Abstract

 According to the celebrated Jaworski Theorem, a finite dimensional aperiodic dynamical system $(X,T)$ embeds in the $1$dimensional cubical shift $([0,1]{}},shift)$. If $X$ admits periodic points (still assuming $dim(X)<\infty$) then we show that periodic dimension $perdim(X,T)<\frac{d}{2}$ implies that $(X,T)$ embeds in the $d$-dimensional cubical shift $(([0,1]{d})},shift)$. 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 $(([0,1]{d})},shift)$. Furthermore we introduce a notion of (Krieger) 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 $mdim(X,T)<\frac{d}{16}$ embeds in the \textit{$(d+1)$}cubical shift $(([0,1]{d+1})},shift)$. Finally we compare these results to a recent sharp theorem joint with Tsukamoto: If $(X,T)$ is an extension of an aperiodic subshift and its mean dimension verifies $mdim(X,T)<\frac{d}{2}$ then it embeds in the $d$-dimensional cubical shift $(([0,1]{d})^{\mathbb{Z}},shift)$.<\PRE>