Publication Date:
2015-10-10
Description:
According to the celebrated Jaworski theorem, a finite-dimensional aperiodic dynamical system $(X,T)$ embeds in the one-dimensional cubical shift $([0,1]{}^{\mathbb {Z}},\hbox {shift})$ . If $X$ admits periodic points (still assuming $\dim (X) 〈 \infty $ ), then we show in this paper that periodic dimension $\hbox {perdim}(X,T) 〈 {d}/{2}$ implies that $(X,T)$ embeds in the $d$ -dimensional cubical shift $(([0,1]^{d})^{\mathbb {Z}},\hbox {shift})$ . This verifies a conjecture by Lindenstrauss and Tsukamoto for finite-dimensional systems. Moreover, for an infinite-dimensional dynamical system, with the same periodic dimension assumption, the set of periodic points can be equivariantly immersed in $(([0,1]^{d})^{\mathbb {Z}},\hbox {shift})$ . Furthermore, we introduce a notion of markers for general topological dynamical systems, and use a generalized version of the Bonatti–Crovisier tower theorem, to show that an extension $(X,T)$ of an aperiodic finite-dimensional system whose mean dimension obeys $\hbox {mdim}(X,T) 〈 {d}/{16}$ embeds in the $(d+1)$ - cubical shift $(([0,1]^{d+1})^{\mathbb {Z}},\hbox {shift})$ .
Print ISSN:
0024-6115
Electronic ISSN:
1460-244X
Topics:
Mathematics
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