Hölder Continuity of the Lyapunov Exponents of Linear Cocycles over Hyperbolic Maps

The paper proves local Hölder continuity of all Lyapunov exponents near typical, fiber‑bunched cocycles by first establishing a uniform large‑deviations type estimate (via a fiber strong‑mixing Markov operator construction) and then invoking an abstract continuity theorem to derive a Hölder modulus; see Theorem 1.1, Theorem 7.1 and Theorem 7.2, and the exterior‑power reduction to all exponents (Theorems 2.2–2.3) . By contrast, the model’s argument critically asserts uniform exponential convergence of the finite‑time averages (1/n)∫log||∧^k B_n|| dμ to Λ_k(B) from an LDT of the form μ{|(1/n)log||∧^k B_n||−Λ_k|>ε} ≤ Ce^{-kε^2 n}. That conclusion does not follow from such LDT bounds (optimizing ε yields only O(n^{-1/2})-type control on the mean deviation, not an exponential rate). The paper avoids this pitfall and obtains Hölder continuity through the abstract continuity theorem, rather than through an (incorrect) exponential speed of convergence of expectations . The model also misattributes the source of uniform LDT to quasi‑multiplicativity (Park); in the paper, the LDT is obtained via strong mixing of a Markov operator on the projectivized bundle and holonomy reduction, not via Park’s approach .