Fatou Components with Punctured Limit Sets

TL;DR

This paper demonstrates that punctured disks can serve as limit sets for non-recurrent Fatou components in holomorphic endomorphisms of ^2, expanding the known classifications of such dynamical systems.

Contribution

It proves the existence of non-recurrent Fatou components with punctured disk limit sets, filling a gap in the classification of invariant Fatou components.

Findings

Punctured disks can occur as limit sets in non-recurrent Fatou components.

Constructs examples of holomorphic and polynomial endomorphisms with specific Fatou component behaviors.

Shows convergence of orbits to the regular parts of arbitrary analytic sets.

Abstract

We study invariant Fatou components for holomorphic endomorphisms in $\mathbb{P}^2$. In the recurrent case these components were classified by Sibony and the second author in 1995. In 2008 Ueda completed this classification by proving that it is not possible for the limit set to be a punctured disk. Recently Lyubich and the third author classified non-recurrent invariant Fatou components, under the additional hypothesis that the limit set is unique. Again all possibilities in this classification were known to occur, except for the punctured disk. Here we show that the punctured disk can indeed occur as the limit set of a non-recurrent Fatou component. We provide many additional examples of holomorphic and polynomial endomorphisms of $\mathbb{C}^2$ with non-recurrent Fatou components on which the orbits converge to the regular part of arbitrary analytic sets.