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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.

The first author constructed a q-parameterized spherical category $\sC$ over C(q) in [Liu15], whose simple objects are labelled by all Young diagrams. In this paper, we compute closed-form expressions for the fusion rule of $\sC$, using Littlewood-Richardson coefficients, as well as the characters (including a generating function), using symmetric functions with infinite variables.