标题： 隐含的次临界性，辛结构，以及类型I算子的尖锐算术谱结果的普遍性 显示英文标题

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

摘要： 我们利用对偶上循环的结构，开发了几种新的概念和工具，用于研究具有解析势能的一频次准周期算子。 作为应用，我们解决了两个长期存在的算术猜想：频率中尖锐相变的普遍性，以及所有频率下整个（非临界）I型算子的积分态密度的绝对连续性，这是一个包括多个流行显式模型的解析邻域的大开集。 此外，我们证明了所有具有丢番图频率的I型算子的积分态密度的$1/2$-霍尔德连续性的普遍性，部分解决了You的猜想\cite{You}。 一个关键的动力学发现：对偶上循环中心的特殊辛结构，令人惊讶的是，在上循环甚至不存在的极限情况下也成立，使得在基于对偶的论证中，首次从三角多项式过渡到一般的解析势能。 它在其他突出的问题中也适用，迄今为止这些问题的结果仅针对三角多项式的情况。 通过这一结构揭示的隐藏亚临界性，允许为这些复辛上循环定义旋转数，并揭示了偶势能的特殊位置的动力学本质。 显示英文摘要

摘要： 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.