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

Abstract

We prove that the extremal Lyapunov exponents of finite-state Markov $\mathrm{GL}(2,\mathbb{R})$-cocycles are pointwise log-Hölder continuous, jointly in the cocycle matrices and the transition kernel, at every parameter $(A,P)$ satisfying $λ_+(A,P)=λ_-(A,P)$. 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 $1/2$ in the logarithmic modulus can be replaced by $1$.