Abstract.We discuss the application of various concepts from the theory of topological dynamical systems to Delone sets and tilings. In particular, we consider the maximal equicontinuous factor of a Delone dynamical system, the proximality relation and the enveloping semigroup of such systems.Mathematics Subject Classification (2010). 37B50, 52C22.

Abstract. Model sets are always Meyer sets but the converse is generally not true. In this work we show that for a repetitive Meyer multiple sets of R d with associated dynamical system (X, R d ), the property of being a model multiple set is equivalent for (X, R d ) to be almost automorphic. We deduce this by showing that a repetitive Meyer multiple set can always be embedded into a repetitive model multiple set having a smaller group of topological eigenvalues. OutlineIn this paper we address a study of particular point patterns of an Euclidean space. From a general point of view, a point pattern is a collection of points inside some space R d (or some locally compact Abelian group), which obeys some discreteness and relative density properties. In this work we will be turned onto the slight generalization of multiple point patterns, that is, a finite collection of point patterns which may overlap. Thanks to this minor generalization our results become true for symbolic sequences and arrays as well. We precise that the concept of point patterns can be much further generalized, by considering for instance weighted Dirac combs as done in [6], or the even more general concept of translation-bounded measures [2].In the topic of point patterns, probably the most understood and studied objects are the so-called model sets, or model multiple set according to our point of view. Formally, a model multiple set is a point set of an Euclidean space that arises as follows: we consider a 'total space' to be the product of R d (the ambient space) with a locally compact Abelian group H (the internal group), together with a latticeΓ in this product. A model multiple set is then a finite family (Λ i ) i∈I of patterns, each Λ i being obtained as projections on R d of the points of the lattice whose projection in H falls into some compact and topologically regular subset W i of H. The data of the group H together with the latticeΓ is called a cut & project scheme and the family {W i } i∈I is called a window. Any coding of a rotation is a model multiple set: such arrays may be defined as the model multiple sets associated with cut & project schemes and window {W i } i∈I obeying the following three conditions: (i) The internal space H is compact (ii) The window {W i } i∈I covers H (iii) The sets W i have pairwise disjoint interiors.It is also often required that the window admits a trivial redundancy subgroup. Famous examples of such arrays are the Toeplitz sequences and arrays [10], also called elsewhere limit-periodic point sets [7], which are uniquely characterized as being the coding of a rotation over an odometer H.

We consider certain point patterns of an Euclidean space and calculate the Ellis enveloping semigroup of their associated dynamical systems. The algebraic structure and the topology of the Ellis semigroup, as well as its action on the underlying space, are explicitly described. As an example, we treat the vertex pattern of the AmmanBeenker tiling of the plane. 37B50, 37B05

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Abstract. We give a constructive proof that a repetitive Meyer multiple set of R d admits a smallest model multiple set containing it colorwise.Meyer sets are objects of central importance in the mathematical theory of Quasicrystals developed in the last thirty years. A special kind of Meyer sets is the class of so-called model sets, which are point patterns for which a geometric picture is available, the latter highly desirable for the understanding of the point pattern as well as for the computation of its relevant quantities. It is known for a long time that a Meyer set always embeds into at least one model set, but it is also true that a huge collection of model sets will contains a common prescribed Meyer set. Hence a natural question is whether there exist a "smallest" model set containing a given Meyer set, and our aim in this note is to answer this question, in the special case where the Meyer set is repetitive, by the affirmative. The existential result is as follows:Theorem. For any repetitive Meyer set Λ of R d is associated a unique model set Λ such that whenever ∆ is a model set with Λ ⊆ ∆ then one hasWe provide a constructive proof of this result (see Theorem 3.1) by explicitly determining the CPS and window involved in the construction of Λ. The proof will be given in the slightly more general formalism of Meyer multiple set, as it for instance naturally appears in the setting of substitution point sets. Meyer sets and model sets of R dA subset Λ of R d is called a Delone set if it consists of a uniformly discrete collection of points, that is, it admits a uniform separation distance between any two of its points, and if it is relatively dense, meaning that any vector of R d fits at uniformly bounded distance to some point of Λ. A Delone set is called a Meyer set, or is said to have the Meyer property, if there is a finite subset F of R d such thatFinally we call a Meyer multiple set a finite collection Λ = (Λ i ) i∈I of Meyer sets where the support S(Λ) := ∪ i∈I Λ i is again a Meyer set. We will often call such quantity a point pattern along the text. There are several very interesting equivalent formulations of the Meyer property which can be found in [8]. The hull of a Meyer