Mean dimension and an embedding theorem for real flows

TL;DR

This paper develops mean dimension theory for real flows, establishing an embedding theorem that allows flows with mean dimension less than r to be embedded into a specific function space, extending previous results for discrete actions.

Contribution

It introduces mean dimension for real flows and proves an embedding theorem linking flows to a Fourier transform-based function space.

Findings

Established fundamental properties of mean dimension for real flows

Proved an embedding theorem for flows with mean dimension less than r

Connected embedding spaces to previous results for Z-actions

Abstract

We develop mean dimension theory for $\mathbb{R}$-flows. We obtain fundamental properties and examples and prove an embedding theorem: Any real flow $(X,\mathbb{R})$ of mean dimension strictly less than $r$ admits an extension $(Y,\mathbb{R})$ whose mean dimension is equal to that of $(X,\mathbb{R})$ and such that $(Y,\mathbb{R})$ can be embedded in the $\mathbb{R}$-shift on the compact function space $\{f\in C(\mathbb{R},[-1,1])|\;\mathrm{supp}(\hat{f})\subset [-r,r]\}$, where $\hat{f}$ is the Fourier transform of $f$ considered as a tempered distribution. These canonical embedding spaces appeared previously as a tool in embedding results for $\mathbb{Z}$-actions.