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Random holomorphic iterations and degenerate subdomains of the unit disk

Authors:	Linda Keen Nikola Lakic

Institution:	Department of Mathematics, Lehman College and Graduate Center, CUNY, Bronx, New York 10468 ; Department of Mathematics, Lehman College and Graduate Center, CUNY, Bronx, New York 10468

Abstract:	Given a random sequence of holomorphic maps $f_1,f_2,f_3,\ldots$ of the unit disk $\Delta$ to a subdomain , we consider the compositions $\begin{displaymath}F_n=f_1 \circ f_{2} \circ \ldots f_{n-1} \circ f_n.\end{displaymath}$ 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 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.

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