rise to a new dynamical system $(\Xi ^\Gamma , \Gamma)$. We call this latter the *subsystem* associated with $\Lambda _0$. The space $\Xi ^\Gamma $ will have a locally compact totally disconnected topology, and as a result its Ellis enveloping semigroup $E(\Xi ^\Gamma , \Gamma)$ will be a locally compact totally disconnected topological space (for Ellis semigroup of dynamical systems over locally compact space see section $2$). The importance of this semigroup is our setting justified by theorem \[theo:suspension.ellis\], which yields a algebraic isomorphism and homeomorphism $$\begin{aligned}
E(\mathbb{X},\mathbb{R}^d) \simeq  E(\Xi ^\Gamma , \Gamma)\times _{\Gamma}\mathbb{R}^d\end{aligned}$$ where the right hand term is understood as the quotient of $E(\Xi ^\Gamma , \Gamma)\times \mathbb{R}^d$ under a natural diagonal action of $\Gamma$. This theorem shows in particular that the Ellis semigroup $E(\mathbb{X},\mathbb{R}^d)$ is in our context a *matchbox manifold*: It is locally the product of an Euclidean open subset with a totally disconnected space. It also asserts that the non-trivial (and in particular the non-commutative) part of $E(\mathbb{X},\mathbb{R}^d)$ is displayed by the semigroup $E(\Xi ^\Gamma , \Gamma)$.

We will thus from know on focus on the calculation of $E(\Xi ^\Gamma , \Gamma)$. At first, we show the existence of an onto continuous semigroup morphism $$\begin{aligned}
