PAPER FORTUNE TELLERS IN THE COMBINATORIAL DYNAMICS OF SOME GENERALIZED MCMULLEN MAPS WITH BOTH CRITICAL ORBITS BOUNDED

Suzanne Boyd, Kelsey Brouwer

correctmedium 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 3.1 constructs a surjective relation Γ: S1 → J* by (i) transporting the quadratic Carathéodory landing map to J+ via the straightening conjugacy and (ii) inductively pulling angles back on each preimage component, using angle-doubling exactly when the component contains a critical point (2-to-1) and the identity otherwise (1-to-1) . The candidate solution follows the same two-case inductive scheme (Case A/B), with the same base semiconjugacy and pullback rules, differing only in exposition (explicit derivative computation, orientation choices). One minor wording issue in the paper’s “Observation” swaps “critical points” for “critical values” but does not affect the theorem’s correctness .

Referee report (LaTeX)

\textbf{Recommendation:} minor revisions

\textbf{Journal Tier:} specialist/solid

\textbf{Justification:}

The main theorem’s construction is correct and useful: it cleanly organizes external angle assignments across the tree of preimages by combining the quadratic Carathéodory loop on J+ with a two-case pullback that mirrors the local degree (identity vs. doubling). The exposition is effective, with figures supporting the intuition. Minor issues remain: a wording slip conflating critical points with critical values, and a missing one-line justification that the restricted map on a branched preimage component has degree exactly 2. These can be fixed easily and do not affect the result.