AI 中文总结
本文研究随机$C^1$微分同胚的拟不可约性等性质，建立其等价准则，应用于李雅普诺夫指数连续性与膨胀性，推广形式主义至格拉斯曼丛。
AI 中文摘要
我们研究黎曼流形紧不变集上的随机$C^1$（局部）微分同胚的拟不可约性、膨胀性及其余切对偶性质。拟不可约性通过到射影切丛的平稳提升定义：每个提升必须实现关于最大李雅普诺夫指数的Furstenberg–Kifer公式。我们证明，对于遍历平稳测度，这等价于赤道的缺失，即不存在使最大指数下降的非随机不变子丛。在每个平稳测度的第一李雅普诺夫指数为简单的条件下，拟不可约性还等价于垂直大部分收缩、平均收缩、垂直谱间隙以及平稳射影提升的唯一性。随后我们将这些准则应用于最大李雅普诺夫指数的连续性与平均膨胀性，特别地，膨胀性的特征是最大李雅普诺夫指数在所有非随机不变子丛上为正；在拟不可约性条件下，它等价于每个平稳测度的最大李雅普诺夫指数为正。对于余拟不可约性、余赤道、余膨胀性及最小李雅普诺夫指数的连续性，存在对偶结论。我们还描述了膨胀性与余膨胀性的相互作用，并将该形式主义推广到格拉斯曼丛，得到控制李雅普诺夫指数中间和的高维版本。
英文摘要
We study quasi-irreducibility, expansion, and their cotangent duals for random $C^1$ (local) diffeomorphisms on compact invariant sets of a Riemannian manifold. Quasi-irreducibility is defined through stationary lifts to the projective tangent bundle: every lift must realize the Furstenberg--Kifer formula for the top Lyapunov exponent. We prove that, for ergodic stationary measures, this is equivalent to the absence of equators, namely non-random invariant subbundles on which the top exponent drops. Under simplicity of the first Lyapunov exponent for every stationary measure, quasi-irreducibility is also equivalent to vertical mostly contraction, contraction on average, a vertical spectral gap, and uniqueness of stationary projective lifts. We then apply these criteria to continuity of the top Lyapunov exponent and to expansion on average. In particular, expansion is characterized by positivity of the top Lyapunov exponent on all non-random invariant subbundles; under quasi-irreducibility, it is equivalent to positivity of the top Lyapunov exponent for every stationary measure. Dual statements hold for coquasi-irreducibility, coequators, coexpansion, and continuity of the bottom Lyapunov exponent. We also describe the interplay between expansion and coexpansion and extend the formalism to Grassmannian bundles, obtaining higher-dimensional versions controlling intermediate sums of Lyapunov exponents.
