Random Substitution Tilings and Deviation Phenomena

Abstract

Suppose a set of prototiles allows $N$ different substitution rules. In this paper we study tilings of ${\mathbb{R}}^d$ constructed from random application of the substitution rules. The space of all possible tilings obtained from all possible combinations of these substitutions is the union of all possible tilings spaces coming from these substitutions and has the structure of a Cantor set. The renormalization cocycle on the cohomology bundle over this space determines the statistical properties of the tilings through its Lyapunov spectrum by controlling the deviation of ergodic averages of the ${\mathbb{R}}^d$ action on the tiling spaces.

33 pages, comments welcome

References in corpus (2)

• Dynamical systems arising from random substitutions
• Traces of random operators associated with self-affine Delone sets and Shubin's formula

Cited by in corpus (4)

• Spectral cocycle for substitution tilings
• Statistics and gap distributions in random Kakutani partitions and multiscale substitution tilings
• Tilings and traces
• Statistical properties of Lorentz gases on aperiodic tilings, Part 1