A Multiplicative Ergodic Theorem and Nonpositively Curved Spaces
A Multiplicative Ergodic Theorem and Nonpositively Curved Spaces
复制标题
乘法遍历定理和非正弯曲空间
DOI：
10.1007/s002200050750
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发表时间：
1999
影响因子：
2.4
通讯作者：
G. Margulis
中科院分区：
文献类型：
--
作者：
A. Karlsson;G. Margulis
翻译后摘要：我们研究可积上循环u（n，x）在遍历测度保持变换，采取的值在一个半群的非扩张映射的非正弯曲空间Y，例如一个Cartan-Hadamard空间或一致凸Banach空间。证明了对任意y ∈ Y和几乎所有x，存在A ≥ 0和Y中唯一的从y开始的测地线γ（t，x），使得
当Y是对称空间GLN（n）/ON（n）且余圈取值于GLN（n）时，这等价于Oseledec的乘法遍历定理，并给出了两个应用.第一个问题涉及泊松边界的确定，第二个问题涉及希尔伯特-施密特算子。
Abstract:We study integrable cocycles u(n,x) over an ergodic measure preserving transformation that take values in a semigroup of nonexpanding maps of a nonpositively curved space Y, e.g. a Cartan–Hadamard space or a uniformly convex Banach space. It is proved that for any y∈Y and almost all x, there exist A≥ 0 and a unique geodesic ray γ (t,x) in Y starting at y such that
In the case where Y is the symmetric space GLN(ℝ)/ON(ℝ) and the cocycles take values in GLN(ℝ), this is equivalent to the multiplicative ergodic theorem of Oseledec.Two applications are also described. The first concerns the determination of Poisson boundaries and the second concerns Hilbert-Schmidt operators.