Korteweg-de Vries Institute for Mathematics

Article of Han Peters in Annals of Mathematics

Wandering domains in 2 dimensions.

5 January 2016

On 23 October 2015 the article: 'A two-dimensional polynominal mapping with a wandering Fatou component' written by Han Peters was accepted for publication by the Annals of Mathematics. Co-authors are: Matthieu Astorg, Xavier Buff, Romain Dujardin, Jasmin Raissy.

Abstract of the Article

Stability under small perturbations is a central concept in dynamical systems. We say that an initial value is stable if orbits of sufficiently nearby initial values remain arbitrarily close for all time.

In the complex category the set of stable initial values is usually referred to as the Fatou set. For polynomial maps the Fatou set is open and dense, and connected components of the Fatou set are mapped onto connected components. The behavior on the so-called Fatou components therefore describes to a large extend the dynamical behavior of the map.

For rational functions in one complex variable it was shown by Dennis Sullivan [annals of mathematics, 1985] that every Fatou component is eventually mapped onto a periodic cycle. In other words, there are no wandering domains: Fatou components that never become periodic. Sullivan's Non Wandering Domains Theorem was later proved to also hold for polynomials on the real line, as well as for large classes of entire maps.

In higher dimensions the existence of wandering domains remained open until recently, when Han Peters, in collaboration with Matthieu Astorg, Xavier Buff, and Jasmin Raissy from Paul Sabatier University in Toulouse and Romain Dujardin from Marne la Vallee in Paris, constructed two-dimensional polynomial maps with wandering domains, both in the real and in the complex setting.

The construction is based on an idea due to Misha Lyubich from Stony Brook University. The two-dimensional map preserves a family of one-dimensional fibers, and hence gives rise to a one-parameter family of polynomials. This allows one-dimensional bifurcation techniques to be used to prove the existence of wandering domains. The paper of Astorg, Buff, Dujardin, Peters and Raissy will appear in the annals of mathematics.
 

Published by  Korteweg de Vries Institute