Invariant Measures of Non-Uniformly Expanding Maps with Higher Order Critical Set

The paper proves, for generalized Viana skew-products φα,D and all C^D-perturbations φ in a small neighborhood N, (i) existence and uniqueness of an a.c.i.p. on the attractor, (ii) topological mixing in the strong (exactness-on-Λ) sense, and (iii) super‑polynomial decay of correlations and the CLT for Hölder observables. These appear as Theorems A–C, with Λ = φ^2(S^1×I) in the even‑D case, and the mixing proof based on a hyperbolic-times/inducing scheme with bounded distortion and a covering argument (notably Proposition 5.2 and Step 4) . The candidate solution follows the same architecture: nonuniform expansion plus slow recurrence to the critical curve, abundance of hyperbolic times, a Gibbs–Markov/Young-type inducing scheme, and a covering argument giving exactness/mixing, then uniqueness of the a.c.i.p. and statistical properties. Minor differences are present (e.g., the candidate asserts stretched‑exponential return‑time tails, whereas the paper only claims super‑polynomial consequences via ALP), but these do not contradict the paper and are not used to claim stronger mixing than stated. Overall, the logic, key lemmas, and conclusions match closely, so both are correct with substantially the same proof strategy. The paper’s nonuniform expansion and slow recurrence estimates (e.g., bounds on exceptional sets and hyperbolic times) and bounded distortion for the induced map align with the model’s steps .