Spectral cocycle for substitution tilings

Abstract

The construction of spectral cocycle from the case of 1-dimensional substitution flows by Bufetov-Solomyak [arXiv:1802.04783] is extended to the setting of pseudo-self-similar tilings in ${\mathbb R}^d$, allowing expanding similarities with rotations. The pointwise upper Lyapunov exponent of this cocycle is used to bound the local dimension of spectral measures of deformed tilings. The deformations are considered, following Trevi\~no [arXiv:2006.16980], in the simpler, non-random setting. We review some of the results on quantitative weak mixing from [arXiv:2006.16980] in this special case and illustrate them on concrete examples.