Random holomorphic iterations and degenerate subdomains of the unit disk

2005/03/04 by Linda Keen, Keen, Linda, Nikola Lakic +1
Mathematics · #30C70 #Complex Variables (math.CV) #Dynamical Systems (math.DS) #FOS: Mathematics #Primary 32G15 #Secondary 30C60 #math.CV #math.DS #msc:30C60 #msc:30C70 #msc:32G15

paper · pdf · doi:10.48550/arxiv.math/0503079

8 Pages To appear PAMS

arxiv created 2005/03/04 · arxiv updated 2009/12/01

Abstract

Given a random sequence of holomorphic maps f₁,f₂,f₃,... of the unit disk Δ to a subdomain X, we consider the compositions F_n=f₁ ∘ f₂ ∘ ... f_n-1 ∘ f_n. The sequence \F_n\ is called the \em 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 \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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