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A NOTE ON TOPOLOGY OF FRACTAL SQUARES WITH ORDER THREE

JUN LUO

School of Mathematics, Sun Yat-Sen University, Guangzhou 510275, P. R. China

E-mail Address: luojun3@mail.sysu.edu.cn

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and

XIAO-TING YAO

https://orcid.org/0000-0002-0471-2799

School of Mathematics, Sun Yat-Sen University, Guangzhou 510275, P. R. China

E-mail Address: 152717315042@163.com

Corresponding author.

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https://doi.org/10.1142/S0218348X21500055Cited by:3 (Source: Crossref)

Abstract

We consider a family of fractal squares, denoted as $ℱ_{3, 7}$ . Each of them satisfies the set equation $K = \frac{1}{3} (K + 𝒟)$ for some $𝒟 \subset {0, 1, 2}^{2}$ with $# 𝒟 = 7$ . It is known that two of these fractal squares are Lipschitz equivalent if and only if they are isometrically equivalent. The aim of our study is to improve this by replacing Lipschitz equivalence with topological equivalence. To this end, we shall investigate the group $G_{aut} (K)$ of all homeomorphisms of a fractal square $K \in ℱ_{3, 7}$ that has a cut point and show that $# G_{aut} (K) = 2$ or $8$ .

Keywords:

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