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2404.14614

Wandering domains for non-archimedean quadratic rational functions

Víctor Nopal-Coello

correcthigh confidence
Category
Not specified
Journal tier
Specialist/Solid
Processed
Sep 28, 2025, 12:56 AM

Audit review

The paper proves Theorem A—existence of wandering components for R(z) = (z^2 − z)/(bz − 1/a) over a complete, algebraically closed non-archimedean field of residual characteristic 2—via a carefully normalized quadratic family with |a|>1 and |b|=1, a precise local-dynamics trichotomy (Lemma 1.1), and a parameter-lifting construction using ΦxN(β) to realize a non-eventually-periodic itinerary; the argument culminates in Section 2.5 and is coherent and complete . By contrast, the model’s outline works in the incompatible regime |a|<1, asserts heuristic “three radial regimes” with thresholds that contradict the paper’s normalization (e.g., 1<|z|<1/(|a||b|) is empty when |a|>1, |b|=1), and defers to the paper for the hard parameter-coding step without supplying the necessary estimates. It also misplaces the critical-point geometry (the paper has |c1|=|c2|=1/√|a| under its normalization) and derivative control, which the paper handles via exact disk-mapping lemmas rather than the model’s non-expansion heuristic .

Referee report (LaTeX)

\textbf{Recommendation:} minor revisions

\textbf{Journal Tier:} specialist/solid

\textbf{Justification:}

The paper gives a careful, correct construction of wandering domains for a quadratic rational map in residual characteristic 2, filling a genuine gap beyond earlier polynomial examples. The parameter-lifting argument and disk-mapping estimates are solid. The exposition would benefit from small clarifications and a compact summary of the sets and radii used, but the core mathematics is sound and significant.