Tail behaviour of stationary densities for one-dimensional random diffeomorphisms

The paper proves the two scaling laws precisely as lim_{x→x+} ln φ(x)/ln^2(x+−x) = (k+1)/(2 ln λ) in the hyperbolic case and lim_{x→x+} (x+−x)^{r−1} ln φ(x)/ln(x+−x) = r(k+1)/[α(r−1)] in the nonhyperbolic case (Theorems A and B; see (1.9) and (1.11)) . The candidate solution reaches the same constants under the same structural assumptions but via a different, more heuristic route (Volterra reduction + Laplace method in the hyperbolic case; product-of-window widths along the extremal envelope in the nonhyperbolic case). The paper’s argument is rigorous and complete (transfer-operator formula (2.10), Lemma 3.1 bounds, hitting-time scalings) , while the model’s outline has minor slips (e.g., an incorrect monotonicity-based inequality) and omits some regularity justifications, yet it arrives at the correct leading orders.