Topological invariants and Lipschitz equivalence of fractal squares
Topological invariants and Lipschitz equivalence of fractal squares
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DOI：
10.1016/j.jmaa.2017.02.012
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发表时间：
2017-07
影响因子：
1.3
通讯作者：
H. Ruan;Yang Wang
中科院分区：
文献类型：
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作者：
H. Ruan;Yang Wang
分形集通常具有非常复杂的几何结构，分形几何的一个基本问题是表征不同分形集的“相似”程度。分形集的Lipschitz等价常用于分形集几何相似的分类。Lipschitz等价和自相似集收缩比的代数性质之间的有趣联系已经被发现并被广泛分析。然而，除了极少数论文外，Lipschitz等价的研究主要集中在完全分离的自相似集上。对于连通的自相似集，即使对于众所周知的分形模型，如分形平方，这个问题也变得相当具有挑战性。本文引入几何和拓扑方法来研究连通分形平方的Lipschitz等价。特别地，我们完整地刻画了去掉一个或两个分形平方的3阶分形平方的Lipschitz等价。我们还讨论了更一般阶的分形平方的Lipschitz等价。本文首次研究了非平凡连通自相似集的Lipschitz等价性，并对更一般的集合提出了一些有趣的问题。
Fractal sets typically have very complex geometric structures, and a fundamental problem in fractal geometry is to characterize how “similar” different fractal sets are. The Lipschitz equivalence of fractal sets is often used to classify fractal sets that are geometrically similar. Interesting links between Lipschitz equivalence and algebraic properties of contraction ratios for self-similar sets have been uncovered and widely analyzed. However, with the exception of very few papers, the study of Lipschitz equivalence has largely focused on totally disconnected self-similar sets. For connected self-similar sets this problem becomes rather challenging, even for well known fractal models such as fractal squares. In this paper, we introduce geometric and topological methods to study the Lipschitz equivalence of connected fractal squares. In particular we completely characterize the Lipschitz equivalence of fractal squares of order 3 in which one or two squares are removed. We also discuss the Lipschitz equivalence of fractal squares of more general orders. Our paper is the first study of Lipschitz equivalence for nontrivial connected self-similar sets, and it raises also some interesting questions for the more general setting.