Bicritical Rational Maps with a Common Iterate

Sarah Koch, Kathryn Lindsey, Thomas Sharland

correcthigh confidence

Category: math.DS
Journal tier: Strong Field
Processed: Sep 28, 2025, 12:56 AM
arXiv Links: Abstract ↗PDF ↗

Audit review

The uploaded paper proves (i) if bicritical f,g share an iterate then Cf=Cg and Vf=Vg (Theorem 1.1), and (ii) for even degree and non-power maps, sharing any iterate forces f2=g2 and yields an involution μ with g=μ∘f=f∘μ (Theorem 1.2) . The candidate solution reproduces the same deck-group strategy to detect Cf and Vf (Theorem 4.2 and quadratic refinements) and then derives μ and f2=g2 using the same lemmas in Section 8 . Apart from minor phrasing (e.g., using order ≥ d where the paper uses order ≥ 3 for d≥3), the model’s proof is a condensed version of the paper’s argument.

Referee report (LaTeX)

\textbf{Recommendation:} minor revisions

\textbf{Journal Tier:} strong field

\textbf{Justification:}

The manuscript develops a cohesive deck-group approach to determine when two bicritical rational maps that share an iterate must coincide on critical sets and, in even degree, share the second iterate. The main results are natural, the methods are well motivated, and the paper clarifies the geometry behind symmetry loci and common-iterate phenomena. The quadratic case is treated carefully, though some sections are dense and could be made more reader-friendly. I view the work as a solid contribution to complex dynamics and iteration theory.