Hidden subcriticality, symplectic structure, and universality of sharp arithmetic spectral results for type I operators

Abstract: We exploit the structure of dual cocycles to develop several new concepts and tools for the study of one-frequency quasiperiodic operators with analytic potentials. As applications we solve two long-standing arithmetic conjectures: universality of sharp phase transition in frequency and absolute continuity of the integrated density of states for all frequencies for the entire class of (non-critical) type I operators, a large open set including the analytic neighborhood of several popular explicit models. Additionally, we prove the universality of $1/2$-H\"older continuity of the integrated density of states for all type I operators with Diophantine frequencies, partially solving You's conjecture \cite{You}. A key dynamical finding: the special symplectic structure of the center of the dual cocycle that, remarkably, holds also in the limit where the cocycle does not even exist, allows, among other things, to pass for the first time from trigonometric polynomials to general analytic potentials, in the duality-based arguments. It is also applicable in other outstanding problems where the results so far have only been obtained for the trigonometric polynomial case. The hidden subcriticality uncovered by this structure allows to define the rotation numbers for these complex-symplectic cocycles and reveals the dynamical nature of the special place of even potentials.