# 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.

## Key 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 $supp(\nu)$.