Characterization of model sets by dynamical systems

Abstract

It is shown how regular model sets can be characterized in terms of the regularity properties of their associated dynamical systems. The proof proceeds in two steps. First, we characterize regular model sets in terms of a certain map β and then relate the properties of β to those of the underlying dynamical system. As a by-product, we can show that regular model sets are, in a suitable sense, as close to periodic sets as possible among repetitive aperiodic sets.Downloaded

Hexagonal Inflation Tilings and Planar Monotiles

“…The (regular) model set structure of the half-hex tiling, as spelled out in Equation (1) for the fixed points under the inflation rule, implies that there is a "torus parametrisation" map onto a compact Abelian group [16,25]. Here, it is a factor map onto the two-dimensional dyadic solenoid S 2 2 , which is almost everywhere one-to-one by Theorem 5 of [25].…”

Substitution Delone sets with pure point spectrum are inter-model sets

“…inter model multi-colour set) if each Γ i is a model set (resp. inter model set) with respect to the same CPS (see [27], [22] and [3] for more about model sets). One should note here that since π 2 need not be 1 − 1 on L in CPS, the notion of inter model set, which is hemmed in between two such sets differing only by points on the boundary of the window W , arises naturally.…”

“…The following theorem is proved in [3] in Delone sets with one colour. The argument can be extended into Delone multi-colour sets without difficulty.…”

“…In [3] regular model sets are characterized in terms of their associated dynamical systems. Can inter model sets be characterized in terms of their associated dynamical systems as well?…”

“…The notion of inter model sets is introduced in [22] (under the name of model sets) and [3] as a model set satisfying a topological condition which is less restrictive than the boundary condition of a regular model set. In this paper we consider the inter model sets (see Def.…”