Spectral cocycle for substitution tilings

Abstract The construction of a spectral cocycle from the case one-dimensional substitution flows [A. I. Bufetov and B. Solomyak. A for systems translation flows. J. Anal. Math. 141 (1) (2020), 165–205] is extended to setting pseudo-self-similar tilings in ${\mathbb R}^d$ , allowing expanding similarities with rotations. pointwise upper Lyapunov exponent this used bound local dimension measures deformed tilings. deformations are considered, following work Treviño [Quantitative weak mixing random Israel appear], simpler, non-random setting. We review some results special illustrate them on concrete examples.