Spectral cocycle for substitution tilings

The construction of a spectral cocycle from the case of one-dimensional substitution flows [A. I. Bufetov and B. Solomyak. A spectral cocycle for substitution systems and translation flows. J. Anal. Math. 141(1) (2020), 165-205] 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 the work of Trevino [Quantitative weak mixing for random substitution tilings. Israel J. Math., to appear], in the simpler, non-random setting. We review some of the results of Trevino in this special case and illustrate them on concrete examples.

Automated summary

This paper extends the construction of a spectral cocycle from the case of one-dimensional substitution flows 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. Following the work of Trevino, the deformations are considered in the simpler, non-random setting, and some of Trevino's results in this special case are reviewed and illustrated on concrete examples.