Finitely many jumps in the integrated density of states of a linearly repetitive tiling
Let $\mathcal T$ be a linearly repetitive tiling of the plane by polygons, with tile graph $\Gamma_{\mathcal T}$ (one vertex per tile, tiles adjacent when they share an edge), graph Laplacian $\Delta = D - A$ and integrated density of states $k$. A locally supported (finitely supported) eigenfunction of $\Delta$ with eigenvalue $E$ makes $k$ jump at $E$; under suitable assumptions the converse holds too (Klassert, Lenz and Stollmann).
Problem. Does $k$ always have only finitely many jumps? Is there an effective way of bounding their number for a given linearly repetitive tiling?
Source: Damanik, Embree, Fillman and Mei, Question 8.5 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, graph Laplacian, integrated density of states, linear repetitivity, locally supported eigenfunctions
- 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}
}
@article{klassert2003,
author = {Klassert, S. and Lenz, Daniel and Stollmann, Peter},
title = {Discontinuities of the integrated density of states for random operators on {D}elone sets},
journal = {Communications in Mathematical Physics},
volume = {241},
number = {2--3},
pages = {235--243},
year = {2003}
}
@article{damanik2001,
author = {Damanik, David and Lenz, Daniel},
title = {Linear repetitivity. {I}. {U}niform subadditive ergodic theorems and applications},
journal = {Discrete \& Computational Geometry},
volume = {26},
number = {3},
pages = {411--428},
year = {2001}
}
@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-00025, "Finitely many jumps in the integrated density of states of a linearly repetitive tiling", clopen.solutions.