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

Baake, Michael; Frank, Natalie Priebe; Grimm, Uwe and Robinson, Jr., E. Arthur (2019). Geometric properties of a binary non-Pisot inflation and absence of absolutely continuous diffraction. Studia Mathematica, 247 pp. 109–154.

DOI: https://doi.org/10.4064/sm170613-10-3

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 renormalization analysis of the pair correlation functions, we show that the diffraction measure cannot comprise any absolutely 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.

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