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2402.19256

Collet–Eckmann maps in the unicritical family

Magnus Aspenberg, Mats Bylund, Weiwei Cui

correcthigh confidence
Category
Not specified
Journal tier
Strong Field
Processed
Sep 28, 2025, 12:56 AM

Audit review

The paper proves the stated theorem using a Benedicks–Carleson style parameter exclusion plus a new “promotion” step to overcome deep returns, with precise transversality and distortion control; the argument is coherent and complete. By contrast, the model’s outline hinges on parapuzzle a priori bounds and a uniform ‘fixed-fraction’ area loss for child parapuzzles under mere CE. Those bounds are not justified at CE parameters without additional slow-recurrence/complex-bounds hypotheses, and the sketched transversality/area estimates are not established. Hence the model’s proof is incomplete for the general CE setting addressed by the paper.

Referee report (LaTeX)

\textbf{Recommendation:} no revision

\textbf{Journal Tier:} strong field

\textbf{Justification:}

The manuscript establishes full Lebesgue density of hyperbolic parameters near CE points in the unicritical family via a robust parameter-exclusion scheme enhanced by a novel promotion step. The approach overcomes deep-return obstacles without invoking parapuzzle complex bounds. Arguments are precise and modular, with clear intermediate lemmas and a clean final measure estimate.