\begin{psmatrix}[colsep=1.5cm,
rowsep=1.2cm]
E(\Xi ^\Gamma , \Gamma) \; \; & \; \; \mathbb{R}^n 
\psset{arrows=->>,linewidth=0.2pt, labelsep=1pt
,nodesep=0pt}
\ncline{1,1}{1,2}^{^{~\; \; \; \Pi ^*}}
\end{psmatrix}\end{aligned}$$

This morphism is closely related with the parametrization map presented above, and will allows us to understand the convergence of a net in $E(\Xi ^\Gamma , \Gamma)$ by how the corresponding net, through this morphism, converge in $\mathbb{R}^n$.

Our wish know is to find a certain semigroup $S$, together with a certain semigroup morphism from $E(\Xi ^\Gamma , \Gamma)$ into $S$, such that the Ellis semigroup $E(\Xi ^\Gamma , \Gamma)$ embeds in the direct product $S\times \mathbb{R}^n$. In order to simplify the problem we let the *almost canonical property* enter the game. This property consist of a condition on the region $W$ used to obtain $\Lambda _0$ as model set, that is, $W$ must be a polytope of $\mathbb{R}^n$ satisfying a particular condition (see section $4$). With the assumption of almost canonicity for the region $W$ together with the almost automorphic property observed on the dynamical system $(\mathbb{X}_{\Lambda _0}, \mathbb{R}^d)$, we are able to identify the correct semigroup $S$ as the *face semigroup* associated with the polytope $W$ in $\mathbb{R}^n$ (see sections $5$ and $6$ for presentation and results).

