Mixing properties of a class of nonuniformly expanding maps. Application to Hölderian invariance principles.

The paper proves a Hölderian invariance principle for nonuniformly expanding maps with return-time tails m(R>n) ≲ n^{-p}s(n) (p>2, s slowly varying →0), by computing τ-mixing coefficients τϕ(n)=O(n^{1-p}s(n)) and invoking Giraudo’s theorem to obtain Wn(ϕ) ⇒ σ(ϕ)W in H0_{1/2−1/p}, while also ensuring absolute convergence of the covariance series; see Theorem 1.1 and the τ-mixing route via the meeting-time bounds P(T>n)=O(n^{1-p}s(n)) and δ(n) estimates . The candidate solution reaches the same result using a different route: Young towers, martingale/coboundary decomposition (or ASIP/WIP), sharp moment bounds and maximal inequalities to get tightness at the borderline exponent and identify the Brownian limit. Apart from a minor overreach (they take q=p in a moment bound that is typically available only for q<p), the outlined proof is sound and compatible with standard results. Net: both are correct, with different methods.