Random Iteration of Rational Functions

It is a theorem of Denker and Urba'nski ('91) that if $T:\mathbb C\to\mathbb C$isarationalmapofdegreeatleasttwoandif$\phi:\mathbb C\to\mathbb R$isH\ddot{o}ldercontinuousandsatisfiesthe\mathrm{"}thermodynamicexpanding\mathrm{"}condition$P(T,\phi) > \sup(\phi)$,thenthereexistsexactlyoneequilibriumstate$\mu$for$T$and$\phi$,andfurthermore$(\mathbb C,T,\mu)$ 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 $(T,\mathrm{\Omega },\mathbf{\text{P}},\theta )$ is a holomorphic random dynamical system on $\mathbb{C}$ and $\varphi :\mathrm{\Omega }\to H_{\alpha }(\mathbb{C})$ 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 $(\mathbb{X},\mathbb{T},\varphi )$ over $(\mathrm{\Omega },\mathbf{\text{P}},\theta )$. Also included is a general (non-thermodynamic) discussion of random dynamical systems acting on $\mathbb{C}$, generalizing several basic results from the deterministic case.