15 March 2024 On the abominable properties of the almost Mathieu operator with well-approximated frequencies
Artur Avila, Yoram Last, Mira Shamis, Qi Zhou
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Duke Math. J. 173(4): 603-672 (15 March 2024). DOI: 10.1215/00127094-2023-0022


We show that some spectral properties of the almost Mathieu operator with frequency well approximated by rationals can be as poor as at all possible in the class of all one-dimensional discrete Schrödinger operators. For the case of critical coupling, we show that the Hausdorff measure of the spectrum may vanish (for appropriately chosen frequencies) whenever the gauge function tends to zero faster than logarithmically. For arbitrary coupling, we show that modulus of continuity of the integrated density of states can be arbitrary close to logarithmic; we also prove a similar result for the Lyapunov exponent as a function of the spectral parameter. Finally, we show that (for any coupling) there exist frequencies for which the spectrum is not homogeneous in the sense of Carleson, and, moreover, fails the Parreau–Widom condition. The frequencies for which these properties hold are explicitly described in terms of the growth of the denominators of the convergents.


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Artur Avila. Yoram Last. Mira Shamis. Qi Zhou. "On the abominable properties of the almost Mathieu operator with well-approximated frequencies." Duke Math. J. 173 (4) 603 - 672, 15 March 2024. https://doi.org/10.1215/00127094-2023-0022


Received: 19 October 2021; Revised: 24 March 2023; Published: 15 March 2024
First available in Project Euclid: 19 April 2024

MathSciNet: MR4734551
Digital Object Identifier: 10.1215/00127094-2023-0022

Primary: 47B36
Secondary: 37C55 , 39A70 , 47B39 , 81Q10

Keywords: Hausdorff dimension , homogeneity , IDS , quasiperiodic , spectrum

Rights: Copyright © 2024 Duke University Press


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Vol.173 • No. 4 • 15 March 2024
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