Random holomorphic iterations and degenerate subdomains of the unit disk

Abstract

Given a random sequence of holomorphic maps $f_1,f_2,f_3,...$ of the unit disk $\Delta$ to a subdomain $X$, we consider the compositions $$F_n=f_1 \circ f_{2} \circ ... f_{n-1} \circ f_n.$$ The sequence $\{F_n\}$ is called the {\em iterated function system} coming from the sequence $f_1,f_2,f_3,....$ We prove that a sufficient condition on the domain $X$ for all limit functions of any $\{F_n\}$ to be constant is also necessary. We prove the condition is a quasiconformal invariant. Finally, we address the question of uniqueness of limit functions.