DECAY OF CORRELATIONS FOR CRITICALLY INTERMITTENT SYSTEMS

The paper proves that for the random skew product F built from “good” and “bad” one-dimensional maps, if θ=∑b pb ℓb<1 and γ1=log θ / log ℓmax<0, then for any γ∈(γ1,0) and for all f∈L∞, h∈H, the annealed correlations satisfy |Cor_n(f,h)|=O(n^γ). It does so by constructing a Young tower over an inducing domain Y on the uniformly expanding right branch, verifying the tower axioms (t1)–(t6), and estimating the tail of the first-return time ϕ (with m({ϕ̂>n})=O(n^γ) via an integrated-tail relation). This is exactly Theorem 1.3 of the paper and its proof outline (Proposition 4.1; Proposition 3.2(ii)–(iii); Theorem 2.1) . The candidate solution builds the same inducing scheme/tower on a slightly different base Y⊂[1/2,1] and derives a polynomial tail for the return time by analyzing bad blocks and using independence to control E(∏ ℓb^t)=(∑b pb ℓb^t)^k, giving the same admissible range t∈(1,1−γ1) and hence the same decay rate n^{−(t−1)}=n^γ. Their use of Young’s tower machinery and the integrated-tail shift matches the paper’s strategy. Minor differences (e.g., invoking negative Schwarzian/Koebe for distortion, or choosing Y=[1−η,1] instead of (1/2,3/4)) do not affect correctness. Overall, both arguments are consistent and essentially the same in structure, with the paper providing full details and sharper composition estimates .