Topological and Statistical Attractors for Interval Maps

V. Pinheiro

incompletehigh confidence

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

Audit review

The main theorem and dichotomy in the paper match the candidate solution, and the overall structure (homterval analysis, finiteness via returns near the singular set, and classification into cycles of intervals or Cantor sets) is consistent. However, the paper repeatedly states that a Cantor attractor equals a countable union of forward orbits of one-sided critical/discontinuity images (Aj = ⋃ O_f^+(v)), which cannot be correct as written because a Cantor set is uncountable; the closure is missing. This occurs in Theorem A, item (4), and in Proposition 4.7. The candidate solution correctly amends this to Aj = cl(⋃ O_f^+(v)) and otherwise supplies a sound proof outline.

Referee report (LaTeX)

\textbf{Recommendation:} minor revisions

\textbf{Journal Tier:} specialist/solid

\textbf{Justification:}

The work consolidates and extends finiteness and structure results for topological/statistical attractors in interval dynamics, including discontinuous maps, by combining Baire-ergodic tools with classical one-dimensional dynamics. The main theorems are valuable and the arguments are largely correct and clear. The primary issue is the repeated omission of closures in the description of Cantor attractors as unions of forward orbits. Fixing this and a few minor expository points would make the paper publication-ready.