# Random local dynamics

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Lorenzo Guerini

created by daniele on 23 Jan 2018

30 jan 2018
-- 16:00

Aula D'Antoni, Università di Roma Tor Vergata

**Abstract.**

The study of the dynamics of an holomorphic map near a fixed point is a central subject in complex dynamics. In this talk we will consider the corresponding random setting: given a probability measure $\mu$ with compact support on the space of germs of holomorphic maps fixing the origin, we study the iterates $f_n\circ\cdots\circ f_1$, where each $f_i$ is chosen with probability $\mu$. We will see, as in the non-random case, that the stability of the family of the random iterates can be studied by looking at the linear part of the germs in the support of the measure and, in particular, at some quantities commonly known as Lyapunov indexes. A particularly interesting case occurs when all Lyapunov indexes vanish. When this happens stability is equivalent to simultaneous linearizability of all germs in $supp(\mu)$.