{
"contributor": [
"ski, Mariusz Urba",
"Cherry, William",
"Fishman, Lior"
],
"creator": [
"Simmons, David"
],
"date": [
"2012-05"
],
"description": [
"It is a theorem of Denker and Urba\u0144ski that if T:\u2102\u2192\u2102 is a rational map of degree at least two and if \u03d5:\u2102\u2192\u211d is H\u00f6lder continuous and satisfies the \u201cthermodynamic expanding\u201d condition P(T,\u03d5) > sup(\u03d5), then there exists exactly one equilibrium state \u03bc for T and \u03d5, and furthermore (\u2102,T,\u03bc) is metrically exact. We extend these results to the case of a holomorphic random dynamical system on \u2102, using the concepts of relative pressure and relative entropy of such a system, and the variational principle of Bogensch\u00fctz. Specifically, if (T,\u03a9,P,\u03b8) is a holomorphic random dynamical system on \u2102 and \u03d5:\u03a9\u2192 \u210b\u03b1(\u2102) is a H\u00f6lder continuous random potential function satisfying one of several sets of technical but reasonable hypotheses,