Classification of generic random holomorphic dynamical systems associated with analytic families of rational maps (Research on the Theory of Random Dynamical Systems and Fractal Geometry)

Abstract

(1) We introduce the notion of weak mean stability in i.i.d. random (holomorphic) 1-dimensional dynamical systems. (2) If a random holomorphic dynamical system on ℂ^ is weakly mean stable and satisfies some mild assumtions, then for all but countably many z ∈ ℂ^, for a.e. orbit starting with z, the Lyapunov exponent is negative. Note that this statemeりtcannot hold for deterministic dynamics of a single holo. map f on ℂ^ with deg(f)≥2. (3) Given an analytic family Y of ratio叫maps(with some mild conditions), generic random holomorphic dynamical systems (with multiplicative noise) of elements of Y are weakly mean stable. Also, we can classify such generic random holomorphic dynamical systems of elements of Y in terms of averaged behavior and quenched dynamics.