2009.13052
Another Look at the Hofer–Zehnder Conjecture
Erman Çineli, Viktor L. Ginzburg, Başak Z. Gürel
correcthigh confidence
- Category
- math.DS
- Journal tier
- Strong Field
- Processed
- Sep 28, 2025, 12:55 AM
- arXiv Links
- Abstract ↗PDF ↗
Audit review
The paper’s Theorem 3.1 establishes only a correspondence for the shortest arrows between Γ(ϕ) and Γeq(ϕ2), proved via the Z2-equivariant pair-of-pants product and barcode considerations (see the definitions of Γ(ϕ), Γeq(ϕ2), deq, and Theorem 3.1 in , , ). By contrast, the model switches to S1-equivariant language (CP∞, |h|=2) rather than the paper’s Z2 setup (S∞/RP∞, |h|=1), and concludes deq reduces to dFl, implying Γ(ϕ) and Γeq(ϕ2) have identical arrows and lengths. This not only rests on incorrect equivariant data but also contradicts the paper’s explicit statement that Γeq(ϕ2) is obtained from Γ(ϕ2) by adding arrows (deq = dFl + O(h)) and that extra arrows can appear (). The paper even notes that an identification deq = dFl is only expected in a local setting and is unproven ().
Referee report (LaTeX)
\textbf{Recommendation:} minor revisions
\textbf{Journal Tier:} strong field
\textbf{Justification:}
The manuscript offers a clear and effective proof of a variant of the Hofer–Zehnder conjecture via a novel focus on shortest arrows in Floer graphs and the Z2-equivariant pair-of-pants product. The approach is technically sound and complements existing work. Minor clarifications distinguishing Z2- from S1-equivariant contexts and an illustrative example of added arrows in the equivariant graph would further improve clarity.