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Quantitative weak mixing of self-affine tilings

Juan Marshall-Maldonado

TL;DR

This work extends the quantitative analysis of spectral measures for translation actions on tilings from 1D self-similar systems to higher-dimensional self-affine tilings. By relating twisted Birkhoff integrals $S^f_R$ to spectral measures $\sigma_f$ through Fejér kernels, the authors develop a multi-scale framework (tower structure, structure factor, and cocycles) to obtain uniform modulus-of-continuity bounds. For self-similar tilings, they derive uniform logarithmic decay of spectral measures and Cesàro correlations, establishing explicit weak-mixing rates; for self-affine tilings under strongly totally non-Pisot conditions, they obtain analogous, uniform logarithmic bounds leveraging cohomological finiteness and ergodic-average deviation results. These results generalize the Bufetov–Solomyak methodology to higher dimensions, providing explicit quantitative weak mixing statements with potential applications to the spectral study of aperiodic order in materials and mathematical tiling dynamics.

Abstract

We study the regularity of spectral measures of dynamical systems arising from a translation action on tilings of substitutive nature. The results are inspired in the work of Bufetov and Solomyak, where they established a log-Hölder modulus of continuity of one-dimensional self-similar tiling systems. We generalize this result to higher dimensions in the more general setting of self-affine tilings systems. Further analysis leads to uniform estimates in the whole space of spectral parameters, allowing to deduce logarithmic rates of weak mixing.

Quantitative weak mixing of self-affine tilings

TL;DR

This work extends the quantitative analysis of spectral measures for translation actions on tilings from 1D self-similar systems to higher-dimensional self-affine tilings. By relating twisted Birkhoff integrals to spectral measures through Fejér kernels, the authors develop a multi-scale framework (tower structure, structure factor, and cocycles) to obtain uniform modulus-of-continuity bounds. For self-similar tilings, they derive uniform logarithmic decay of spectral measures and Cesàro correlations, establishing explicit weak-mixing rates; for self-affine tilings under strongly totally non-Pisot conditions, they obtain analogous, uniform logarithmic bounds leveraging cohomological finiteness and ergodic-average deviation results. These results generalize the Bufetov–Solomyak methodology to higher dimensions, providing explicit quantitative weak mixing statements with potential applications to the spectral study of aperiodic order in materials and mathematical tiling dynamics.

Abstract

We study the regularity of spectral measures of dynamical systems arising from a translation action on tilings of substitutive nature. The results are inspired in the work of Bufetov and Solomyak, where they established a log-Hölder modulus of continuity of one-dimensional self-similar tiling systems. We generalize this result to higher dimensions in the more general setting of self-affine tilings systems. Further analysis leads to uniform estimates in the whole space of spectral parameters, allowing to deduce logarithmic rates of weak mixing.
Paper Structure (16 sections, 23 theorems, 94 equations)

This paper contains 16 sections, 23 theorems, 94 equations.

Table of Contents

  1. Introduction
  2. Acknowledgments
  3. Background on tiling spaces
  4. Tiling dynamical systems
  5. Substitutions
  6. Spectral theory
  7. Twisted Birkhoff integrals
  8. Relation between $\sigma_f$ and $S^f_R(\boldsymbol{X}, \boldsymbol{\omega})$
  9. Preliminary results
  10. Tower structure
  11. Bound for the structure factor
  12. Spectrum of self-affine tilings expansions
  13. Modulus of continuity for strongly totally non-Pisot substitutions
  14. Uniform bounds: self-similar case
  15. Local estimate at $\boldsymbol{\omega} = {\boldsymbol{0}}$
  16. ...and 1 more sections

Key Result

Theorem 2.1

If every tiling in $\mathfrak{T}_{\mathcal{A}}$ has f.l.c., then the full-tiling endowed with Dist is a compact metric space. Also, the map $\phi : \mathbb{R}^d \times \mathfrak{T}_\mathcal{A} \longrightarrow \mathfrak{T}_\mathcal{A}$, $\phi({\boldsymbol{s}},\boldsymbol{X}) = \phi_{{\boldsymbol{s}}}

Theorems & Definitions (44)

  • Theorem 2.1: see robinson2004symbolic, Theorem 2.9
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.4: B. Solomyak, SolUniq Theorem 1.1
  • Definition 2.5
  • Proposition 2.6: praggastis1999numeration, Proposition 1.2
  • Theorem 2.7: see lee2003consequences, Theorem 4.1
  • Theorem 2.8: see einsiedler2017functional, Theorem 9.58
  • Definition 2.9
  • Lemma 2.10
  • ...and 34 more