Random holomorphic iterations and degenerate subdomains of the unit disk

Given a random sequence of holomorphic maps $f_1,f_2,f_3,\ldots$ of the unit disk $\Delta$to a subdomain $X$, we consider the compositions

The sequence $\{F_n\}$ is called the iterated function system coming from the sequence $f_1,f_2,f_3,\ldots.$ 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 that the condition is a quasiconformal invariant. Finally, we address the question of uniqueness of limit functions.