Cut and projects, substitutions and patterns that are both

Abstract

Aperiodically ordered patterns provide interesting examples of dynamical systems as well as models for physical quasicrystals. Most notable examples come from just two construction techniques: substitution rules and the cut and project method. In fact, despite their discoveries being down to just one or neither technique, many famous tilings (including the Penrose, Ammann–Beenker and hat tilings) turned out to be both substitutive and cut and projects. It is of interest to know when substitutive patterns are also cut and project (with regular window) because this implies pure point diffraction. The Pisot Conjecture essentially studies this question, starting from a substitution pattern. In this talk I will explain recent joint work with Edmund Harriss and Henna Koivusalo which considers the reverse direction: starting with a cut and project scheme, we find simple conditions to also be ‘substitutive’, and an even simpler condition when the window of the cut and project scheme is a polytope.