Orbits Inside Basins of Attraction of Skew Products

TL;DR

This paper investigates the behavior of orbits within basins of attraction for polynomial skew products in complex dynamics, extending previous results to a broader class of such maps.

Contribution

It demonstrates that a key property regarding Kobayashi discs and the set of points landing on the fixed point holds for many polynomial skew products, generalizing earlier findings.

Findings

Kobayashi discs of bounded radius intersect the set S in many skew products.

The property fails for some skew products but holds for a large class.

Extends understanding of orbit structure in complex dynamical systems.

Abstract

A basic problem in complex dynamics is to understand orbits of holomorphic maps. One problem is to understand the collection of points $S$ in an attracting basin whose forward orbits land exactly on the attracting fixed point. In the paper [13], the second author showed that for holomorphic polynomials in $\mathbb C$, there is a constant $C$ so that all Kobayashi discs of radius $C$ must intersect this set $S$. In the paper [15], the second author showed that there are holomorphic skew products in $\mathbb {C}^2$ where this result fails. The main result of this paper is to show that for a large class of polynomial skew products, this result nevertheless holds.