Schrödinger operators with potentials generated by hyperbolic transformations: II. Large deviations and Anderson localization

The paper proves ULD (Theorem 2.10) by constructing stable/unstable holonomies for the Schrödinger cocycle in the LC/SH regime, showing uniqueness of the u-state for E ∈ Jη (Lemma 3.4), and establishing an exponential LDT for Birkhoff sums on the skew-product via a martingale-difference argument that leverages the bounded distortion property (Lemma 3.9), culminating in Theorems 3.6 → 3.5 → 2.10. No spectral-gap or “projective transfer operator” machinery, nor the Avalanche Principle, is used in this part of the paper. By contrast, the candidate solution hinges on an unsubstantiated spectral gap for a projective transfer operator and appeals to the Avalanche Principle to bridge subadditive and additive quantities. These steps are neither established nor needed in the paper’s proof and introduce missing assumptions (e.g., uniform projective contraction). Hence, the paper’s argument is correct and complete for Theorem 2.10, while the model’s outline is not justified as stated. See the paper’s statement of Theorem 2.10 and PLE/ULD definitions, the u-state uniqueness, the reduction to past dependence, and the LDT scheme via martingale differences and disintegration leading to Theorems 3.6 and 3.5, then to Theorem 2.10 .