Random Iteration of Rational Maps

Abstract

Random and non-autonomous iteration has been a subject of interest in Mathematics that has received some attention in the last few decades. The earliest paper on random iteration in the complex setting was written by Fornaess and Sibony. They have shown that given a family of functions $\{f_c\}_{c \in \W}$ where $\W$ is a small open set, for almost every z the random iteration is stable on a subset of $\W^\N$ of full probability measure. Later, Hiroki Sumi further extended these results to a more general situation using rational semigroups. We will show that the results of Fornaess and Sibony can be extended using the concept of non-generic points. Then we describe the connection between Sumi's kernel Julia set and non-generic points. In the third chapter, we will look at seed iteration. This is where a function $f(w,z)$ is composed in the second variable to get a function $f^n(w,z)$ and then we set $z=w$ to get a sequence of functions $F_n(w)$. We will study the properties of the corresponding Julia and Fatou sets of the sequence $F_n(w)$. Furthermore, we will look at evidence that there may be basins of attraction and sub-invariant domains contained inside the space of analytic functions over a domain $U$, similar to what we see in classical iteration theory.