Ergodic theory of random transformations
Ergodic theory of random transformations
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DOI：
10.2307/2288883
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发表时间：
1986
期刊：
影响因子：
--
通讯作者：
Y. Kifer
中科院分区：
文献类型：
--
作者：
Y. Kifer
动力系统的遍历理论，即对单个变换的迭代进行定性分析，是目前发展得很好的理论。1945年，S. Ulam和J. von Neumann在他们的简短笔记[44]中建议研究遍历定理，用于更一般的情况，当人们依次应用随机选择的不同变换时。他们的计划在1951年由S. Kakutani b[23]完成。两篇论文都考虑了具有公共不变测度的变换的情况。最近大野b[38]注意到这种情况是过度的。遍历定理只是遍历理论的开始。进一步的主要发展是熵和特征指数的概念。本书的目的是研究进化过程的各种遍历理论性质，这些性质是由根据某种概率分布从某一类随机选择的转换的独立应用所产生的。这本书展示了随机变换的遍历理论的第一个系统的处理，即，独立随机映射的组成行动的分析。这种设置允许一个统一的方法来解决动力系统，随机矩阵的乘积和随机微分方程产生的随机流的许多问题。”
Ergodic theory of dynamical systems i.e., the qualitative analysis of iterations of a single transformation is nowadays a well developed theory. In 1945 S. Ulam and J. von Neumann in their short note [44] suggested to study ergodic theorems for the more general situation when one applies in turn different transforma tions chosen at random. Their program was fulfilled by S. Kakutani [23] in 1951. 'Both papers considered the case of transformations with a common invariant measure. Recently Ohno [38] noticed that this condition was excessive. Ergodic theorems are just the beginning of ergodic theory. Among further major developments are the notions of entropy and characteristic exponents. The purpose of this book is the study of the variety of ergodic theoretical properties of evolution processes generated by independent applications of transformations chosen at random from a certain class according to some probability distribution. The book exhibits the first systematic treatment of ergodic theory of random transformations i.e., an analysis of composed actions of independent random maps. This set up allows a unified approach to many problems of dynamical systems, products of random matrices and stochastic flows generated by stochastic differential equations."