title: Random Iteration of Rational Functions
creator: Simmons, David
contributor: ski, Mariusz Urba
contributor: Cherry, William
contributor: Fishman, Lior
publisher: University of North Texas
date: 2012-05
language: English
description: It is a theorem of Denker and Urbański that if T:ℂ→ℂ is a rational
map of degree at least two and if ϕ:ℂ→ℝ is Hölder continuous and satisfies the
“thermodynamic expanding” condition P(T,ϕ) > sup(ϕ), then there exists exactly
one equilibrium state μ for T and ϕ, and furthermore (ℂ,T,μ) is metrically
exact. We extend these results to the case of a holomorphic random dynamical
system on ℂ, using the concepts of relative pressure and relative entropy of
such a system, and the variational principle of Bogenschütz. Specifically, if
(T,Ω,P,θ) is a holomorphic random dynamical system on ℂ and ϕ:Ω→ ℋα(ℂ) is a
Hölder continuous random potential function satisfying one of several sets of
technical but reasonable hypotheses, then there exists a unique equilibrium
state of (X,P,ϕ) over (Ω,Ρ,θ).
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