Monotonicity, global symplectification and the stability of Dry Ten Martini Problem
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
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2026
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| _version_ | 1866911575499079680 |
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| author | Li, Xianzhe Xu, Disheng Zhou, Qi |
| author_facet | Li, Xianzhe Xu, Disheng Zhou, Qi |
| contents | For any fixed irrational frequency and trigonometric-polynomial potential, we show that every type I energy with positive Lyapunov exponent that satisfies the gap-labelling condition is a boundary of an open spectral gap. As a corollary, for the almost-Mathieu operator in the supercritical regime the "all spectral gaps are open" property is robust under a small trigonometric-polynomial perturbation at any irrational frequency. The proof introduces a geometric, all-frequency approach built from three ingredients: (i) the projective action on the Lagrangian Grassmannian and an associated fibred rotation number, (ii) monotonicity of one-parameter families of (Hermitian) symplectic cocycles, and (iii) a partially hyperbolic splitting with a two-dimensional center together with a global symplectification (holonomy-driven parallel transport). This provides a partial resolution to the stability of the Dry Ten Martini Problem in the supercritical regime, and answers a question by M. Shamis regarding the survival of periodic gaps. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_02222 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Monotonicity, global symplectification and the stability of Dry Ten Martini Problem Li, Xianzhe Xu, Disheng Zhou, Qi Dynamical Systems Mathematical Physics Spectral Theory For any fixed irrational frequency and trigonometric-polynomial potential, we show that every type I energy with positive Lyapunov exponent that satisfies the gap-labelling condition is a boundary of an open spectral gap. As a corollary, for the almost-Mathieu operator in the supercritical regime the "all spectral gaps are open" property is robust under a small trigonometric-polynomial perturbation at any irrational frequency. The proof introduces a geometric, all-frequency approach built from three ingredients: (i) the projective action on the Lagrangian Grassmannian and an associated fibred rotation number, (ii) monotonicity of one-parameter families of (Hermitian) symplectic cocycles, and (iii) a partially hyperbolic splitting with a two-dimensional center together with a global symplectification (holonomy-driven parallel transport). This provides a partial resolution to the stability of the Dry Ten Martini Problem in the supercritical regime, and answers a question by M. Shamis regarding the survival of periodic gaps. |
| title | Monotonicity, global symplectification and the stability of Dry Ten Martini Problem |
| topic | Dynamical Systems Mathematical Physics Spectral Theory |
| url | https://arxiv.org/abs/2601.02222 |