2106.06374
A Fixed Point Theorem for Twist Maps
Peizheng Yu, Zhihong Xia
correctmedium confidence
- Category
- Not specified
- Journal tier
- Specialist/Solid
- Processed
- Sep 28, 2025, 12:56 AM
- arXiv Links
- Abstract ↗PDF ↗
Audit review
The paper proves, sharply, that a twist map with the intersection property has at least one fixed point; Carter’s example shows one is the best possible. The model’s existence argument is broadly sound if one cites Franks, but it incorrectly strengthens the conclusion to “at least two” fixed points, which is contradicted by the paper’s discussion of Carter’s example.
Referee report (LaTeX)
\textbf{Recommendation:} minor revisions
\textbf{Journal Tier:} specialist/solid
\textbf{Justification:}
The note provides a clean, self-contained proof that twist maps with the intersection property have at least one fixed point. The approach is classical yet clearly organized, using chain recurrence, a careful interpolation to a rigid-boundary model, and Brouwer's translation theorem. Given existing work (Franks 1988; Carter 1982), the contribution is primarily expository with some useful clarifications; a few statements (e.g., orientability assumptions and commutation of lifts) could be made more explicit.