SUBSYSTEMS WITH SHADOWING PROPERTY FOR Zk-ACTIONS

Lin Wang, Xinsheng Wang, Yujun Zhu

correcthigh confidence

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

Audit review

The paper proves Theorems A–C under the Basic assumption by constructing uniformly (partially) hyperbolic (nonautonomous) subsystems inside a thin tube around Lv and then invoking ready-made shadowing/quasi-shadowing lemmas and an expansivity-lifting proposition; the candidate solution proves the same statements via a Lyapunov–Perron fixed-point scheme for a nonautonomous exponential dichotomy and a center-compensated quasi-shadowing argument, plus a suspension reduction. The logical endpoints agree, and the technical steps are compatible with the paper’s hypotheses, though the model’s Step 2 (tube ordering) would benefit from an explicit proof of the finite-increment/comparability claim that the paper achieves by constructing a sequence with controlled coordinate increments.

Referee report (LaTeX)

\textbf{Recommendation:} minor revisions

\textbf{Journal Tier:} specialist/solid

\textbf{Justification:}

The manuscript establishes clean criteria for when directional subsystems of a smooth Z\^k-action exhibit (quasi-)shadowing, under a uniform Lyapunov spectrum hypothesis. The combination of thickened geometric tubes, nonautonomous hyperbolicity, and a suspension reduction is well executed and connects discrete and flow settings. The results are correct and useful to the subdynamics literature. Minor clarifications (explicit construction of the tube sequences and parameter dependencies) would improve readability.