Random Iteration of Rational Functions

Abstract

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 $\varphi :\mathbb{C}\to \mathbb{R}$ is Hölder continuous and satisfies the "thermodynamic expanding" condition $P(T,\varphi )>\sup (\varphi )$, then there exists exactly one equilibrium state $\mu$ for $T$ and $\varphi$, and furthermore $(\mathbb{C},T,\mu )$ 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 $(T,\Omega ,\text{P},\theta )$ is a holomorphic random dynamical system on $\mathbb{C}$ and $\varphi :\Omega \to H_{\alpha }(\mathbb{C})$ 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 $(\mathbb{X},\mathbb{T},\varphi )$ over $(\Omega ,\text{P},\theta )$. Also included is a general (non-thermodynamic) discussion of random dynamical systems acting on $\mathbb{C}$, generalizing several basic results from the deterministic case.