A two-dimensional polynomial mapping  with a wandering Fatou component

Matthieu Astorg; Xavier Buff; Romain Dujardin; Han Peters; Jasmin Raissy

doi:https://doi.org/10.4007/annals.2016.184.1.2

A two-dimensional polynomial mapping with a wandering Fatou component

Pages 263-313 from Volume 184 (2016), Issue 1 by Matthieu Astorg, Xavier Buff, Romain Dujardin, Han Peters, Jasmin Raissy

Abstract

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

Keywords

Fatou set, holomorphic dynamics, parabolic implosion, polynomial mappings, skew-products, wandering Fatou components

Mathematical Subject Classification

Primary: 37F10, 37F50

DOI

https://doi.org/10.4007/annals.2016.184.1.2

3505180

zbMATH

06605832

Milestones

Received: 16 December 2014
Revised: 25 September 2015
Accepted: 23 October 2015
Published online: 1 July 2016

Authors

Matthieu Astorg

Institut de Mathématiques de Toulouse, Université de Toulouse, CNRS, Toulouse, France

Xavier Buff

Institut de Mathématiques de Toulouse, Université de Toulouse, CNRS, Toulouse, France

Romain Dujardin

Laboratoire d'Analyse et de Mathématiques Appliquées, Université Paris-Est Marne-la-Vallée, Champs-sur-Marne, France

Han Peters

KdV Institute for Mathematics University of Amsterdam, The Netherlands

Jasmin Raissy

Institut de Mathématiques de Toulouse, Université de Toulouse, CNRS, Toulouse, France