Geometric properties of a binary non-Pisot inflation and absence of absolutely continuous diffraction

Michael Baake, Natalie Priebe Frank, Uwe Grimm, E. Arthur Robinson Jr.
2019 Studia Mathematica  
2019). Geometric properties of a binary non-Pisot inflation and absence of absolutely continuous diffraction. Studia Mathematica, 247 pp. 109-154. For guidance on citations see FAQs. Abstract. One of the simplest non-Pisot substitution rules is investigated in its geometric version as a tiling with intervals of natural length as prototiles. Via a detailed renormalisation analysis of the pair correlation functions, we show that the diffraction measure cannot comprise any absolutely continuous
more » ... utely continuous component. This implies that the diffraction, apart from a trivial Bragg peak at the origin, is purely singular continuous. En route, we derive various geometric and algebraic properties of the underlying Delone dynamical system, which we expect to be relevant in other such systems as well. Proof. Since our inflation rule is aperiodic, we have local recognisability [32] . This means that each tile in any (fixed) element of the hull lies inside a unique level-1 supertile that is identified by a local rule. Concretely, each patch of type 0111 constitutes a supertile of type 0, while each tile of type 0 that is followed by another 0 (to the right) stands for a supertile of type 1. Below, we simply say supertile, as no level higher than 1 will occur in this proof. Due to the inflation structure, it is also clear that the relative frequency (meaning relative to Λ) of two supertiles of type i and j with distance z (from i to j) is given by 1 λ ν ij z λ . This
doi:10.4064/sm170613-10-3 fatcat:snriaptgy5e2nhpfz2tjbj4qve