Random Iteration of Rational Functions

TL;DR

This paper extends classical results on equilibrium states for rational maps to the setting of holomorphic random dynamical systems on the complex plane, establishing existence and uniqueness under certain conditions.

Contribution

It generalizes deterministic thermodynamic formalism results to random systems on or the first time, using relative pressure and entropy concepts.

Findings

Existence and uniqueness of equilibrium states for holomorphic random systems.

Extension of exactness results to random dynamical systems.

General discussion on random systems acting on or broader applicability.

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.