$C^*$-Algebras (a recent source is [@Put], see references therein), and also perform the computation of their cohomology and K-theory groups [@FoHuKe] [@GaHuKe] [@Put] as well as the asymptotic exponent of their complexity function [@Ju]. We show that in our situation it is possible to completely describe elements of the Ellis semigroup, their action onto the underlying space, as well as the algebraic and topological structure of this semigroup. The type of calculation made here can be compared with the calculation performed in [@Pik] about Sturmian and Sturmian-like systems (see also example $4.5$ of [@Gl]). We also show that for those dynamical systems the Ellis semigroup is of first class on the sense of the dichotomy of [@GlMe], that is, almost canonical model sets have tame systems.

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In order to construct a model set of $\mathbb{R}^d$, one begin by considering a higher dimensional Euclidean space $\mathbb{R}^{n+d}$ together with a lattice $\Sigma$ in it, a well as an embedded $d$-dimensional slope, usually placed in an ’irrational’ manner, which is thought as the space $\mathbb{R}^d$ itself. Such an environment used to construct a model set is called a *cut $ \& $ project scheme*. Then a second step is to consider 