Mean dimension and an embedding theorem for real flows
Volume 251 / 2020
Fundamenta Mathematicae 251 (2020), 161-181
MSC: 37B05; 54H20.
DOI: 10.4064/fm597-2-2020
Published online: 27 March 2020
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 $\{f\in C(\mathbb {R},[-1,1]) : \operatorname{supp} (\hat {f})\subset [-r,r]\}$, 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.
