Universal hierarchical structure of quasiperiodic eigenfunctions

TL;DR

This paper reveals a universal hierarchical structure in the eigenfunctions of almost Mathieu operators across all frequencies in the localization regime, connecting their behavior to continued fraction expansions and confirming key conjectures.

Contribution

It provides exact exponential asymptotics for eigenfunctions and transfer matrices, proving the frequency transition conjecture and describing non-regularity phenomena in hyperbolic cocycles.

Findings

Universal structure governed by continued fractions

Proof of the frequency transition conjecture

Explicit description of non-regularity phenomena

Abstract

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.