arXiv subjects

E. Arthur Robinson, Jr.

Publications and source records attributed to E. Arthur Robinson, Jr..

2 recordsLinked to original sources

arXiv2017

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

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.

math.DS↗

arXiv2005

Generalized $β$-expansions, substitution tilings, and local finiteness

For a fairly general class of two-dimensional tiling substitutions, we prove that if the length expansion $β$ is a Pisot number, then the tilings defined by the substitution must be locally finite. We also give a simple example of a two-dimensional substitution on rectangular tiles, with a non-Pisot length expansion $β$, such that no tiling admitted by the substitution is locally finite. The proofs of both results are effectively one-dimensional and involve the idea of a certain type of generalized $β$-transformation.

math.DS↗