测度空间中度量的动力学及其渐近不变量
Dynamics of metrics in measure spaces and their asymptotic invariants
摘要：
我们讨论柯尔莫哥洛夫熵和西奈的定义，然后定义一个变形的熵，称为{\it标度熵};这也是一个度量不变的措施保持行动的组，这是更强大的比普通熵。为了定义它，我们引入了测度空间中度量的$ε$-熵的概念，这也是A. n. Kolmogorov稍早。我们建议用度量或半度量的迭代代替熵理论中传统的可测划分技术。这使我们的关键思想，本文这是我们希望是答案的老问题：什么是自然的背景下，人们应该考虑熵的措施，保持行动的群体？关于它的推广的同样的问题--标度熵，以及更一般的遍历论的问题。也就是说，我们提出了一个研究纲领，称为{测度空间中度量的\it渐近动力学}，其中，例如，广义熵被理解为{\it与动力系统相关联的度量空间序列的渐近豪斯多夫维数}。可以假设，动力系统作为一个整体的度量同构问题也得到了一个新的几何解释。
英文摘要：
We discuss the Kolmogorov's entropy and Sinai's definition of it; and then define a deformation of the entropy, called {\it scaling entropy}; this is also a metric invariant of the measure preserving actions of the group, which is more powerful than the ordinary entropy. To define it, we involve the notion of the $ε$-entropy of a metric in a measure space, also suggested by A. N. Kolmogorov slightly earlier. We suggest to replace the techniques of measurable partitions, conventional in entropy theory, by that of iterations of metrics or semi-metrics. This leads us to the key idea of this paper which as we hope is the answer on the old question: what is the natural context in which one should consider the entropy of measure-preserving actions of groups? the same question about its generalizations--scaling entropy, and more general problems of ergodic theory. Namely, we propose a certain research program, called {\it asymptotic dynamics of metrics in a measure space}, in which, for instance, the generalized entropy is understood as {\it the asymptotic Hausdorff dimension of a sequence of metric spaces associated with dynamical system.} As may be supposed, the metric isomorphism problem for dynamical systems as a whole also gets a new geometric interpretation.
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