关于Oseledets子空间的不可约性
On irreducibility of Oseledets subspaces
摘要：
对于可逆实$n$乘$n$矩阵的上循环，乘法遍历定理给出了$\mathbb{R}^n$的Oseledets子空间分解;也就是说，在底空间中的每个点之上，$\mathbb{R}^n$被写为等变子空间的直和，每个等变子空间对应于上循环的李雅普诺夫指数。很自然地，我们会问这些和数是否可以进一步分解为等变子空间，也就是说，Oseledets子空间是否是可约的。证明了一个定理，给出了平凡等变子空间$\mathbb{R}^2$和$\mathbb{C}^2$对于$O_2(\mathbb{R})$ -值上圈不可约的充分条件，并给出了满足这些条件的例子.
英文摘要：
For a cocycle of invertible real $n$-by-$n$ matrices, the Multiplicative Ergodic Theorem gives an Oseledets subspace decomposition of $\mathbb{R}^n$; that is, above each point in the base space, $\mathbb{R}^n$ is written as a direct sum of equivariant subspaces, one for each Lyapunov exponent of the cocycle. It is natural to ask if these summands may be further decomposed into equivariant subspaces; that is, if the Oseledets subspaces are reducible. We prove a theorem yielding sufficient conditions for irreducibility of the trivial equivariant subspaces $\mathbb{R}^2$ and $\mathbb{C}^2$ for $O_2(\mathbb{R})$-valued cocycles and give explicit examples where the conditions are satisfied.
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