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Orbits Inside Basins of Attraction of Skew Products

John Erik Fornaess, Mi Hu

TL;DR

This paper addresses how orbits in attracting basins of holomorphic maps behave with respect to backward dynamics, asking whether a uniform Kobayashi-distance bound to a preimage orbit landing at the fixed point exists. It introduces a broad class $\mathcal{A}$ of polynomial skew products in which the Kobayashi geometry of the basin can be controlled, via a technical condition $\mathcal{C}$ and slice-wise analysis, to guarantee a uniform bound: for every $(z,w)$ in the basin $\Omega$, there is a point in the same Fatou component within distance $C$ that maps to the attracting fixed point after finitely many steps. This extends one-dimensional results to a large family of two-dimensional skew products, while acknowledging counterexamples outside $\mathcal{A}$. The paper also provides a concrete explicit example showing $\mathcal{A}$ is nonempty, illustrating the applicability of the main theorem.

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.

Orbits Inside Basins of Attraction of Skew Products

TL;DR

This paper addresses how orbits in attracting basins of holomorphic maps behave with respect to backward dynamics, asking whether a uniform Kobayashi-distance bound to a preimage orbit landing at the fixed point exists. It introduces a broad class of polynomial skew products in which the Kobayashi geometry of the basin can be controlled, via a technical condition and slice-wise analysis, to guarantee a uniform bound: for every in the basin , there is a point in the same Fatou component within distance that maps to the attracting fixed point after finitely many steps. This extends one-dimensional results to a large family of two-dimensional skew products, while acknowledging counterexamples outside . The paper also provides a concrete explicit example showing is nonempty, illustrating the applicability of the main theorem.

Abstract

A basic problem in complex dynamics is to understand orbits of holomorphic maps. One problem is to understand the collection of points 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 , there is a constant so that all Kobayashi discs of radius must intersect this set . In the paper [15], the second author showed that there are holomorphic skew products in 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.
Paper Structure (6 sections, 10 theorems, 5 equations)

This paper contains 6 sections, 10 theorems, 5 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Kobayashi metric
  4. A class $\mathcal{A}$ of skew products
  5. The dynamics of skew products of class $\mathcal{A}$
  6. An example where the Theorem applies

Key Result

Theorem 3.1

Suppose that $F\in \mathcal{A}$. Then there exists a constant $C$ such that for every point $(z,w)\in \Omega$ there is a point $(z_0,w_0)$ in the same Fatou component $\Omega^j$ as $(z,w)$ and an integer $N$ such that the Kobayashi distance between $(z,w)$ and $(z_0,w_0)$ in $\Omega^j$ is at most $C

Theorems & Definitions (20)

  • Definition 2.1
  • Definition 2.2
  • Theorem 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • Lemma 3.5
  • Lemma 3.6
  • ...and 10 more