Non-bulging Fatou components for transcendental skew-products

TL;DR

This paper studies how perturbations affect the bulging behavior of Fatou components in transcendental skew-products, revealing that bulging depends on more than just fiber dynamics.

Contribution

It demonstrates that orbitally unbounded components can be non-bulging under certain perturbations, and well-behaved perturbations can induce bulging in attracting fibers.

Findings

Orbitally unbounded components are non-bulging with appropriate perturbations.

Well-behaved perturbations can cause bulging in attracting fibers.

Bulging depends on factors beyond fiber dynamics and one-dimensional coordinate behavior.

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.