Geometric properties of a binary non-Pisot inflation and absence of absolutely continuous diffraction
Volume 247 / 2019
Studia Mathematica 247 (2019), 109-154
MSC: Primary 37A30, 42A38, 37B50; Secondary 37H15, 52C23.
DOI: 10.4064/sm170613-10-3
Published online: 23 November 2018
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 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.
