with respect to the basis $(\tilde{e_1}, \tilde{e_2}, \tilde{e_3}, \tilde{e_4})$. Hence we get a regular octagonal window $W_{oct}$ of the form

(-6.2,-2.4)(6,3) (0,-2)(2,-2) (2,-2)(3.41,-0.59) (3.41,-0.59)(3.41,1.41) (3.41,1.41)(2,2.83) (2,2.83)(0,2.83) (0,2.83)(-1.41,1.41) (-1.41,1.41)(-1.41,-0.59) (-1.41,-0.59)(0,-2) (2,-2)(0.59,-0.59) (0.59,-0.59)(-1.41,-0.59) (0.59,-0.59)(0.59,1.41) (0.59,1.41)(-1.41,1.41) (0.59,1.41)(2,2.83) (0,2.83)(1.41,1.41) (1.41,1.41)(3.41,1.41) (1.41,1.41)(1.41,-0.59) (1.41,-0.59)(3.41,-0.59) (1.41,-0.59)(0,-2) (-1.41,1.41)(0,0) (0,0)(1.41,1.41) (3.41,1.41)(2,0) (2,0)(0.59,1.41) (3.41,-0.59)(2,0.82) (2,0.82)(0.59,-0.59) (-1.41,-0.59)(0,0.83) (0,0.83)(1.41,-0.59) (2,0.82)(0,0.83) (0,0)(2,0) (0,0)(0,-2) (2,0)(2,-2) (2,0.82)(2,2.83) (0,0.83)(0,2.83)

(1.8,0.2) (1.85,0.14)[$e_1^*$]{} (0,0) (0,0.15) (-1.41,1.41) (-1.65,1.5)[$e_2^*$]{} (0,-2) (0.25,-1.96)[$e_3^*$]{} (1.41,1.41) (1.45,1.43)[$e_4^*$]{}

Then the vertex point pattern appearing in the Ammann-Beenker tiling is given by the non-singular model set $\mathfrak{P}\left( W_{oct} -\dfrac{e_1^*+e_2^*+e_3^*+e_4^*}{2}\right)$.

 {#section:ellis.semigroup}

Ellis semigroup and equicontinuity
----------------------------------

Let us consider a compact dynamical system, that is, a compact (Hausdorff) space $\mathbb{X}$ together with an action of a group $T$ by homeomorphisms. Following [@Aus] we let

\[def:ellis.semigroup\] The Ellis semigroup $E(\mathbb{X},T)$ is the pointwise closure of the group of homeomorphisms given by the $T$-action in the space $\mathbb{X}^\mathbb{X}$ of self-mappings on $\mathbb{X}$.

The Ellis semigroup $E(\mathbb{X},T)$ is then a family of transformations on the space $\mathbb{X}$ which are pointwise limits of homeomorphisms coming from the $T$-action, and is stable under composition. Moreover it is a compact (Hausdorff) space when endowed with the pointwise convergence topology coming from $\mathbb{X}^\mathbb{X}$. In case the acting group is Abelian then, although the Ellis 