Dynamics of metrics in measure spaces and scaling entropy

IF 16.4 1区化学 Q1 CHEMISTRY, MULTIDISCIPLINARY

Accounts of Chemical Research Pub Date : 2023-11-24 DOI:10.4213/rm10103e

A. Vershik, Georgii A Veprev, P. Zatitskii

引用次数: 0

Abstract

This survey is dedicated to a new direction in the theory of dynamical systems, the dynamics of metrics in measure spaces and new (catalytic) invariants of transformations with invariant measure. A space equipped with a measure and a metric which are naturally consistent with each other (a metric triple, or an mm-space) defines automatically the notion of its entropy class, thus allowing one to construct a theory of scaling entropy for dynamical systems with invariant measure, which is different and more general in comparison to the Shannon-Kolmogorov theory. This possibility was hinted at by Shannon himself, but the hint went unnoticed. The classification of metric triples in terms of matrix distributions presented in this paper was proposed by Gromov and Vershik. We describe some corollaries obtained by applying this theory. Bibliography: 88 titles.

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度量空间中的度量动态和缩放熵

本研究致力于动态系统理论的一个新方向，即度量空间中的度量动力学和具有不变度量的变换的新（催化）不变式。一个空间如果配备了相互自然一致的度量和度量（度量三重空间或毫米空间），就会自动定义其熵类的概念，从而可以为具有不变度量的动力系统构建一种缩放熵理论，这种理论与香农-科尔莫戈罗夫理论不同，而且更具一般性。香农本人曾暗示过这种可能性，但这一暗示并未引起人们的注意。本文提出的以矩阵分布为基础的度量三元分类是由格罗莫夫和弗尔希克提出的。我们描述了应用这一理论得到的一些推论。参考书目：88 种。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Accounts of Chemical Research 化学-化学综合

CiteScore

31.40

自引率

1.10%

发文量

312

审稿时长

2 months

期刊介绍： Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance. Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.