Rescaled expansive measures for flows

Yun Yang

incompletemedium confidence

Category: Not specified
Journal tier: Strong Field
Processed: Sep 28, 2025, 12:57 AM
arXiv Links: Abstract ↗PDF ↗

Audit review

The paper’s main claim (existence of rescaled expansive invariant measures under h_μ(φ)>0) is plausible and largely consistent with its stated strategy, but the proof of the key decay estimate (Theorem 4.4) appears to implicitly require an integrability hypothesis on ln||X|| that is not stated in the theorem, and the text contains what looks like a dimensional/typographical error in the partition diameter bound. The candidate’s solution reaches the same conclusion by a different route (via classical Brin–Katok for flows and a reduction to non-rescaled Bowen–Walters balls), but it contains two substantive issues: a flawed “a.e. centers to all centers” step (upper semicontinuity is used in the wrong direction) and an incorrect statement about singular centers (Γ_ε(x)=Sing(X) for x singular). With small but nontrivial repairs—adding the missing integrability/partition hypothesis on the paper side, and replacing the model’s Step 2 by the paper’s equivalence Theorem 1.3 and fixing the singular-center claim—both arguments can be made sound.

Referee report (LaTeX)

\textbf{Recommendation:} major revisions

\textbf{Journal Tier:} strong field

\textbf{Justification:}

The contribution is interesting and potentially impactful, extending expansive-measure phenomena to flows with singularities and offering a rescaled Brin–Katok formula. However, the proof of the central decay estimate appears to rely on an unstated integrability hypothesis and contains a likely typographical/dimensional error in a key partition diameter bound. These issues should be corrected, and the role of Theorem 1.3 in upgrading a.e. statements to for-all-x should be made explicit in the proof of Theorem 1.4. With these revisions, the paper would meet the standards of correctness and clarity for publication.