Random Iteration of Rational Functions

Abstract

It is a theorem of Denker and Urba\'nski ('91) that if $T:\mathbb C\to\mathbb C$ is a rational map of degree at least two and if $\phi:\mathbb C\to\mathbb R$ is H\"older continuous and satisfies the "thermodynamic expanding" condition $P(T,\phi) > \sup(\phi)$, then there exists exactly one equilibrium state $\mu$ for $T$ and $\phi$, 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 Bogensch\"utz ('92/'93). Specifically, if $(T,\Omega,\textbf P,\theta)$ is a holomorphic random dynamical system on $\mathbb C$ and $\phi:\Omega\to H_\alpha(\mathbb C)$ is a H\"older continuous random potential function satisfying one of several sets of technical but reasonable hypotheses, then there exists a unique equilibrium state of $(\mathbb X,\mathbb T,\phi)$ over $(\Omega,\textbf P,\theta)$. Also included is a general (non-thermodynamic) discussion of random dynamical systems acting on $\mathbb C$, generalizing several basic results from the deterministic case.