"Counting tiles in substitution tilings"
Garber, AlexeyFor classical substitution tilings on finite sets of prototiles, the order of growth of the number of tiles in a large supertile is governed by the Perron-Frobenius eigenvalue. The order of the second term in the counting function can be either exponential or polynomial times exponential where the exponent comes from the second largest eigenvalue of the substitution matrix. In this talk we will discuss that for substitutions on infinite sets of prototiles the second term may behave differently. Particularly, we will show that it may behave as the sequence of Catalan numbers or as ratio of exponent and half-integer power for a certain family of substitutions. The talk is based on a joint work with Dirk Frettlöh (Bielefeld University) and Neil Mañibo (Open University, UK).