2404.10247
Orientation Preserving Homeomorphisms of the Plane Having BP-Chain Recurrent Points
Jiehua Mai, Kesong Yan, Fanping Zeng
correcthigh confidence
- Category
- Not specified
- Journal tier
- Specialist/Solid
- Processed
- Sep 28, 2025, 12:56 AM
- arXiv Links
- Abstract ↗PDF ↗
Audit review
The paper proves that an orientation-preserving plane homeomorphism with a BP-chain recurrent point has a fixed point, via a complete construction that modifies the map near a finite segment of an ε-chain to produce a periodic orbit while preserving the fixed-point set, then invokes Brouwer’s plane theorem to obtain a contradiction. The candidate solution sketches a different strategy using a periodic disk-chain criterion, but it contains a key gap: it covers only K = cl(W) by free disks and then (incorrectly) claims that both f(x_{s_j-1}) and x_{s_j} lie in K, even though x_{s_j} need not be in W or cl(W). This misuses the Lebesgue number argument. It also leaves unaddressed standard technical hypotheses required by the disk-chain criterion (e.g., pairwise disjointness of free disks).
Referee report (LaTeX)
\textbf{Recommendation:} minor revisions \textbf{Journal Tier:} specialist/solid \textbf{Justification:} The manuscript offers a compact, self-contained proof of Brouwer’s plane fixed-point theorem and a strengthening via BP-chain recurrence. The arguments are careful and geometrically transparent, with quantitative separation estimates and a clean surgery producing an auxiliary map with a periodic orbit. A few minor editorial clarifications would further streamline the presentation, but the mathematics appears sound.