Random holomorphic iterations and degenerate subdomains of the unit disk

Details Random holomorphic iterations and degenerate subdomains of the unit disk Random holomorphic iterations and degenerate subdomains of the unit disk

2005-03-04

arxiv.org/abs/math/0503079v1

Mathematics

Complex Variables

8 Pages To appear PAMS

Scientific paper

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.

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