Random substitution tilings and deviation phenomena

Scott Schmieding; Rodrigo Treviño

doi:10.3934/dcds.2021020

Article Contents

2021, Volume 41, Issue 8: 3869-3902. Doi: 10.3934/dcds.2021020

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Random substitution tilings and deviation phenomena

Scott Schmieding^1, and
Rodrigo Treviño^2, ,

1.
University of Denver, Denver, CO, USA
2.
University of Maryland, College Park, MD, USA

^* Corresponding author
^* Corresponding author

Received: May 2020

Revised: October 2020

Early access: January 2021

Published: August 2021

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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.

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Mathematics Subject Classification: Primary:52C23, 37H15.

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