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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.
@article{10_4007_annals_2018_187_3_3,
author = {Svetlana Jitomirskaya and Wencai Liu},
title = {Universal hierarchical structure of quasiperiodic eigenfunctions},
journal = {Annals of mathematics},
pages = {721--776},
publisher = {mathdoc},
volume = {187},
number = {3},
year = {2018},
doi = {10.4007/annals.2018.187.3.3},
mrnumber = {3779957},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.4007/annals.2018.187.3.3/}
}
TY - JOUR AU - Svetlana Jitomirskaya AU - Wencai Liu TI - Universal hierarchical structure of quasiperiodic eigenfunctions JO - Annals of mathematics PY - 2018 SP - 721 EP - 776 VL - 187 IS - 3 PB - mathdoc UR - http://geodesic.mathdoc.fr/articles/10.4007/annals.2018.187.3.3/ DO - 10.4007/annals.2018.187.3.3 LA - en ID - 10_4007_annals_2018_187_3_3 ER -
%0 Journal Article %A Svetlana Jitomirskaya %A Wencai Liu %T Universal hierarchical structure of quasiperiodic eigenfunctions %J Annals of mathematics %D 2018 %P 721-776 %V 187 %N 3 %I mathdoc %U http://geodesic.mathdoc.fr/articles/10.4007/annals.2018.187.3.3/ %R 10.4007/annals.2018.187.3.3 %G en %F 10_4007_annals_2018_187_3_3
Svetlana Jitomirskaya; Wencai Liu. Universal hierarchical structure of quasiperiodic eigenfunctions. Annals of mathematics, Tome 187 (2018) no. 3, pp. 721-776. doi: 10.4007/annals.2018.187.3.3
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