Expansivity and strong structural stability for composition operators on Lp spaces

Martina Maiuriello

correcthigh confidence

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

Audit review

The paper’s statements and proofs match standard operator-dynamical facts and are internally consistent. In contrast, the candidate solution contains two substantive errors in Problem A: (i) it incorrectly deduces the uniform limit lim_n μ(f^{-n}(B))/μ(B)=∞ from a single-step doubling (only along multiples of n0), and (ii) it gives a flawed “dichotomy” argument for uniform (two-sided) expansivity. In Problem B, the candidate’s direct shadowing⇒strong structural stability construction is not justified (continuity/homeomorphism of the tracing map is not proved), and the strong structural stability⇒shadowing interpolation via a global small Lipschitz perturbation is not secured on general L^p spaces. The paper’s arguments instead rely on precise equivalences (via [6, Prop. 19]) and established structural-stability/shadowing machinery, and are correct.

Referee report (LaTeX)

\textbf{Recommendation:} no revision

\textbf{Journal Tier:} specialist/solid

\textbf{Justification:}

The paper cleanly characterizes expansive behaviors of composition operators and clarifies the relation between strong structural stability and shadowing in a key dissipative setting. The proofs are concise, correct, and grounded in appropriate prior results. The work consolidates and extends the shift-like viewpoint for composition operators. I find the presentation sound and self-contained for readers in linear dynamics/operator ergodic theory.