(see section $6$).

The embedding morphism $$\begin{aligned}
\begin{psmatrix}[colsep=1.5cm,
rowsep=1.2cm]
E(\Xi ^\Gamma , \Gamma) \; \;  & \; \;  \mathfrak{H}_W\times \mathbb{R}^n
\psset{arrows=->,linewidth=0.2pt, labelsep=1pt
,nodesep=0pt}
\ncline{1,1}{1,2}
\end{psmatrix}\end{aligned}$$ is made from the observation that a neighborhood basis of any transformation $g\in E(\Xi ^\Gamma , \Gamma)$ is provided by the vector $w_g:= \Pi ^*(g)$ of $\mathbb{R}^n$ together with a certain cone $C_g\in \mathfrak{T}_W$, in the sense that a net in $\Gamma$ converges to $g$ in the Ellis semigroup $E(\Xi ^\Gamma , \Gamma)$ (such a net exists by construction) if and only if the corresponding net in $\mathbb{R}^n$ converges to $w_g$ and eventually lies into $C_g+w_g$. In this sense the cone $C_g$ provides the direction a net must follow in order to converge to the transformation $g$. This allows us to calculate the corresponding image subsemigroup in $\mathfrak{H}_W\times \mathbb{R}^n$, which is the aim of theorem \[theo:principal.interne\], proved to be a finite disjoint union of *subgroups* of $\mathbb{R}^n$. Moreover the topology of $E(\Xi ^\Gamma , \Gamma)$ is here completely described by a geometric criterion of convergence for nets.

Finally, we fusion theorems \[theo:suspension.ellis\] and \[theo:principal.interne\] to formulate our main theorem (see the statement \[theo:principal\]), setting the existence of an embedding semigroup morphism $$\begin{aligned}
\begin{psmatrix}[colsep=1.5cm,
rowsep=1.2cm]
E(\mathbb{X} 