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

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.

40 pages, 3 figures, 1 table

References in corpus (1)

• Singular substitutions of constant length

Cited by in corpus (13)

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• Uniformly distributed orbits in ${\mathbb{T}}^d$ and singular substitution dynamical systems
• Absence of absolutely continuous diffraction spectrum for certain S-adic tilings
• Flow views and infinite interval exchange transformations for recognizable substitutions
• Hyperuniformity and anti-hyperuniformity in one-dimensional substitution tilings