Pesin theory for transcendental maps and applications

The paper’s Theorem A (and its stronger natural-extension version, Theorem 5.1) is clearly stated and proved under the hypotheses log|f'| ∈ L^1(ω_U), positive Lyapunov exponent χ_ωU(f)>0, and thin singular values in ∂U. The proof uses Rokhlin’s natural extension, ergodicity/recurrence of the boundary map, a summable control of harmonic measure near the singular set (Lemma 3.10), Borel–Cantelli, and Koebe distortion, yielding well-defined inverse branches on a uniform disk and exponential shrinking, and guaranteeing visits to any positive-measure set along a subsequence . The candidate solution mirrors this program in spirit (inner-function model, natural extension, summable windows, Borel–Cantelli, Koebe), but it crucially asserts the identity ∫∂D log|g'| dλ = ∫∂U log|f'| dω_U and treats their common value as a Lyapunov exponent for g. The paper explicitly lists this equality (χ_λ(g) = χ_ωU(f)) as an open question in the transcendental setting, even noting that proving it would simplify the positivity assumption for χ_ωU(f) . While the rest of the model’s outline could be salvaged without that claim (working entirely with f on ∂U, as the paper does), as written it depends on a step the paper identifies as open. Hence: paper correct; model, as stated, incorrect on a key point.