### Abstract

For the family of maps Fλ(z) = z^{n} + λ/z^{n} where n ≥ 3, it is known that there is a McMullen domain surrounding the origin in the parameter plane. This domain is then surrounded by infinitely many “Mandelpinski” necklaces I_{k} for k = 0, 1, 2, …. These are simple closed curves surrounding the McMullen domain and passing through exactly (n − 2)n^{k} + 1 centers of baby Mandelbrot sets and the same number of centers of Sierpinski holes. When n = 2 there is no such McMullen domain in the parameter plane. However, we show in this paper that there do exist Mandelpinski necklaces I^{k} in this case. Now these necklaces converge down to the origin. And, consistent with the formula for higher values of n, each I^{k} passes through the centers of only one Mandelbrot set and one Sierpinski hole.

Original language | English (US) |
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Title of host publication | Difference Equations, Discrete Dynamical Systems and Applications - ICDEA 2012 |

Editors | Jim M. Cushing, Alberto A. Pinto, Saber Elaydi, Lluis Alseda i Soler |

Publisher | Springer New York LLC |

Pages | 63-72 |

Number of pages | 10 |

ISBN (Print) | 9783662529263 |

DOIs | |

State | Published - Jan 1 2016 |

Event | 18th International Conference on Difference Equations and Applications, ICDEA 2012 - Barcelona, Spain Duration: Jul 23 2012 → Jul 27 2012 |

### Publication series

Name | Springer Proceedings in Mathematics and Statistics |
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Volume | 180 |

ISSN (Print) | 2194-1009 |

ISSN (Electronic) | 2194-1017 |

### Other

Other | 18th International Conference on Difference Equations and Applications, ICDEA 2012 |
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Country | Spain |

City | Barcelona |

Period | 7/23/12 → 7/27/12 |

### Keywords

- Complex dynamics
- Julia set
- Mandelbrot set
- Mandelpinski necklace
- Rational map
- Sierpinski hole

### ASJC Scopus subject areas

- Mathematics(all)

## Fingerprint Dive into the research topics of 'Simple Mandelpinski necklaces for z<sup>2</sup> + λ/z<sup>2</sup>'. Together they form a unique fingerprint.

## Cite this

^{2}+ λ/z

^{2}. In J. M. Cushing, A. A. Pinto, S. Elaydi, & L. A. i Soler (Eds.),

*Difference Equations, Discrete Dynamical Systems and Applications - ICDEA 2012*(pp. 63-72). (Springer Proceedings in Mathematics and Statistics; Vol. 180). Springer New York LLC. https://doi.org/10.1007/978-3-662-52927-0_5