mean dimension theory of continuous flows. We introduce the notion of metric mean dimension for continuous flows to characterize the complexity of flows with infinite topological entropy. For continuous flows, we establish variational principles for metric mean dimension in terms of local $\epsilon$-entropy function and Brin-Katok $\epsilon$-entropy; For a class of special flow, called uniformly Lipschitz flow, we establish variational principles for metric mean dimension in terms of Kolmogorov-Sinai $\epsilon$-entropy, Brin-Katok's $\epsilon$-entropy and Katok's $\epsilon$-entropy.