混合马尔可夫拟周期上循环的李雅普诺夫指数的解析性
Analyticity of Lyapunov Exponents for Mixed Markov Quasi-Periodic Cocycles
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中文总结 AI 辅助
该研究针对混合马尔可夫拟周期上循环，通过转移算子、纤维维数归纳等方法，证明最大李雅普诺夫指数对转移概率的依赖具有解析性。
中文摘要 AI 辅助
我们研究混合马尔可夫拟周期上循环的最大李雅普诺夫指数对转移概率的依赖关系。假设转移矩阵是本原的，环面上的马尔可夫延拓沿有向环是非共振的，且最大李雅普诺夫指数是单的。我们首先通过正向马尔可夫转移算子和对初始马尔可夫状态一致的收缩估计，在不可约情形下证明该结果。随后，如同Bezerra-Sánchez-Tall的伯努利论证中那样，通过沿可测不变截面分解为限制和商丛上循环并对纤维维数进行归纳，得到可约情形。即使原矩阵上循环是连续的，可测不变截面也未必允许连续平凡化，因此我们针对具有本质有界逆的本质有界可测丛上循环，表述不可约解析情形。
英文摘要
We study the dependence of the top Lyapunov exponent of a mixed Markov quasi-periodic cocycle on the transition probabilities. The transition matrix is assumed primitive, the Markov extension on the torus is assumed non-resonant along directed cycles, and the top Lyapunov exponent is assumed simple. We first prove the result in the irreducible case by a forward Markov transfer operator and contraction estimates uniform over the initial Markov state. The reducible case is then obtained, as in the Bernoulli argument of Bezerra-Sánchez-Tall, by decomposing along measurable invariant sections into restricted and quotient bundle cocycles and inducting on the fiber dimension. A measurable invariant section need not admit a continuous trivialization, even when the original matrices cocycle are continuous. We therefore formulate the irreducible analytic case for essentially bounded measurable bundle cocycles with essentially bounded inverses.
