2205.03794
ON PARTIAL MAPS DERIVED FROM FLOWS
Tomoharu Suda
correctmedium confidence
- Category
- Not specified
- Journal tier
- Specialist/Solid
- Processed
- Sep 28, 2025, 12:56 AM
- arXiv Links
- Abstract ↗PDF ↗
Audit review
The paper’s Main Theorem A states that types of boundary points (A, B, C and A-1/2/3) are preserved under topological equivalence h with B = h(A) for regular open sets. The proof reduces to Lemma 3.11, which shows h(dom EA) = dom EB, h(dom PĀ) = dom PB̄, and h(EA(x)) = EB(h(x)), h(PĀ(x)) = PB̄(h(x)) (hence types are preserved), and then concludes Theorem 3.9 directly . The candidate solution proves the same facts in a slightly more explicit way: it introduces an explicit orbit-wise time-change τx and shows T^e_B(h(x)) = τx(T^e_A(x)), T^r_{B̄}(h(x)) = τx(T^r_{Ā}(x)), whence h∘EA = EB∘h and h∘PĀ = PB̄∘h, preserving domains and fixed-point status. This is essentially the same mechanism as the paper’s argument, just phrased via the explicit time reparameterization. The candidate also (correctly) notes that regularity is preserved by h via h(∂A) = ∂B and h(Ā) = B̄, which is consistent with the paper’s assumption that both A and B are regular. Definitions of EA, PĀ, and the type classification, as well as the statement of Theorem 3.9, match the paper exactly .
Referee report (LaTeX)
\textbf{Recommendation:} minor revisions \textbf{Journal Tier:} specialist/solid \textbf{Justification:} The work provides a coherent framework for partial maps derived from flows and a topological classification of boundary points, proving invariance under topological equivalence. The core arguments (especially Lemma 3.11 leading to Theorem 3.9) are correct and well-motivated. Minor clarifications on definitions and terminology would make the exposition even clearer.