Non-bulging Baker domains for transcendental skew products

TL;DR

The paper analyzes Baker domains in transcendental skew products $F(z,w)=(f(z,w), g(w))$, focusing on how higher-order perturbations in the first coordinate influence bulging of these domains. It develops a bulging criterion based on the order $\rho_1(h)$ of the perturbation and constructs both bulging and non-bulging examples: bulging occurs when $\rho_1(h)<1$ (or under a finite $\rho_1(h)$ with a super-attracting $g$), while a non-bulging Baker domain is engineered via a Runge-based approximation of the perturbation $h$. This extends Lilov’s polynomial-skew product results to the transcendental setting, revealing that nontrivial first-coordinate terms can decisively alter global Fatou components. The results deepen the understanding of how cross-fiber dynamics govern two-dimensional Fatou components and provide constructive methods to realize both bulging and non-bulging Baker domains.

Abstract

In this paper we show that Baker domains of transcendental skew products can either bulge or not, depending on the higher order terms. This is in contrast to polynomial skew products where all Fatou components with bounded orbits of an invariant attracting fiber do bulge.

Figures (1)

• Figure 1: The first disks $D_k$ together with $H_2$.

Theorems & Definitions (31)

• Definition 1.1: Bulging
• Theorem 1.2: Bulging for polynomial skew products, Lilov
• Theorem 1.3: Lilov
• Definition 2.1
• Lemma 2.2: ABFP2019
• Definition 2.3
• Lemma 2.4
• Lemma 2.5
• proof
• Proposition 3.1