Complex dynamics of Markov systems of families of rational maps (Integrated Research on the Theory of Random Dynamical Systems)

Abstract

We consider the random holomorphic dynamical systems on the Riemann sphere whose choices of maps are related to Markov chains. Our motivation is to generalize the facts which hold in i.i. d. random holomorphic dynamical systems. Especially, we focus on the function Gamma_{infty} which represents the probability of tending to infinity. We show some sufficient conditions which make Gamma_{infty} continuous on the whole space and we characterize the Julia sets in terms of the function Gamma_{infty} under certain assumptions.