Mean dimension and an embedding theorem for real flows

Abstract

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 r 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 is an element of C(R, [-1,1]) : supp((f) over cap) subset of [-r , r]}, where (f) over cap 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.