Lipschitz Equivalence of Self-Similar Sets: Algebraic and Geometric Properties
Lipschitz Equivalence of Self-Similar Sets: Algebraic and Geometric Properties
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DOI：
10.1090/conm/600/11963
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发表时间：
2013-03
期刊：
影响因子：
--
通讯作者：
H. Rao;H. Ruan;Yang Wang
中科院分区：
文献类型：
--
作者：
H. Rao;H. Ruan;Yang Wang
本文综述了自相似集的Lipschitz等价研究的最新进展。Lipschitz等价性是分形几何中的一个重要性质，因为它保留了分形集的许多关键性质。Falconer和Marsh[on the Lipschitz等价of Cantor Sets，Texttit{Mahematika}，Textbf{39}(1992)，223-233]的一个基本结果基于自相似集的压缩比的代数性质建立了Lipschitz等价的条件。最近，该领域还取得了其他实质性进展。本文是对该领域的一次全面调查。它总结了该领域的重要和有趣的结果。此外，我们还对用于证明一些关键结果的几种重要技术进行了详细讨论。我们希望这篇论文将提供一个很好的主要结果和技术的概述，并为任何有兴趣研究该领域问题的人提供一个友好的切入点。
In this paper we provide an up-to-date survey on the study of Lipschitz equivalence of self-similar sets. Lipschitz equivalence is an important property in fractal geometry because it preserves many key properties of fractal sets. A fundamental result by Falconer and Marsh [On the Lipschitz equivalence of Cantor sets, \textit{Mathematika}, \textbf{39} (1992), 223--233] establishes conditions for Lipschitz equivalence based on the algebraic properties of the contraction ratios of the self-similar sets. Recently there has been other substantial progress in the field. This paper is a comprehensive survey of the field. It provides a summary of the important and interesting results in the field. In addition we provide detailed discussions on several important techniques that have been used to prove some of the key results. It is our hope that the paper will provide a good overview of major results and techniques, and a friendly entry point for anyone who is interested in studying problems in this field.