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On Self-Similarity Of Bounded Regions In The Plane

Author

Listed:
  • 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)

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.

Suggested Citation

  • Xuemin Wang & Kan Jiang & Lifeng Xi, 2025. "On Self-Similarity Of Bounded Regions In The Plane," FRACTALS (fractals), World Scientific Publishing Co. Pte. Ltd., vol. 33(07), pages 1-5.
    • Handle: RePEc:wsi:fracta:v:33:y:2025:i:07:n:s0218348x25500537
    DOI: 10.1142/S0218348X25500537
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