Weak Mean Stability in Random Holomorphic Dynamical Systems (Integrated Research on the Theory of Random Dynamical Systems)

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 hat{mathb{C} is weakly mean stable, then for any xinhat{mathbb{C}}, the orbit of the Dirac measure at x under the iterations of the dual map of the transition operator converges to a periodic cycle of probability measures. (3) If a random holomorphic dynamical system on hat{mathb{C} is weakly mean stable and satisfies some mild assumtions, then for all but countably many zinhat{mathbb{C}}, for 50 a.e. orbit starting with z, the Lyapunov exponent is negative. Note that the statements of (2) and (3) cannot hold for deterministic dynamics of a single rational map f with deg(f)geq 2. (4) In many holomorphic families of rational maps (including random relaxed Newton's methods family), generic random dynamical systems satisfy the statements of (2) and (3). We can apply this to random relaxed Newton's method to find a root of any polynomial.