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Finer filtration for matrix-valued cocycle based on Oseledec's multiplicative ergodic theorem

Kefeng Liu

Dynamical Systems mathscidoc:1912.43121

arXiv preprint arXiv:1308.6111, 2013.8
We consider a measurable matrix-valued cocycle A: Z+ X Rd d, driven by a measurepreserving transformation T of a probability space (X,, ), with the integrability condition log+ A (1,) L1 (). We show that for -ae x X, if limn 1 n log A (n, x) v= 0 for all v Rd\{0}, then the trajectory {A (n, x) v} n= 0 is far away from 0 (ie lim supn A (n, x) v> 0) and there is some nonzero v such that lim supn A (n, x) v v. This improves the classical multiplicative ergodic theorem of Oseledec. We here present an application to linear random processes to illustrate the importance.
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@inproceedings{kefeng2013finer,
  title={Finer filtration for matrix-valued cocycle based on Oseledec's multiplicative ergodic theorem},
  author={Kefeng Liu},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191221111651982202681},
  booktitle={arXiv preprint arXiv:1308.6111},
  year={2013},
}
Kefeng Liu. Finer filtration for matrix-valued cocycle based on Oseledec's multiplicative ergodic theorem. 2013. In arXiv preprint arXiv:1308.6111. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191221111651982202681.
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