THERE ARE NOT MANY PERIODIC ORBITS IN BUNCHES FOR ITERATION OF COMPLEX QUADRATIC POLYNOMIALS OF ONE VARIABLE

Feliks Przytycki

correctmedium confidence

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

Audit review

The paper’s Theorems 1.5 and 1.6 are proved with a complete, self-contained argument that avoids the unproven “pairwise unlinking” property of orbit portraits and instead relies on distortion control, ‘no bottom returns’, preservation of cyclic order of truncated external rays, and a direct counting of angle-arc trajectories; it also acknowledges and fixes an earlier gap. By contrast, the model’s solution crucially invokes the pairwise-unlinked orbit-portrait property to deduce uniqueness, a step the paper explicitly states cannot be justified in this setting. The model also asserts univalence of f^k on a disc from the mere exclusion of the critical point 0, which does not rule out precritical points of f^k. Hence the model’s proof is incomplete/incorrect, while the paper’s proof is correct.

Referee report (LaTeX)

\textbf{Recommendation:} minor revisions

\textbf{Journal Tier:} strong field

\textbf{Justification:}

The paper establishes sharp bounds on the number of periodic orbits near Cremer points for quadratic polynomials, with techniques that are robust and carefully justified. It recognizes and corrects a gap in an earlier version by avoiding the need for a difficult unlinking property. The exposition is generally clear and logically structured. A few technical steps could be marginally expanded for readability, hence minor revisions.