Two Instances of Chaos in Deterministic and Quantum Dynamical Systems
arXiv:2607.26538
The thesis studies two kinds of chaos: it proves stable ergodicity for certain linear toral automorphisms, and shows that eigenvector delocalization is typical for deterministic Schrödinger operators on large graphs under regularity conditions on the IDS.
Abstract
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 and . 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.
PhD 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