Joint work with Lei Jin.
We develop mean dimension theory for $R$-flows. We obtain fundamental properties and examples and prove an embedding theorem:
Any real flow $(X,R)$ of mean dimension strictly less than $a$ admits an extension $(Y,R)$ whose mean dimension is equal to that of $(X,R)$ and such that $(Y,R)$ can be embedded in the $R$-shift on the compact function space $\{f:R\to [-1,1] | {\rm supp}(\hat{f})\subset [-a,a]\}$, 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 Z-actions.