Weyl Mean Equicontinuity and Weyl Mean Sensitivity of a Random Dynamical System

Yuan Lian

incompletemedium confidence

Category: Not specified
Journal tier: Specialist/Solid
Processed: Sep 28, 2025, 12:56 AM
arXiv Links: Abstract ↗PDF ↗

Audit review

The paper’s corollary asserts the WME/WMS dichotomy under a minimal skew product with finite base, but its proof skips a needed step: showing that the absence of any Weyl–mean sensitive point implies the whole RDS is Weyl–mean equicontinuous; it moves from “one transitive point is not WMS” to “the system is WME” without justification. The model’s solution also has gaps: it incorrectly derives a pointwise dichotomy from D̂ ≤ D, mishandles fiber membership in D̂ω, and relies on an unproved upper semicontinuity step to propagate sensitivity. Both need revisions to be complete.

Referee report (LaTeX)

\textbf{Recommendation:} major revisions

\textbf{Journal Tier:} specialist/solid

\textbf{Justification:}

The paper contributes a natural extension of Weyl–mean (in)sensitivity to random dynamical systems and gives a plausible dichotomy under a minimal skew-product with finite base. Its treatment of sensitivity propagation is solid. However, the proof of the main corollary omits crucial steps: a pointwise WME/WMS dichotomy is asserted rather than proved (despite the two metrics differing), and the transition from a single WME point to uniform WME for the RDS needs an RDS-specific argument. Clarifying definitions (domain of the fiberwise metric) and adding the missing lemmas would make the result correct and self-contained.