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Rings Around Sierpinski Holes

Author

Listed:
  • HYEGYONG JANG

    (Department of Mathematics, University of Science, Pyongyang, D. P. R. Korea)

  • SINHYOK PAK

    (Department of Mathematics, University of Science, Pyongyang, D. P. R. Korea)

  • SONGIL RI

    (Department of Mathematics, University of Science, Pyongyang, D. P. R. Korea)

Abstract

For the family of rational maps Fλ(z) = zn + λ/zn where n ≥ 2, it is known that there are infinitely many Mandelpiński necklaces 𠒮k with k ∈ ℕ around the McMullen domain surrounding the origin in the parameter λ-plane. In this paper, we prove the existence of infinitely many these rings with a number of 𠒮k for fixed k outside the Mandelpiński necklace 𠒮1. The ring 𠒮2 is a simple closed curve meeting 𠒮1 at n − 1 points, such that it passes through exactly n2 − 1 centers of Sierpinski holes and n(n − 1) superstable parameter values. For each k ≥ 3, 𠒮k passes through precisely alternating 2n superstable parameter values and the same number of centers of Sierpiński holes. There exist (n − 1) disjoint rings 𠒮k+1 not meeting 𠒮k−1 and surrounding the centers of Sierpiński holes lying on 𠒮k, in the exterior and interior of a curve 𠒮k−1, respectively. The number of such rings 𠒮k for fixed k is 2k−3(n − 1)k−1.

Suggested Citation

  • Hyegyong Jang & Sinhyok Pak & Songil Ri, 2022. "Rings Around Sierpinski Holes," FRACTALS (fractals), World Scientific Publishing Co. Pte. Ltd., vol. 30(06), pages 1-12, September.
    • Handle: RePEc:wsi:fracta:v:30:y:2022:i:06:n:s0218348x22501274
    DOI: 10.1142/S0218348X22501274
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