Random local complex dynamics

TL;DR

This paper explores the behavior of random compositions of holomorphic maps near a fixed point, revealing how stability and linearizability depend on the linear parts of the maps and the Lyapunov indices.

Contribution

It extends complex dynamical systems theory to a random setting, analyzing stability and linearizability of compositions of holomorphic germs.

Findings

Stability is mainly determined by the linear parts of the germs.

Vanishing Lyapunov indices imply simultaneous linearizability.

The study provides insights into the stability of random holomorphic dynamical systems.

Abstract

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 $\nu$ with compact support on the space of germs of holomorphic maps fixing the origin, we study the compositions $f_n\circ\cdots\circ f_1$, where each $f_i$ is chosen independently with probability $\nu$. 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 indices vanish, in which case stability implies simultaneous linearizability of all germs in $supp(\nu)$.