Random Dynamical Systems of Regular Polynomial Maps on $mathbb{C}$$^{2}$ (Integrated Research on Random Dynamical Systems and Multi-Valued Dynamical Systems)

Abstract

We introduce the notion of mean stability in i.i.d. random (holomorphic) 2-dimensional dynamical systems. We can see that a generic random dynamical system of regular polynomial maps on ℙ² (the complex 2-dimensional projective space) having an attractor in the line at infinity, is mean stable. If a random holomorphic dynamical system on ℙ² is mean stable then for each z ∈ ℙ² for a.e. orbit starting with z, the Lyapunov exponent is negative. If a random holomorphic dynamical system on ℙ² is mean stable, then for any z ∈ ℙ², the orbit of the Dirac measure at z under the iterations of the dual map of the transition operator converges to a periodic cycle of probability measures. Note that the above statements cannot hold for deterministic dynamics of a single regular polynomial map f with deg(f) ≥ 2. We see many randomness-induced phenomena (phenomena in random dynamical systems which cannot hold for iteration dynamics of single maps). In this talk, we have seen randomness-induced order.