Orbits Inside Basins of Attraction of Skew Products

John Erik Fornæss, Mi Hu

correcthigh confidence

Category: Not specified
Journal tier: Specialist/Solid
Processed: Sep 28, 2025, 12:56 AM
arXiv Links: Abstract ↗PDF ↗

Audit review

The paper’s main theorem (Theorem 3.1) explicitly proves the uniform “C and N” statement for class A skew products using a careful analysis of slice maps, branched coverings, and the constructed graphs of preimages of the strong stable manifold, via Lemmas 3.2–3.8 and a final global argument . By contrast, the candidate solution hinges on the claim that (0,0) is not a critical value of F^n for any n, so that one can choose a small polydisk avoiding all critical values and pull back biholomorphically; but the paper’s framework explicitly involves branched covers between slices (degree d, with branch points in a compact set), so critical values can occur and must be handled (not avoided) . The model’s proof thus omits essential complications addressed by the paper.

Referee report (LaTeX)

\textbf{Recommendation:} no revision

\textbf{Journal Tier:} specialist/solid

\textbf{Justification:}

The manuscript presents a clean, self-contained criterion (class A with Condition C) that ensures a uniform Kobayashi bound to points whose forward iterates land exactly at the attracting fixed point in C\^2 skew products. The proof is careful about branched coverings between slices and builds a robust global argument via preimage graphs and an exhaustion, avoiding overly delicate metric estimates. The result positively resolves a natural two-dimensional analogue of a one-variable phenomenon for a broad class of maps, while clarifying why known counterexamples lie outside the class.