LOCAL DISTRIBUTIONAL CHAOS

Francisco Balibrea, Lenka Rucká

correcthigh confidence

Category: Not specified
Journal tier: Specialist/Solid
Processed: Sep 28, 2025, 12:56 AM
arXiv Links: Abstract ↗PDF ↗

Audit review

The paper establishes (i) that every interval map with positive entropy has a perfect uncountable set of DC1-points, and (ii) the existence of a triangular map with positive entropy (hence DC2) but no DC2-point. Both results are supported by concrete constructions and careful use of semi-conjugacy and fiber dynamics. The model’s Part (1) gives a different but sound horseshoe-based proof. However, the model’s Part (2) argument fatally assumes one can choose ε so small that the x-projection of a ball around a point of the Cantor set Q lies inside Q—impossible since Q has empty interior—and then derives a contradiction from B ⊂ f^n(B). The paper’s proof avoids this and gives a correct, detailed obstruction to the envelope inclusion A ⊂ F^n(A).

Referee report (LaTeX)

\textbf{Recommendation:} minor revisions

\textbf{Journal Tier:} specialist/solid

\textbf{Justification:}

The paper clearly strengthens the local DC1-point existence theory for interval maps and gives a sharp negative result for DC2-points in triangular maps with positive entropy. The constructions are classical yet deftly adapted to the local setting. A minor clarification would improve rigor in the transfer from an iterate back to the original map; otherwise the exposition is clear and the results are correct.