Mean dimension and an embedding problem: an example
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 conjecture formulated here that if \((X,T)\) has \(\mathrm{mdim}(X,T)<\frac12 D\) and \(\dim(\{x\mid T^nx=x\})<\frac12 nD\) for all \(n\geq1\) then \((X,T)\) can be embedded into \(\Sigma_D\). In this paper, for any \(D\geq1\), a minimal topological dynamical system with mean dimension equal to \(\frac12 D\) is constructed that cannot be embedded into the full shift with alphabet \([0,1]^D\), showing that if the conjecture holds then the condition \(\mathrm{mdim}(X,T)<\frac12 D\) is optimal. The construction makes use of topological machinery developed by \textit{J. Matoušek} [Using the Borsuk-Ulam theorem. Lectures on topological methods in combinatorics and geometry. Written in cooperation with Anders Björner and Günter M. Ziegler. Berlin: Springer (2003; Zbl 1016.05001)] as well as some ideas from the earlier constructions by Lindenstrauss and Weiss [op. cit.].
scientific article; zbMATH DE number 6308393