RICE UNIVERSITY Positive Lyapunov Exponent for Ergodic Schr¨ odinger Operators by Helge Kr¨ uger A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE Doctor of Philosophy Approved, Thesis Committee: David Damanik, Professor, Chair Mathematics Michael Boshernitzan, Professor Mathematics Mark Embree, Professor Computational and Applied Mathematics Houston, Texas April 2010