UC Irvine

We consider quasiperiodic Jacobi and Schr\"odinger operators of both a single- and multi-frequency. These operators appear very naturally in condensed matter physics, where they have seen applications in the study of Graphene and the Quantum Hall effect. The prototypical example of the single-frequency operator is the almost Mathieu operator (AMO). While much is known about the AMO, less may be said about the multifrequency analogues and even some perturbed single-frequency models. This thesis has one recurring theme: properties of the Lyapunov exponent (LE); moreover, this thesis may be split into two parts. First, we explore continuity of the LE for multifrequency analytic quasiperiodic cocycles and positivity of the LE for single-frequency analytic quasiperiodic Schr\"odinger operators with an additional background potential. We then derive upper bounds in quantum dynamics as a consequence of LE regularity, and explore the fractal properties of spectral measures. In addressing the first part, we prove joint continuity of the LE for non-identically singular multifrequency analytic quasiperiodic cocycles in both cocycle and frequency, and we prove that the (lower) LE for single frequency analytic quasiperiodic Schr\"odinger operators with added background potential can be made uniformly positive by taking a sufficiently large coupling constant independent of the background. The former is accomplished by adapting an inductive argument of Bourgain which was originally used to derive similar results for $SL(2,\C)$-cocycles. The latter involves complexification in phase of the associated transfer matrix and appealing to various properties of analytic and subharmonic functions. In the second part, we first derive a lite version of dynamical localization: under suitable assumptions on the frequency and LE, (time-averaged) moments of the position operator grow no faster than a power of the logarithm. The main achievement here is that our notion is stable under perturbations and holds for all values of the phase and an arithmetically defined set of frequencies of full measure. We then extend the Jitomirskaya-Last power-law subordinacy theory and Last theory of quantum dynamics to encompass a more general version of Hausdorff dimension. This allows us to study fine dimensional properties of spectral measures, particularly `zero-dimensional' measures.