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Mean dimension and a sharp embedding theorem: extensions of aperiodic subshifts

Published online by Cambridge University Press:  25 June 2013

YONATAN GUTMAN  and
MASAKI TSUKAMOTO
YONATAN GUTMAN
Affiliation:
Institute of Mathematics, Polish Academy of Sciences, ul. Śniadeckich 8, 00-956 Warszawa, Poland email [email protected]
MASAKI TSUKAMOTO
Affiliation:
Department of Mathematics, Kyoto University, Kyoto 606-8502, Japan email [email protected]
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Abstract

We show that if $(X, T)$ is an extension of an aperiodic subshift (a subsystem of $(\mathop{\{ 1, 2, \ldots , l\} }\nolimits ^{ \mathbb{Z} } , \mathrm{shift} )$ for some $l\in \mathbb{N} $) and has mean dimension $\mathrm{mdim} (X, T)\lt (D/ 2), D\in \mathbb{N} $, then it can be equivariantly embedded in $(\mathop{(\mathop{[0, 1] }\nolimits ^{D} )}\nolimits ^{ \mathbb{Z} } , \mathrm{shift} )$. The result is sharp. If $(X, T)$ is an extension of an aperiodic zero-dimensional system then it can be equivariantly embedded in $(\mathop{(\mathop{[0, 1] }\nolimits ^{D+ 1} )}\nolimits ^{ \mathbb{Z} } , \mathrm{shift} )$.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 34 , Issue 6 , December 2014 , pp. 1888 - 1896
Copyright
© Cambridge University Press, 2013 

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