Analyticity of the Lyapunov Exponents of Random Products of Matrices

The paper proves real-analyticity of the top Lyapunov exponent under L1>L2 by (i) establishing exponential contraction of an average Hölder constant k_α for the projective Markov operators (via wedge-power estimates and uniform-in-v bounds) and (ii) showing the limit of iterates Q^n applied to a fixed observable is a uniform limit of polynomials along complex lines, hence Gateaux-holomorphic and therefore holomorphic in a TV-neighborhood (Theorem 1.1 for i.i.d.; Theorem 1.2 for Markov) . The key technical ingredient is Lemma 4.2/Proposition 4.3 (Markov) and their Bernoulli analogues, which yield v_α(Q^n φ) ≤ k_α(·)^n v_α(φ) with k_α(·) < 1 uniformly on a neighborhood, obtained by bounding ∫(s1 s2/||A_n v||^2)^α/2 and using submultiplicativity plus continuity in the TV norm . Holomorphic dependence then follows from continuity + Gateaux holomorphy along lines μ+zν (or K+zL), since each Q^n is polynomial in z and the limit is uniform in the neighborhood . By contrast, the model’s solution hinges on a false pointwise claim: δ(g·x,g·y) ≤ (σ2/σ1)(g) δ(x,y) for the angular metric. This would force orthogonal directions to contract, which fails even in d=2 (orthogonal images remain orthogonal, giving ratio 1 while σ2/σ1<1). Consequently the model’s Doeblin–Fortet bound [Q_μ^n φ]_α ≤ E[(σ2/σ1)(G_n)^α][φ]_α and the ensuing spectral-gap/analyticity argument are invalid. The paper avoids this pitfall by working with the averaged Hölder constant k_α and wedge-power estimates, and by using a direct Gateaux-holomorphic limit argument rather than Kato’s spectral projectors . The support-restricted Markov statement (S(K0)) is also handled carefully by continuity of L1 under support inclusion and induction on invariant sections, as stated in Section 4 .