Random local complex dynamics

The study of the dynamics of an holomorphic map near a fixed point is a central topic in complex dynamical systems. In this paper, we will consider the corresponding random setting: given a probability measure with compact support on the space of germs of holomorphic maps fixing the origin, we study the compositions, where each is chosen independently with probability. As in the deterministic case, the stability of the family of the random iterates is mostly determined by the linear part of the germs in the support of the measure. A particularly interesting case occurs when all Lyapunov exponents vanish, in which case stability implies simultaneous linearizability of all germs in.