$\Lambda .t := \Lambda -t$, that is, by shifting any point pattern site by site. This provides so a dynamical system $(\mathbb{X}, \mathbb{R}^d)$. In order to figure out what exactly consists this space, we invoke the following beautiful result:

\[theo:parametrization.map\] [@Sch] Let $\mathbb{X}$ be the hull associated with a cut $\&$ project scheme $(\mathsf{H},\Sigma,\mathbb{R}^d)$ and some window. Then $\mathbb{X}$ is compact and the dynamical system $(\mathbb{X}, \mathbb{R}^d)$ is minimal. Each $\Lambda \in \mathbb{X}$ satisfy inclusions of the form $$\begin{aligned}
\mathfrak{P} (w_\Lambda +\mathring{W})-t_\Lambda \subseteq \Lambda \subseteq \mathfrak{P} (w_\Lambda +W)-t_\Lambda\end{aligned}$$ where $(w_\Lambda , t_\Lambda)\in \mathsf{H}\times \mathbb{R}^d$ is unique up to an element of $\Sigma$, thus defining a factor map $$\begin{aligned}
\begin{psmatrix}[colsep=1.5cm,
rowsep=0cm]
 \pi : \; \mathbb{X}\; \; & \; \;  \mathsf{H}\times _{\Sigma} \mathbb{R}^d
\psset{arrows=->>,linewidth=0.2pt, labelsep=1.5pt
,nodesep=0pt}
\ncline{1,1}{1,2}
\end{psmatrix}\end{aligned}$$ with $\mathsf{H}\times _{\Sigma} \mathbb{R}^d$ the compact Abelian group quotient of $\mathsf{H}\times \mathbb{R}^d$ by the lattice $\Sigma$.\
The map $\pi$ is injective precisely on the collection of non-singular model sets of $\mathbb{X}$.

By factor map we mean here a continuous, onto and $\mathbb{R}^d$-equivariant map, where on the compact Abelian group $\mathsf{H}\times _{\Sigma} \mathbb{R}^d$ the space $\mathbb{R}^d$ acts through $[w,t]_{\Sigma}.s:= [w,t+s]_{\Sigma}$. In the context of real cut $\&$ project schemes the compact Abelian group is given 