A NOTE ON TOPOLOGY OF FRACTAL SQUARES WITH ORDER THREE

Jun Luo () and Xiao-Ting Yao
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Jun Luo: School of Mathematics, Sun Yat-Sen University, Guangzhou 510275, P. R. China
Xiao-Ting Yao: School of Mathematics, Sun Yat-Sen University, Guangzhou 510275, P. R. China

FRACTALS (fractals), 2021, vol. 29, issue 01, 1-11

Abstract: We consider a family of fractal squares, denoted as ℱ3,7. Each of them satisfies the set equation K = 1 3(K + 𠒟) for some 𠒟⊂{0, 1, 2}2 with #𠒟 = 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 Gaut(K) of all homeomorphisms of a fractal square K ∈ℱ3,7 that has a cut point and show that #Gaut(K) = 2 or 8.

Keywords: Fractal Square; Irreducible Cut; Fiber (search for similar items in EconPapers)
Date: 2021
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DOI: 10.1142/S0218348X21500055

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