Hierarchy and cut and project sets

Abstract

I will introduce the two main methods of construction of aperiodically ordered patterns: by substitution (or ‘inflate, subdivide’) rules, creating patterns which may be thought of as ‘upside-down fractals’, and the cut and project method, wherein one takes an irrational slice of a periodic lattice. Many well-known aperiodic patterns, such as the Penrose tilings or recently discovered hat/spectre families of tilings (at least with particular parameters for edge-lengths), can be constructed through both procedures. I will introduce a generalised notion of a ‘pattern’ and its associated dynamical system, which gives a unified framework for studying tilings, point sets and more in the translationally FLC (finite local complexity) case. We also define when such a pattern is ‘substitutional’ in a general sense. I will explain recent results (in preparation) which establish an equivalence between unique composition of the hierarchy of such patterns and aperiodicity of their associated dynamical system, mildly generalising a well-known result of Solomyak (still only in the FLC case, but extending here to non-minimal patterns). I will then explain recent results, joint with Harriss and Koivusalo, on when a cut and project set with Euclidean total space is substitutional. We obtain a particularly simple and checkable criterion in the case that the window is assumed to be polytopal.