Zero measure Cantor spectrum for aperiodic metric tiling graphs
Abstract
We study a family of aperiodic metric graphs generated by one-dimensional tilings. We prove that, generically, the spectrum of the associated Kirchhoff Laplacian has Lebesgue measure zero and, apart from a discrete set of isolated eigenvalues, is a generalized Cantor set. For a large subclass of these graphs, we further show that isolated eigenvalues are generically absent. This extends several well-known results from the setting of discrete ergodic Schrödinger operators on ℤ to a large family of metric graphs. Earlier results of this kind for metric graphs typically assume equilateral edges and exploit symmetries that reduce the analysis to a Sturm-Liouville operator on the real line. By contrast, the graph families considered here need not have equilateral edges or the symmetries needed for such a reduction and therefore require a different analytical approach.
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BibTeX
@misc{band2026zero,
title = {{Zero measure Cantor spectrum for aperiodic metric tiling graphs}},
author = {Ram Band and Gilad Sofer},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.02650},
eprint = {2610.02650},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.02650},
}
RIS
TY - UNPB
TI - Zero measure Cantor spectrum for aperiodic metric tiling graphs
AU - Ram Band
AU - Gilad Sofer
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2610.02650
UR - https://arxiv.org/abs/2610.02650
AB - We study a family of aperiodic metric graphs generated by one-dimensional tilings. We prove that, generically, the spectrum of the associated Kirchhoff Laplacian has Lebesgue measure zero and, apart from a discrete set of isolated eigenvalues, is a generalized Cantor set. For a large subclass of these graphs, we further show that isolated eigenvalues are generically absent. This extends several well-known results from the setting of discrete ergodic Schrödinger operators on ℤ to a large family of metric graphs. Earlier results of this kind for metric graphs typically assume equilateral edges and exploit symmetries that reduce the analysis to a Sturm-Liouville operator on the real line. By contrast, the graph families considered here need not have equilateral edges or the symmetries needed for such a reduction and therefore require a different analytical approach.
ER -