Hölder continuity of the boat–star integrated density of states at the spectral edges
Let $\Gamma_\bigstar$ be the tile graph of the boat–star tiling: Penrose's pentagon tiling P1, generated by the six-tile substitution of Grünbaum and Shephard (Definition 3.1 of the source), with one vertex per tile and tiles adjacent when they share an edge. Let $\Sigma_\bigstar$ be the spectrum of its graph Laplacian $\Delta = D - A$, and $k_\bigstar$ its integrated density of states, normalised per tile. $k_\bigstar$ jumps at $E = 4$, by at least $(65 - 29\sqrt5)/10 \approx 0.0154$ (Theorem 1.4 of the source).
Problem. On a neighbourhood of $\min\Sigma_\bigstar = 0$, or of $\max\Sigma_\bigstar$, is $k_\bigstar$ $\alpha$-Hölder continuous for some $\alpha \in (0, 1)$? Is it Lipschitz continuous?
Source: Damanik, Embree, Fillman and Mei, Question 8.6 in “Discontinuities of the integrated density of states for Laplacians associated with Penrose and Ammann–Beenker tilings”, Experimental Mathematics 33 (2024); arXiv:2209.01443.
Imported from arxiv.org/abs/2209.01443.
Classification
- Primary
- SP — Spectral Theory
- Secondary
- DS, MP
- Keywords
- aperiodic order, Hölder continuity, integrated density of states, Penrose tiling, spectral edges
- Literature checked
- Oct. 1, 2026 by clemens-admin
References
@article{damanik2024,
author = {Damanik, David and Embree, Mark and Fillman, Jake and Mei, May},
title = {Discontinuities of the integrated density of states for {L}aplacians associated with {P}enrose and {A}mmann--{B}eenker tilings},
journal = {Experimental Mathematics},
volume = {33},
number = {4},
pages = {588--610},
year = {2024},
doi = {10.1080/10586458.2023.2206589},
eprint = {2209.01443},
archivePrefix = {arXiv}
}
@book{grunbaum1987,
author = {Gr{\"u}nbaum, Branko and Shephard, G. C.},
title = {Tilings and Patterns},
publisher = {W. H. Freeman and Company},
address = {New York},
year = {1987}
}
@article{lenz2005,
author = {Lenz, Daniel and Stollmann, Peter},
title = {An ergodic theorem for {D}elone dynamical systems and existence of the integrated density of states},
journal = {Journal d'Analyse Math{\'e}matique},
volume = {97},
pages = {1--24},
year = {2005}
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00026, "Hölder continuity of the boat–star integrated density of states at the spectral edges", clopen.solutions.