Extremal Lyapunov exponents in random dynamics

Both the paper and the candidate solution correctly set up the word-averaged cocycle and obtain a subadditive structure ensuring existence (a.e.) and L1 convergence of 1/(nN^n)∑_{|ω|=n} log norms (and the analogous “−” side). The paper proves this via an N-subadditive Kingman-type theorem (Theorem 3.10) and then deduces Theorem 3.8 by instantiating Φ_n with the word-sums . The model proves it via the Akcoglu–Krengel subadditive ergodic theorem for positive contractions, using the averaging operator P. However, both arguments fall short at the claimed T_j-invariance step: the paper asserts invariance from equality of integrals ∫Φ∘T_j=∫Φ (which does not imply Φ∘T_j=Φ a.e.) , and the model uses an L1 inequality to conclude that each ∥Λ∘T_j−Λ∥_1=0 from ∥PΛ−Λ∥_1=0, which is not logically valid. Thus, while existence and integral identities are established, the individual T_j-invariance remains unproven in both.