Fatou Components of Attracting Skew Products: From Polynomial Classification to Transcendental Complexity

Abstract

We study the Fatou components of holomorphic skew products $F \colon \mathbb{C}^2 \to \mathbb{C}^2, F(z,w) = (f(z) + r(z,w), g(w))$ near an attracting invariant fiber, introducing a refined classification of components on the fiber: bulging, partial bulging, and nowhere bulging. For polynomial skew products, we give a complete characterization of when the basin at infinity is non-bulging, and show that in this case non-bulging implies nowhere bulging, ruling out partial bulging entirely. In the transcendental setting, the behavior is richer: we construct partially bulging wandering domains and Baker domains, show that any orbitally unbounded Fatou component can be made nowhere bulging by choosing an appropriate function $r$, and prove that for any nonconstant $r$ there exists an entire function $f$ with an escaping wandering domain that fails to be bulging. Bulging thus depends on the interplay between fiber dynamics and perturbation.