equivalent, among patterns admitting the Meyer property, with the property that continuous eigenfunctions forms a separating family for a full measure subset of the associated hull [@BaaLenMo] [@LeeMo]. A third striking result is that *pure point diffractivity* of a pattern [@Hof], with is truly of statistical nature [@Mo3], is known to be equivalent to the existence of a basis of eigenfunctions for the Hilbert space provided by the hull together with a certain ergodic measure (there is a widely developed literature about this aspect of patterns, see for instance [@LeeMoSo] [@BaaLe] and references therein). These statements have been shown under various mild assumptions on the considered pattern.

A certain form of this eigenvalue problem for a point pattern can be addressed, on a topological point of view, by the knowledge of the *Ellis enveloping semigroup* of its dynamical system $(\mathbb{X}, \mathbb{R}^d)$. This semigroup has been introduced for dynamical systems by Ellis and Gottschalk [@EllGo] as a way to study actions of a group on a compact space from an algebraic point of view. In a series of papers, Glasner investigated this semigroup for fairly general dynamical systems (see the review [@Gl] and references therein), and him together with Megrelishvili showed 