Fatou Components of Attracting Skew Products: From Polynomial Classification to Transcendental Complexity
Abstract
We study the Fatou components of holomorphic skew products F : C 2 → 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.
Related
- Superintegrability of a resonant ghostly tri-Hamiltonian model: polynomial first integrals and transcendental completion
- Why Shallow Neural Networks Learn Sinusoidal Components First: An Orthogonal Polynomial Dynamics Framework
- Complexity of finding color-avoiding connected components
- Classification of pairs of second-order Hamiltonian operators and hydrodynamic type systems in six components
- Analysis of Polynomial Threshold Functions on Random Regular Graphs: Computational Complexity of Detecting Noisy Random Lifts
BibTeX
@misc{potthink2026fatou,
title = {{Fatou Components of Attracting Skew Products: From Polynomial Classification to Transcendental Complexity}},
author = {Tom Potthink},
howpublished = {HAL},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.16583},
eprint = {2610.16583},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05799718v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Fatou Components of Attracting Skew Products: From Polynomial Classification to Transcendental Complexity
AU - Tom Potthink
PY - 2026
DA - 2026/10/12
PB - HAL
DO - 10.48550/arxiv.2610.16583
UR - https://hal.science/hal-05799718v1
AB - We study the Fatou components of holomorphic skew products F : C 2 → 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.
ER -