Introduction to Fatou components in holomorphic dynamics

2023/02/06 by Buff, Xavier, Jasmin Raissy, Raissy, Jasmin
Mathematics · Physics and Astronomy · #37F10 #37F80 #Advanced Differential Equations and Dynamical Systems #Complex Variables (math.CV) #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Quantum chaos and dynamical systems

paper · pdf · doi:10.48550/arxiv.2302.02669

openalex publication_date 2023/02/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 ℂ².

Related

• A two-dimensional polynomial mapping with a wandering Fatou component
• The wandering domain problem for attracting polynomial skew products
• Fatou components of elliptic polynomial skew products
• A polynomial skew-product with a wandering Fatou-disk
• Fatou components of attracting skew products