A GENERAL DYNAMICAL THEORY OF SCHWARZ REFLECTIONS, B-INVOLUTIONS, AND ALGEBRAIC CORRESPONDENCES

Yusheng Luo, Mikhail Lyubich, Sabyasachi Mukherjee

correctmedium confidence

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

Audit review

The paper proves existence, uniqueness (up to Möbius), and combinatorial mating statements for degree-d anti-polynomials with the Nielsen map of the ideal (d+1)-gon reflection group, exactly as asserted, via a framework of polygonal Schwarz reflections, degenerate anti-polynomial-like restrictions, David surgery, and parameter-slice straightening. The candidate solution mirrors this structure (circle conjugacy E_d, topological mating, realization/straightening to a global Schwarz reflection, uniqueness, and lamination push-forward) and reaches the same conclusions. Minor issues in the candidate outline (unnecessary reliance on local connectivity in general and a unicritical citation) do not alter the final claims, which coincide with the paper’s theorems.

Referee report (LaTeX)

\textbf{Recommendation:} minor revisions

\textbf{Journal Tier:} strong field

\textbf{Justification:}

The manuscript establishes a robust, general framework linking polygonal Schwarz reflections and matings of anti-polynomials with ideal polygon reflection groups, extending and unifying prior special cases. The existence, uniqueness, and combinatorial mating results in key dynamical classes appear correct and significant, with David surgery and conformal removability providing the needed analytic backbone. Minor revisions would improve clarity regarding analytic prerequisites and the relationship between the degenerate-slice straightening perspective and the explicit surgery constructions.