AI 中文总结
该论文通过两个项目，分别证明了特定条件下N维环面遍历线性自同构的稳定遍历性，以及大有限图上确定性薛定谔算子本征向量离域化的拓扑常见性，推广了相关已有结果。
AI 中文摘要
本论文包含两个独立项目，分别属于光滑动力系统与谱理论领域，共同聚焦于经典和量子动力系统中的混沌机制与统计行为。第一个项目研究光滑动力系统：我们证明，所有具有二维中心的N维环面的遍历线性自同构都是稳定遍历的，包括所有N≤5及N=7维的遍历自同构，这推广了Rodriguez-Hertz的先前结果，后者要求线性自同构的特征多项式满足额外代数条件。第二个项目研究薛定谔算子的谱理论：我们证明，若积分态密度（IDS）满足合适的正则性条件，则在给定大有限图上的确定性薛定谔算子空间中，大多数本征向量的离域化是拓扑常见的，该结果推广了Avila和Damanik的近期定理；我们还通过证明Thouless公式的一个变体，描述了满足我们判据的灵活图族。
英文摘要
This thesis consists of two distinct projects situated in the areas of smooth dynamics and spectral theory, respectively. They are united by a common interest in mechanisms of chaos and statistical behavior in classical and quantum dynamical systems. The first concerns smooth dynamics. We prove that all ergodic linear automorphisms of the N-dimensional torus with two-dimensional center are stably ergodic, including all ergodic automorphisms in dimensions $N \leq 5$ and $N = 7$ . This generalizes a previous result of Rodriguez-Hertz, which required an additional algebraic condition on the characteristic polynomial of the linear automorphism. The second project deals with spectral theory of Schrödinger operators. We prove that delocalization of most eigenvectors is topologically common in the space of deterministic Schrödinger Operators on a given large finite graph, provided that the IDS satisfies a suitable regularity condition. This result generalizes a recent theorem of Avila and Damanik. We also describe a flexible family of graphs satisfying our criterion, by proving a variant of the Thouless formula.
CommentsPhD thesis, University of Zurich, defended July 2026. Cumulative thesis: Chapter 2 corresponds to arXiv:2603.23778; Chapter 3 is based on a forthcoming manuscript. Collaborations and contributions are disclosed in the thesis
