A two-dimensional electron gas in a generic potential and a transverse magnetic field exhibits very intriguing fractal spectra with various crossings and avoided crossings in the band structure. These crossings are characterized by integral topological invariants. These features are key towards understanding some recent experiments in the organic conductors which exhibit cascades of field-induced spin density waves and will have novel consequences in experiments involving quantum dot arrays. In the incommensurate limit, a two-dimensional electron gas exhibits a cascade of metal-insulator transitions which at zero temperature has a "devil's fork" phase diagram. A large class of models will be studied to demonstrate that this fractal phase diagram is universal. Some of the quantum chaos models will be examined and their experimental consequences will be explored. In addition, topological aspects of quantum chaos will be studied in order to obtain a unified picture of a large class of systems with competing periodicities. A general formalism to study the finite temperature properties of systems with competing lengths will be developed. This method will be applied to calculate thermodynamical properties of field-induced spin density waves systems and the magnetic Fibonacci and other periodic superlattices. A novel aspect of the Fibonacci lattices is the power law behavior with modulations in the thermodynamic properties. %%% Theoretical research will be conducted on a variety of physical and model systems which have competing periodicities. This competition between various natural periods of the systems results in highly nonlinear behavior. Special mathematical techniques need to be applied to understand the resulting behavior. Physical systems to be studied include the two-dimensional electron gas which forms at semiconductor interfaces exposed to high magnetic fields, quantum dot arrays, and magnetic waves induced in organic conductors. The chaotic limit will also be studied. The results of this work will yield generic techniques to handle these type of systems and will assist in interpreting the behavior of these technologically important materials.
