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2408.01353

Finite Dynamical Laminations

Forrest M. Hilton

correctmedium confidence
Category
math.DS
Journal tier
Specialist/Solid
Processed
Sep 28, 2025, 12:56 AM

Audit review

The paper’s Theorem 6.3 states precisely the target claim and proves it by: (i) showing the base laminations yield an invariant laminational equivalence via Theorems 5.11 and 5.12; (ii) ruling out rotation curves using Lemma 6.2 together with the Siegel-gap classification; and (iii) invoking Kiwi’s realization theorem to obtain a polynomial with connected Julia set. It then upgrades ‘impressions intersect’ to ‘rays land exactly as prescribed’ using standard landing results for rational rays and Kiwi’s Corollary 1.2, thereby concluding that each non-singular FDL class is realized as an exact co-landing set . The candidate solution follows the same strategy: construct a rational laminational equivalence from the FDL, apply Kiwi’s realization, and use the landing theorems for rational rays to conclude exact co-landing. Differences are largely presentational: the model cites Kiwi’s earlier preprint and Milnor, and adds a (plausible but unneeded) parameter refinement to choose a Misiurewicz map; the paper cites Kiwi (2004) and Douady–Hubbard. Minor issues: the model informally says a leaf’s forward orbit is finite (read: its image set is finite); the paper’s Lemma 6.2 proof is terse. Neither issue alters the main conclusion.

Referee report (LaTeX)

\textbf{Recommendation:} minor revisions

\textbf{Journal Tier:} specialist/solid

\textbf{Justification:}

The manuscript ties finite dynamical laminations to realizable polynomial dynamics using a coherent framework (pullback trees, FDLs) and established results (Kiwi’s realization). The main theorem mirrors known principles but is useful and well-motivated. Some arguments are terse—most notably Lemma 6.2 and the NR exclusion—and should be expanded or more precisely referenced. With minor clarifications, the work will be a solid contribution for readers working with laminations and parameter space.