in [@GlMe] that a dichotomy occurs on the Ellis semigroup $E(\mathbb{X}, \mathbb{R}^d)$: It is either sequentially compact or contains a topological copy of $\beta \mathbb{N}$ the $Stone-\check{C}ech$ compactification of the integers. The former situation admits several equivalent formulations and when it occurs the underlying dynamical system is called *tame* [@Gl0]. Tame systems are dynamically simple: Indeed it is proved in [@Gl2] that they are uniquely ergodic, almost automorphic and measurably conjugated with a Kronecker system. In the vocabulary of eigenfunctions this means that continuous eigenfunctions of the system separates a residual subset of full measure in the underlying compact space. In the topic of point patterns this just asserts that, following the characterization of model sets provided in [@BaaLenMo], a point pattern admitting the Meyer property and with a tame dynamical system must be a regular model set.

In this work we propose to provide a qualitative description of the Ellis semigroup of dynamical systems systems associated with particular point patterns, the *almost canonical model sets*. These particular patterns are relevant in the crystallographic sense, as well as very accessible mathematically: One can get a complete picture of the hull $\mathbb{X}_{\Lambda_0}$ of such patterns [@Le], as well as there associated 