  According to the celebrated Jaworski Theorem, a finite dimensional aperiodic
dynamical system $(X,T)$ embeds in the $1$-dimensional cubical shift
$([0,1]^{\mathbb{Z}},shift)$. If $X$ admits periodic points (still assuming
$\dim(X)<\infty$) then we show in this paper that periodic dimension
$perdim(X,T)<\frac{d}{2}$ implies that $(X,T)$ embeds in the $d$-dimensional
cubical shift $(([0,1]^{d})^{\mathbb{Z}},shift)$. 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
$(([0,1]^{d})^{\mathbb{Z}},shift)$. 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
$mdim(X,T)<\frac{d}{16}$ embeds in the $(d+1)$-cubical shift
$(([0,1]^{d+1})^{\mathbb{Z}},shift)$.
