A two-dimensional polynomial mapping with a wandering Fatou component

TL;DR

This paper constructs polynomial endomorphisms of complex and real two-dimensional spaces that have wandering Fatou components, using parabolic implosion techniques and an innovative approach inspired by Lyubich.

Contribution

It demonstrates the existence of wandering Fatou components in polynomial skew-products of C^2 and extends these examples to real spaces, introducing new methods in complex dynamics.

Findings

Existence of wandering Fatou components in polynomial endomorphisms of C^2.

Construction of real examples with wandering domains in R^2.

Application of parabolic implosion techniques and Lyubich's ideas.

Abstract

We show that there exist polynomial endomorphisms of C^2, possessing a wandering Fatou component. These mappings are polynomial skew-products, and can be chosen to extend holomorphically of P^2(C). We also find real examples with wandering domains in R^2. The proof is based on parabolic implosion techniques, and is based on an original idea of M. Lyubich.