Fractals and Hyperspaces
Fractals and Hyperspaces
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DOI：
10.1007/bfb0089156
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发表时间：
1992-09
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影响因子：
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通讯作者：
K. Wicks
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文献类型：
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作者：
K. Wicks
面向所有对分形、超空间、定点理论、平铺和非标准分析感兴趣的读者，这本书以原始和可访问的方式展示了它的主题，并附有许多数字。书的第一部分发展了关于豪斯多夫度量和Vietoris拓扑的某些超空间理论，作为自相似性和分形性的基础。它的一个主要特点是用非标准分析方法对某些已知结果进行了新的证明，比以往的证明更加灵活。JE哈钦森的不变集理论（由较小的图像本身组成的集）的发展，研究了当这样一个集是平铺的图像和分类的许多不变集，无论是定期或剩余。书的最后和最原始的部分介绍了”视图”的概念作为研究给定空间内集合结构的框架的一部分。这导致了新的，优雅的概念（定义纯粹拓扑）的自相似性和分形：特别是，作者表明，许多不变集是“视觉分形”，即有无限的细节在一定意义上。这些想法有相当大的进一步发展的空间，并列出了问题和研究路线。
Addressed to all readers with an interest in fractals, hyperspaces, fixed-point theory, tilings and nonstandard analysis, this book presents its subject in an original and accessible way complete with many figures. The first part of the book develops certain hyperspace theory concerning the Hausdorff metric and the Vietoris topology, as a foundation for what follows on self-similarity and fractality. A major feature is that nonstandard analysis is used to obtain new proofs of some known results much more slickly than before. The theory of JE Hutchinson's invariant sets (sets composed of smaller images of themselves) is developed, with a study of when such a set is tiled by its images and a classification of many invariant sets as either regular or residual. The last and most original part of the book introduces the notion of a" view" as part of a framework for studying the structure of sets within a given space. This leads to new, elegant concepts (defined purely topologically) of self-similarity and fractality: in particular, the author shows that many invariant sets are" visually fractal", ie have infinite detail in a certain sense. These ideas have considerable scope for further development, and a list of problems and lines of research is included.