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

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...

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Bibliographic Details
Published in:Topology and its applications 2024-12, Vol.358, p.109115, Article 109115
Main Authors: Ni, Tianjia, Wen, Zhiying
Format: Article
Language:English
Subjects:
Hyperbolic graph
IFS
Metrization
Quotient space
Self-similar set
Topological self-similar set
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Summary: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.
ISSN:0166-8641
DOI:10.1016/j.topol.2024.109115