Random Holomorphic Iterations and Degenerate Subdomains of the Unit Disk
نویسندگان
چکیده
Given a random sequence of holomorphic maps f1, f2, f3, . . . of the unit disk ∆ to a subdomain X, we consider the compositions Fn = f1 ◦ f2 ◦ . . . fn−1 ◦ fn. The sequence {Fn} is called the iterated function system coming from the sequence f1, f2, f3, . . . . We prove that a sufficient condition on the domain X for all limit functions of any {Fn} to be constant is also necessary. We prove the condition is a quasiconformal invariant. Finally, we address the question of uniqueness of limit functions.
منابع مشابه
Random holomorphic iterations and degenerate subdomains
We first study necessary and sufficient conditions for subdomains of the unit disk to be degenerate under random holomorphic iterations; that is, any random composition of holomorpic maps from the domain to the subdomain has only constant functions as accumulation points. Sufficient conditions based on properties of the hyperbolic metric were known for a subdomain of the unit disk to be degener...
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