RINGS AROUND SIERPINSKI HOLES

Hyegyong Jang, Sinhyok Pak and Songil Ri
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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

FRACTALS (fractals), 2022, vol. 30, issue 06, 1-12

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.

Keywords: Mandelpinski Necklaces; Baby Mandelbrot Set; Sierpinski Holes; McMullen Family (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1142/S0218348X22501274

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