Substitution tilings

Abstract

The two main approaches to constructing aperiodically ordered patterns are by the cut and project method and substitution. A substitution is a rule for replacing tiles with larger patches of tiles in a way which can be iterated, defining infinite and highly ordered tilings in the limit. Such objects are even of interest in one dimension, where one may consider symbolic substitutions and the languages they generate. In the case of a primitive substitution over a finite alphabet, the associated dynamical hull of patterns generated, which is acted upon by translations, is uniquely ergodic. I will explain the basics of the theory and then present some new results, joint with Neil Mañibo and Dan Rust, on extending to substitutions defined over infinite prototile sets which are equipped with a compact Hausdorff topology.