2501.07484
Hyperbolic components and iterated monodromy of polynomial skew-products of C2
Virgile Tapiero
correcthigh confidence
- Category
- math.DS
- Journal tier
- Specialist/Solid
- Processed
- Sep 28, 2025, 12:56 AM
- arXiv Links
- Abstract ↗PDF ↗
Audit review
The paper’s Theorem 1.2 and Corollary 1.3 are correctly proved via admissible parameters, iterated monodromy, and structural stability, and they allow connected components of J(f_λ) to wind with degree ≥1 over S^1. The candidate solution incorrectly claims each component projects with degree 1 by imposing a global inverse-branch labeling and a global graph transform over S^1, which fails in presence of nontrivial monodromy; the paper’s Example 1.6 exhibits a single loop winding three times, contradicting the model’s stronger claim.
Referee report (LaTeX)
\textbf{Recommendation:} minor revisions
\textbf{Journal Tier:} specialist/solid
\textbf{Justification:}
The paper extends the quadratic classification of hyperbolic components in skew-product families to higher degrees by introducing a natural invariant (an algebraic braid) attached to each unbounded component Λ⊂D′ and proving the surjectivity of the induced map to conjugacy classes in Sd. The arguments are sound, combining admissibility and contraction in Poincaré metric with an iterated monodromy analysis and structural stability. Exposition is generally clear; a few clarifications (e.g. early intuition for admissibility and examples illustrating winding number) would improve accessibility.