Invariant Measures for Degenerate Random Perturbations of Discrete-Time Dynamical Systems
Tanya Yarmola

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 {\it 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 {\it 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 $\mathbb{T}^2$
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.