  According to a conjecture of Lindenstrauss and Tsukamoto, a topological
system $(X,T)$ embeds in the $d$-dimensional cubical shift
$(([0,1]^d)^\mathbb{Z},$shift) if its mean dimension and periodic dimension
verify mdim$(X,T)<d/2$ and perdim$(X,T)<d/2$. If $(X,T)=(\prod\limits_{i\in
\mathbb{N}}X_i,\prod\limits_{i\in \mathbb{N}}T_i)$ ($(X_i, T_i)$ dynamical
systems), and $\liminf\limits_{n\to +\infty} {\rm perdim}(X^{(n)},T^{(n))}) <
\frac{d}{2}$, then $(X,T)$ embeds in $(([0,1]^d)^\mathbb{Z},$shift).
