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Non-bulging Fatou components for transcendental skew-products

Tom Potthink, Jasmin Raissy

Abstract

In this paper, we investigate the bulging of escaping or oscillating Fatou components on invariant fibers for general skew-products, with a focus on the dependence on the perturbation. We show that any orbitally unbounded component is non-bulging for an appropriate choice of perturbation, whereas sufficiently well-behaved perturbations can render it bulging when the fiber is attracting. Our results highlight that bulging is influenced by more than just the dynamics on the fiber and in the one-dimensional coordinate, contrasting sharply with established results for non-escaping Fatou components.

Non-bulging Fatou components for transcendental skew-products

Abstract

In this paper, we investigate the bulging of escaping or oscillating Fatou components on invariant fibers for general skew-products, with a focus on the dependence on the perturbation. We show that any orbitally unbounded component is non-bulging for an appropriate choice of perturbation, whereas sufficiently well-behaved perturbations can render it bulging when the fiber is attracting. Our results highlight that bulging is influenced by more than just the dynamics on the fiber and in the one-dimensional coordinate, contrasting sharply with established results for non-escaping Fatou components.

Paper Structure

This paper contains 4 sections, 12 theorems, 57 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Non-bulging Fatou Components
  4. Bulging Wandering Domains

Key Result

Theorem 1.1

Let $f$ and $g$ be nonconstant entire functions with $g(0) = 0$. Furthermore, assume that there exists $z_0 \in \mathbb{C}$ with $f^{n_k}(z_0) \to \infty$ as $k\to\infty$ for some subsequence of iterates. Then there exists $h$ entire such that the point $(z_0, 0)$ belongs to the Julia set of the ske

Figures (1)

  • Figure 1: The first sets of the construction if $n_0 = 0$ and $n_1 = 4$.

Theorems & Definitions (27)

  • Theorem 1.1
  • Corollary 1.2
  • Lemma 2.1: Baker1984
  • Definition 2.2: wu1967normal
  • Definition 2.3: Arosio2019
  • Definition 2.4
  • Definition 2.5
  • Lemma 2.6
  • Lemma 2.7: Arosio2019
  • Definition 2.8
  • ...and 17 more