Non-bulging Fatou components for transcendental skew-products

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.