ON SELF-SIMILARITY OF BOUNDED REGIONS IN THE PLANE

Xuemin Wang (), Kan Jiang () and Lifeng Xi
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Xuemin Wang: School of Information Technology, Zhejiang Fashion Institute of Technology, Ningbo 315211, P. R. China
Kan Jiang: Department of Mathematics, Ningbo University, P. R. China
Lifeng Xi: Department of Mathematics, Ningbo University, P. R. China

FRACTALS (fractals), 2025, vol. 33, issue 07, 1-5

Abstract: Wen posed the following question at the Chinese Conference on Fractal Geometry and Dynamical Systems 2023: under what conditions is a quadrangle a self-similar set with the open set condition? Xu, Xi and Jiang [On self-similarity of quadrangle, Fractals 32(5) (2024) 2450096] proved that for any trapezoid, if the ratio of the lengths of two bases is rational, then the trapezoid is a self-similar set with the open set condition. Motivated by this result and Wen’s question, it is natural to consider the self-similarity of a bounded planar region with positive two-dimensional Lebesgue measure. In this paper, we partially address this problem as follows. Suppose Γ ⊂ ℠2 is a C2 non-degenerate Jordan curve in the plane such that any point on Γ has non-zero curvature. Let D be a bounded closed region with Γ as its boundary. Then D is not a self-similar set. Similar result can be obtained in ℠3.

Keywords: Self-Similar Sets; Similitude; Jordan Region (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1142/S0218348X25500537

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