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

Lingrui Ge; Svetlana Jitomirskaya; Jiangong You

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arXiv · 2308.09321

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

Lingrui Ge·Svetlana Jitomirskaya·Jiangong You

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Abstract

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.

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BibTeX RIS

Lingrui Ge, Svetlana Jitomirskaya, Jiangong You. 2023-08-18. Kotani theory, Puig's argument, and stability of The Ten Martini Problem. https://arxiv.org/abs/2308.09321

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