IRRATIONAL FATOU COMPONENTS IN NON-ARCHIMEDEAN DYNAMICS

Juan Rivera-Letelier

correcthigh confidence

Category: Not specified
Journal tier: Strong Field
Processed: Sep 28, 2025, 12:56 AM
arXiv Links: Abstract ↗PDF ↗

Audit review

The paper’s Theorem A states exactly the target claim and proves it by computing diameters for a Benedetto-family wandering domain (Main Theorem) and then exhibiting a Cantor set of possible diameter exponents whose Q-span is all of R, guaranteeing a choice outside any proper divisible subgroup D ≥ |K*|; hence an irrational Fatou disk . The candidate solution reproduces this strategy: it invokes the same family, the itinerary-driven diameter computation, and the Cantor-set/Q-span argument to avoid D. Notational slips aside (using κ as a parameter name and informal Aκ/Bκ partition), the reasoning aligns with the paper’s proof and depends on the same main ingredients (existence of controlled itineraries and explicit diameter formula) .

Referee report (LaTeX)

\textbf{Recommendation:} minor revisions

\textbf{Journal Tier:} strong field

\textbf{Justification:}

This work gives the first example of an irrational Fatou component in the non-Archimedean setting, leveraging explicit computations in the Benedetto family and a clever Cantor-set argument to defeat any prescribed divisible subgroup containing the value group. The approach is technically refined yet conceptually clear, building on and extending well-established machinery for realizing itineraries and tracking diameters through wild ramification. The results resolve a natural gap in the geometric picture of wandering components. Minor expository tweaks would further aid readability, but the mathematical core appears sound and significant.