Denote by \(\mathcal{O}(\mathbb{C}^m,0)\) the space of all germs of holomorphic maps from \(\mathbb{C}^m\) to itself, fixing the origin. Let \(\nu\) be a Borel probability measure on \(\mathcal{O}(\mathbb{C}^m,0)\), with compact support. Let \(\{f_i\}_{i=1}^\infty\) be a sequence in \(\mathcal{O}(\mathbb{C}^m,0)\), chosen independently with probability \(\nu\). The random local complex dynamics in this paper is to study the compositions \(f_n\circ\cdots \circ f_1\).
In this random setting, the authors introduce the notion of Lyapunov exponents and classify the measure \(\nu\) into four classes: attracting, repelling, neutral and semi-attracting, according to the Lyapunov exponents. The first two cases are relatively easy. For the neutral case, the authors give a necessary and sufficient condition for the origin to lie in the random Fatou set. The precise statement is given in Theorem 1.1 and Theorem 3.3, and its proof is given in Subsection 4.2. The semi-attracting case is treated in Section 5.
Several examples are given to illustrate the results, and in the appendix the authors discuss the inductive limit topology on \(\mathcal{O}(\mathbb{C}^m,0)\).