Mean dimension and an embedding problem: an example
type: Item
instance of: scholarly article
title: Mean dimension and an embedding problem: an example
zbMATH Open document ID: 1301.37011
author: Elon Lindenstrauss
author: Masaki Tsukamoto
published in: Israel Journal of Mathematics
publication date: 2014-06-25T00:00:00Z
review text: The theory of mean dimension was introduced by \textit{M. Gromov} [Math. Phys. Anal. Geom. 2, No. 4, 323--415 (1999; Zbl 1160.37322)] and developed by \textit{E. Lindenstrauss} [Publ. Math., Inst. Hautes Étud. Sci. 89, 227--262 (1999; Zbl 0978.54027)] and \textit{E. Lindenstrauss} and \textit{B. Weiss} [Isr. J. Math. 115, 1--24 (2000; Zbl 0978.54026)] as a non-trivial invariant for topological dynamical systems with infinite topological entropy. It has powerful applications in embedding problems because it behaves well with respect to topological embedding: if \((X,T)\) can be embedded in \((Y,S)\) then \(\mathrm{mdim}(X,T)\leq \mathrm{mdim}(Y,S)\). The natural simplest example in this theory is the full shift \(\Sigma_D\) with alphabet \([0,1]^D\), which has mean dimension equal to \(D\). In particular, in order for \((X,T)\) to be embedded into \(\Sigma_D\) it is necessary that \(\mathrm{mdim}(X,T)\leq D\). Clearly it is also necessary that the topological dimension of the set of points of period \(n\) does not exceed \(nD\), so for this and other reasons attention naturally turns to the problem of deciding when a minimal system can embed into \(\Sigma_D\). Lindenstrauss and Weiss [ibid.] constructed a minimal system with mean dimension exceeding one, which therefore cannot be embedded into \(\Sigma_1\). Lindenstrauss [op. cit.] made progress on the very difficult question of finding partial converses to the necessary condition for embedding into \(\Sigma_D\) by showing that there is some constant \(c\geq\frac{1}{36}\) with the property that if \((X,T)\) is an extension of an infinite minimal system and has \(\mathrm{mdim}(X,T)<cD\), then \((X,T)\) can be embedded into \(\Sigma_D\). The classical (purely topological rather than topological-dynamical) Menger-Nöbeling theorem, stating that any compact metric space of topological dimension no more than \(D\) can be topologically embedded into \([0,1]^{2D+1}\), is well known to be sharp. This lends some support to the…
reviewed by: Thomas B. Ward
Mathematics Subject Classification ID: 37B40
Mathematics Subject Classification ID: 54H20
Mathematics Subject Classification ID: 37C45
zbMATH DE Number: 6308393
zbMATH Keywords: mean dimension
zbMATH Keywords: embedding
zbMATH Keywords: minimal dynamical system
MaRDI profile type: Publication
full work available at URL: s11856-013-0040-9
OpenAlex ID: W2064756795
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DOI: 10.1007/S11856-013-0040-9
description: scientific article; zbMATH DE number 6308393