Substitutions, cut and projects and both

Abstract

I will introduce a notion of a pattern – usually, a tiling or Delone set – being “substitutional” with respect to an inflation map. This definition is very similar to that of being “pseudo self-affine”, using the concept of local derivability. Whilst this essentially corresponds to familiar definitions, when applied to substitution tilings and substitution Delone (multi)sets, it has several practical advantages when it comes to proving results for such patterns, and in classifying when a pattern is or is not substitutional. I will briefly discuss a general recognisability result for substitutional patterns. Then, I will explain recent work with Harriss and Koivusalo, giving a classification of exactly when a cut and project set (defined by a Euclidean cut and project scheme) defines substitutional patterns, in simple terms of the cut and project data. In addition, we may describe which substitution maps the hull admits, and which elements are fixed by some inflative substitution map i.e., the pseudo self-affine points. We obtain very simple criteria in the case that the window is polytopal.