Behavior of Random Dynamical Systems of a Complex Variable

Abstract

In this thesis we examine some methods of adding noise to the discrete dynamical system z → z^2 + c, in the complex plane. We compare the "; Traditional Random Iteration "; : choosing a sequence of c-values and applying that sequence of maps to the entire plane, versus what we introduce as "; Noisy Random Iteration "; : for each z and for each iterate calculated, we choose a different c-value. We examine two methods of choices for c: (1) Uniform distribution on a neighborhood of c, versus (2) a Bernoulli choice from two values {a,b}, with varying probability p in [0,1] that c=a. We show the results of computer investigations, provide definitions and prove some initial results about Noisy Random Iteration. Finally, we leave the audience with some open questions and directions for future research.