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Mean dimension and an embedding theorem for real flows

Volume 251 / 2020

Yonatan Gutman, Lei Jin 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.

Authors

  • Yonatan GutmanInstitute of Mathematics
    Polish Academy of Sciences
    Śniadeckich 8
    00-656 Warszawa, Poland
    e-mail
  • Lei JinCenter for Mathematical Modeling
    University of Chile
    and UMI 2807 - CNRS
    e-mail

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