Random Iteration of Rational Functions

It is a theorem of Denker and Urbański ('91) that if $T:\mathbb C\to\mathbb C$ is a rational map of degree at least two and if $ϕ:\mathbb C\to\mathbb R$ 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 $(\mathbb C,T,μ)$ is metrically exact. We extend these results to the case of a holomorphic random dynamical system on $\mathbb C$, using the concepts of relative pressure and relative entropy of such a system, and the variational principle of Bogenschütz ('92/'93). Specifically, if $(T,Ω,\textbf P,θ)$ is a holomorphic random dynamical system on $\mathbb C$ and $ϕ:Ω\to H_α(\mathbb C)$ 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 $(\mathbb X,\mathbb T,ϕ)$ over $(Ω,\textbf P,θ)$. Also included is a general (non-thermodynamic) discussion of random dynamical systems acting on $\mathbb C$, generalizing several basic results from the deterministic case.

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