Improved estimates of statistical properties in some non-uniformly hyperbolic dynamical systems

The paper’s Theorem 5 precisely states the three implications relating the geometric return tails μ̂(Hn) and the Young tower base-hitting tails μΔ0(An): (a) μ̂(Hn) ≪ r(n) ⇒ μΔ0(An) ≪ r(n); (b) μ̂(Hn) ≫ n^{-a} ⇒ μΔ0(An) ≫ μ̂(Hn); and (c) μ̂(Hn) ≍ r(n) ⇒ μΔ0(An) ≍ r(n) (see Theorem 5 and surrounding definitions of Dn, Hn, An) . The paper’s proofs are built on a careful excursion decomposition and a key Proposition 7 derived from conditions (C1)–(C5), notably the widths assumption (C4) and the Growth Lemma (C5) for standard families, to control “crowding” of long excursions without losing logarithmic factors, yielding the upper bound (a) and the lower bound (b) via a three-term decomposition of Hn and quantitative estimates . By contrast, the candidate solution asserts the identity h = Rσ on the base and uses the inequality {Rσ > n} ⊂ {R > n/σ}, which treats the base return time as a product. In the CMZ/Young-tower setting, the base return time is, in general, the sum of excursion lengths across σ laps (i.e., a sum of R’s), not their product; the paper’s proof explicitly organizes the analysis via the longest (and second-longest) excursions and avoids the product reduction invoked by the candidate . Moreover, the candidate’s “uniform tail-comparison lemma” for μB(R>n) under arbitrary base-conditioning is nontrivial and is neither stated nor used in the paper; the paper instead proves and uses Proposition 7 to control intersections Rk ∩ T̂−m π(Hk^q) with precise k- and m-dependence, which is essential to eliminate logarithmic losses in (a) . Consequently, the model’s argument relies on a wrong structural identity and on an unproven uniform tail-comparison step, while the paper’s argument is coherent and complete under (C1)–(C5).