Introduction to Fatou components in holomorphic dynamics

TL;DR

This survey introduces the classification of Fatou components in holomorphic dynamics, covering rational maps and recent advances in polynomial skew-products in complex dimension two, including wandering domains.

Contribution

It provides a comprehensive overview of Fatou components, highlighting recent results on wandering domains for polynomial endomorphisms in two complex dimensions.

Findings

Classification of Fatou components for rational maps

Recent results on wandering domains in complex dimension two

Key steps in constructing wandering domains for polynomial endomorphisms

Abstract

This survey is an introduction to the classification of Fatou components in holomorphic dynamics. We start with the description of the Fatou and Julia sets for rational maps of the Riemann sphere, and finish with an account of the recent results on Fatou components for polynomial skew-products in complex dimension two, where we focus on the key steps in the construction giving the existence of a wandering domain for a polynomial endomorphism of $\mathbb{C}^2$.