  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.
