Spectral cocycle for substitution tilings

TL;DR

This paper extends the spectral cocycle framework from 1D substitution flows to higher-dimensional pseudo-self-similar tilings with rotations, using Lyapunov exponents to analyze spectral measures and weak mixing properties.

Contribution

It introduces a generalized spectral cocycle for higher-dimensional tilings with rotations and applies Lyapunov exponents to study spectral measures and weak mixing.

Findings

Bounded local dimension of spectral measures for deformed tilings.

Illustration of weak mixing results on concrete examples.

Extension of spectral cocycle construction to higher dimensions with rotations.

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.