Two Instances of Chaos in Deterministic and Quantum Dynamical Systems

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 Schr\"odinger operators. We prove that delocalization of most eigenvectors is topologically common in the space of deterministic Schr\"odinger 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.