2404.09076
Polynomial escape rates via maximal large deviations
Yaofeng Su
correctmedium confidence
- Category
- math.DS
- Journal tier
- Specialist/Solid
- Processed
- Sep 28, 2025, 12:56 AM
- arXiv Links
- Abstract ↗PDF ↗
Audit review
The paper proves that for the LSV map g_α with an open hole H bounded away from 0 and with nonempty interior, the survival probability satisfies μ(τ_H>n) ≈ n^{1−1/α} (Theorem 3.1), via a first-return Young tower over Y=(a_m,1], a lifting argument, a lower bound from return-time tails, and an upper bound combining exponential avoidance of a small Markov subhole on the base with a maximal large deviations estimate for return counts . The candidate solution reaches the same conclusion: it matches the tower setup and lower bound, and for the upper bound either appeals directly to Demers–Fernández (2016) for cylinder holes and sandwiches a general hole H between cylinder holes, or argues via exponential avoidance under the induced Gibbs–Markov map. However, the model’s intermediate bound uses a flawed combinatorial step (it asserts “at least one R_j ≥ n/k” when actually one has “at least one R_j ≤ n/k”), and it incorrectly calls the tail of R “summable” for all 0<α<1; these missteps are not needed because the cited cylinder-hole result plus monotonicity already yields the correct exponent. The paper’s proof is complete and internally consistent, and it explicitly relaxes the Markov-hole requirement compared to prior work .
Referee report (LaTeX)
\textbf{Recommendation:} minor revisions \textbf{Journal Tier:} specialist/solid \textbf{Justification:} This short note gives a streamlined derivation of polynomial escape rates for intermittent systems, notably recovering the LSV result for general open holes away from the neutral point and extending to a higher-dimensional solenoid example. The approach via maximal large deviations is technically light and potentially extensible. The argument is sound and the presentation concise, though a few steps (e.g., the reduction to a small Markov subhole and the precise maximal large deviations input) would benefit from minor elaboration for clarity.