Sufficient condition for a topological self-similar set to be a self-similar set

IF 0.6 4区数学 Q3 MATHEMATICS

Topology and its Applications Pub Date : 2024-10-24 DOI:10.1016/j.topol.2024.109115

Tianjia Ni , Zhiying Wen

引用次数: 0

Abstract

A self-similar set always possesses a self-similar topological structure coded by the shift space (symbolic space), which is considered as the coordinate system for this set. On the contrary, it is known that given a compact set K with self-similar topological structure, there may not exist a metric d such that

(K, d)

is a self-similar set with the same topological structure. We provide an easy-to-use sufficient condition for the existence of such metric d in terms of the associated graph with respect to the self-similar topological structure. Therefore, one can easily construct a required self-similar set from the shift space by specifying the topological structure.

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拓扑自相似集合是自相似集合的充分条件

自相似集合总是具有由移位空间（符号空间）编码的自相似拓扑结构，移位空间被视为该集合的坐标系。相反，已知给定一个具有自相似拓扑结构的紧凑集合 K，可能不存在一个度量 d，使得 (K,d) 是一个具有相同拓扑结构的自相似集合。我们提供了一个易于使用的充分条件，即在相关图中存在与自相似拓扑结构有关的度量 d。因此，只要指定拓扑结构，就能轻松地从移位空间构造出所需的自相似集合。

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

Topology and its Applications 数学-数学

CiteScore

1.20

自引率

33.30%

发文量

251

审稿时长

6 months

期刊介绍： Topology and its Applications is primarily concerned with publishing original research papers of moderate length. However, a limited number of carefully selected survey or expository papers are also included. The mathematical focus of the journal is that suggested by the title: Research in Topology. It is felt that it is inadvisable to attempt a definitive description of topology as understood for this journal. Certainly the subject includes the algebraic, general, geometric, and set-theoretic facets of topology as well as areas of interactions between topology and other mathematical disciplines, e.g. topological algebra, topological dynamics, functional analysis, category theory. Since the roles of various aspects of topology continue to change, the non-specific delineation of topics serves to reflect the current state of research in topology. At regular intervals, the journal publishes a section entitled Open Problems in Topology, edited by J. van Mill and G.M. Reed. This is a status report on the 1100 problems listed in the book of the same name published by North-Holland in 1990, edited by van Mill and Reed.