Random substitution tilings and deviation phenomena

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