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We solve the ten martini problem (Cantor spectrum with no condition onirrational frequencies, previously only established for the almost Mathieu) fora large class of one-frequency quasiperiodic operators, includingnonperturbative 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 ofindependent interest: Kotani theory, for a class of finite-range operators overgeneral minimal underlying dynamics, making the first step towards andproviding a partial solution of the Kotani-Simon problem, simplicity of pointspectrum for the same class, and the all-frequency version of Puig's argument.

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