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Universal hierarchical structure of quasiperiodic eigenfunctions

Svetlana Jitomirskaya UC Irvine Wencai Liu UC Irvine

Functional Analysis Mathematical Physics mathscidoc:2212.12001

Ann. of Math., 187, (3), 2018.5
We determine exact exponential asymptotics of eigenfunctions and of corresponding transfer matrices of the almost Mathieu operators for all frequencies in the localization regime. This uncovers a universal structure in their behavior, governed by the continued fraction expansion of the frequency, explaining some predictions in physics literature. In addition it proves the arithmetic version of the frequency transition conjecture. Finally, it leads to an explicit description of several non-regularity phenomena in the corresponding non-uniformly hyperbolic cocycles, which is also of interest as both the first natural example of some of those phenomena and, more generally, the first non-artificial model where non-regularity can be explicitly studied.
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@inproceedings{svetlana2018universal,
  title={Universal hierarchical structure  of  quasiperiodic eigenfunctions },
  author={Svetlana Jitomirskaya, and Wencai Liu},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20221206111723921834741},
  booktitle={Ann. of Math.},
  volume={187},
  number={3},
  year={2018},
}
Svetlana Jitomirskaya, and Wencai Liu. Universal hierarchical structure of quasiperiodic eigenfunctions . 2018. Vol. 187. In Ann. of Math.. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20221206111723921834741.
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