Log-Höder continuity at zero Lyapunov gap for finite state Markov $GL(2)$-cocycles

We prove that the extremal Lyapunov exponents of finite-state Markov $\mathrm{G}\mathrm{L}(2,\mathbb{R})$-cocycles are pointwise log-Hölder continuous, jointly in the cocycle matrices and the transition kernel, at every parameter $(A,P)$ satisfying ${\lambda }_{+}(A,P)={\lambda }_{-}(A,P)$. Perturbations are taken within a fixed transition graph. The main new ingredient is a Markov perpetuity estimate for the nonsplit triangular case, obtained through a martingale–coboundary decomposition. In the conformal case, the exponent $1\mathrm{/}2$ in the logarithmic modulus can be replaced by $1$.