Tilings, traces and triangles

Rodrigo Treviño

Published 2019-06-02Version 1

This paper deals with random substitutions on a finite set of prototiles. The assumptions on the types of substitution rules allowed are very weak, leading to very general constructions. Using renormalization tools applied to elements from operator algebras we establish upper and lower bounds on the rate of deviations of ergodic averages for the uniquely ergodic $\mathbb{R}^d$ action on the tiling spaces obtained from such tilings. We apply the results to obtain statements about the convergence rates for integrated density of states for random Schr\"odinger operators obtained from aperiodic tilings in the construction.