Random holomorphic iterations and degenerate subdomains of the unit disk

Keen, Linda; Lakic, Nikola

doi:10.1090/S0002-9939-05-08280-8

Random holomorphic iterations and degenerate subdomains of the unit disk
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by Linda Keen and Nikola Lakic PDF

Proc. Amer. Math. Soc. 134 (2006), 371-378 Request permission

Abstract:

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 \[ F_n=f_1 \circ f_{2} \circ \ldots f_{n-1} \circ f_n.\] 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.

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