AI 中文总结
该研究证明有限状态马尔可夫GL(2)上同调在零李雅普诺夫间隙处的极值李雅普诺夫指数满足对数-赫尔德连续性，提出非分裂三角情形的马尔可夫永续性估计方法，共形情形下对数模指数可优化为1。
AI 中文摘要
我们证明，有限状态马尔可夫GL(2,R)-上同调的极值李雅普诺夫指数，在满足λ₊(A,P)=λ₋(A,P)的所有参数(A,P)处，关于上同调矩阵与转移核是逐点对数-赫尔德连续的，且扰动取自固定转移图内。主要新要素是针对非分裂三角情形的马尔可夫永续性估计，通过鞅-上同调分解得到；在共形情形下，对数模中的指数1/2可替换为1。
英文摘要
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 $1$.
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