Abstract
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, 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 Treviño [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 Treviño in this special case and illustrate them on concrete examples.
| Original language | English |
|---|---|
| Pages (from-to) | 1629-1672 |
| Number of pages | 44 |
| Journal | Ergodic Theory and Dynamical Systems |
| Volume | 44 |
| Issue number | 6 |
| DOIs | |
| State | Published - 18 Jun 2024 |
Bibliographical note
Publisher Copyright:© The Author(s), 2023. Published by Cambridge University Press.
Funding
We are grateful to Lorenzo Sadun for patiently explaining to us the subtleties of the AP-complex for PSS tilings and their deformations. The images for Kenyon’s tilings in § were constructed using Sage code developed by Mark Van Selous for the Laboratory of Experimental Mathematics at Maryland. The research of B.S. was supported in part by the Israel Science Foundation grant 911/19. R.T. acknowledges support from the Simons Collaboration Grant #712227.
| Funders | Funder number |
|---|---|
| Simons Collaboration | 712227 |
| Israel Science Foundation | 911/19 |
Keywords
- spectral cocycle
- substitution tiling
- tiling cohomology