Yotam Smilansky, Rutgers University

Abstract

A colored partition of a set in Rd is its representation as a disjoint union of subsets, referred to as tiles, where each tile is also assigned a color. In the talk, we will consider sequences of colored partitions defined using multiscale substitution rules on finite collections of colored prototiles. In the substitution process, which generalizes a construction first introduced by Kakutani, tiles of maximal volume in a given partition are replaced by colorful patterns consisting of rescaled copies colored prototiles, thus defining the next partition in the sequence. Tiles that appear in the process are modeled by a flow on a directed weighted graph, and distributional and statistical questions on sequences of partitions are reformulated as questions on the distribution of paths on graphs. Under a natural incommensurability assumption, special properties of the poles of the Laplace transforms of graph counting functions imply various explicit statistical results. In addition, computer experiments reveal the beautiful patterns in which these poles appear in the complex plane, patterns which seem to be closely related to Diophantine properties of the generating substitution rule.