Invariant measures for degenerate random perturbations of discrete-time dynamical systems

Random perturbations of dynamical systems is an important tool in modeling noise and other types of uncontrolled fluctuations. In many real-life systems, fluctuations do not occur everywhere or uniformly in all directions some of these situations can be modeled by localized, degenerate noise. In this thesis, we focus on random perturbations of discrete-time dynamical systems that occur in a single direction we call these rank one perturbations. The aim of this work is to study whether such perturbations lead to invariant measures that are absolutely continuous with respect to Lebesgue measure.This study is based on the premise that when the dynamics are rich enough, they tend to mix the different directions after a number of iterates. Thus barring unfortunate coincidences, one should expect even rank one perturbations to have absolutely continuous invariant measures. We introduce a very general condition that ensures that all the invariant measures for the perturbed system are absolutely continuous.For systems with hyperbolic properties, this condition relates to roughly speaking ruling out tangencies of certain types with the stable foliation. For hyperbolic toral automorphisms and Anosov Diffeomorphisms with codimension 1 stable manifolds this condition is generic in the class of rank one perturbations.To demonstrate what pathological behavior can occur when conditions of this type are not met, we give an example of a rank one perturbation of the Cat Map that produces a "global statistical attractor" in the form of a line segment. The Cat Map is well known to have strong hyperbolic and mixing properties, yet in our example, all initial distributions on T2 are attracted to a piece of local stable manifold.In the final chapter of this thesis, some of the ideas above are applied to a class of billiard maps derived from the 2-dimensional periodic Lorentz gas. Existence and uniqueness of absolutely continuous invariant densities are proved for a certain family of rank one perturbations.