Statistical properties of equilibrium states for fiber-bunched matrix cocycles and applications

The paper’s Theorem A states exactly what the candidate proves: for a Hölder, fiber-bunched, 1-typical cocycle A over a mixing SFT, there exists t*>0 such that for each t in (−t*,∞) there is a unique Gibbs equilibrium state μ_t for tΦ_A, and t↦P(σ,tΦ_A)′=λ1(μ_t,A) is real-analytic on (−t*,t*) . The paper’s proof strategy matches the model’s: (i) for t>0, use quasi-multiplicativity under 1-typicality to get uniqueness/Gibbs (Lemma 7.1 and references therein) ; (ii) for |t| small (including t<0), build a transfer operator on Σ×PR^{d−1} with a spectral gap (Proposition 5.7) and deduce existence, uniqueness and Gibbs, as well as analyticity of pressure and the derivative identity via ρ′_t/ρ_t . Fiber-bunching supplies holonomies and bounded distortion for log∥A^n∥ on cylinders (eq. (4.9)), which both arguments employ . The final assembly of the two regimes into Theorem A is given explicitly via Theorems 6.9 and 6.10 in the paper . The candidate’s outline follows these same ingredients with only minor stylistic differences (e.g., including g-weights is a technical device in the paper), so both are correct and essentially the same.