Hölder continuity of Lyapunov exponents for non-invertible and non-compact random cocycles

The paper proves local Hölder continuity of the top Lyapunov exponent in the Wasserstein metric near a quasi-irreducible measure with a spectral gap (Theorem 2.1), via a spectral/Markov-operator method yielding uniform strong mixing on Hölder observables and a careful truncation/tail analysis of ψ_v̂(g)=log∥gv∥ (see the statement and proof outline in Sections 2 and 7, together with the spectral theory developed in Section 6 ). The candidate solution establishes the same conclusion by a coupling-and-contraction route, deriving a recursion for projective angles and a telescoping sum bound. Aside from a minor constant/exponent slip (a factor r^{1−α} should replace r^{α−1} in the final normalization) and an over-terse justification of the expectation-to-exponent limit, the argument is logically sound and compatible with the paper’s hypotheses. Hence, both are correct, but they use substantially different proof strategies.