Kotani theory, Puig's argument, and stability of The Ten Martini Problem

We solve the ten martini problem (Cantor spectrum with no condition on irrational frequencies, previously only established for the almost Mathieu) for a large class of one-frequency quasiperiodic operators, including nonperturbative analytic neighborhoods of several popular explicit families. The proof is based on the structural analysis of dual cocycles as introduced in [35]. As a part of the proof, we develop several general ingredients of independent interest: Kotani theory, for a class of finite-range operators over general minimal underlying dynamics, making the first step towards and providing a partial solution of the Kotani-Simon problem, simplicity of point spectrum for the same class, and the all-frequency version of Puig's argument.