Simple Mandelpinski necklaces for z2 + λ/z2

Daniel Cuzzocreo, Robert L. Devaney*

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

For the family of maps Fλ(z) = zn + λ/zn 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 Ik for k = 0, 1, 2, …. These are simple closed curves surrounding the McMullen domain and passing through exactly (n − 2)nk + 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 Ik in this case. Now these necklaces converge down to the origin. And, consistent with the formula for higher values of n, each Ik passes through the centers of only one Mandelbrot set and one Sierpinski hole.

Original languageEnglish (US)
Title of host publicationDifference Equations, Discrete Dynamical Systems and Applications - ICDEA 2012
EditorsJim M. Cushing, Alberto A. Pinto, Saber Elaydi, Lluis Alseda i Soler
PublisherSpringer New York LLC
Pages63-72
Number of pages10
ISBN (Print)9783662529263
DOIs
StatePublished - Jan 1 2016
Event18th International Conference on Difference Equations and Applications, ICDEA 2012 - Barcelona, Spain
Duration: Jul 23 2012Jul 27 2012

Publication series

NameSpringer Proceedings in Mathematics and Statistics
Volume180
ISSN (Print)2194-1009
ISSN (Electronic)2194-1017

Other

Other18th International Conference on Difference Equations and Applications, ICDEA 2012
CountrySpain
CityBarcelona
Period7/23/127/27/12

Keywords

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

ASJC Scopus subject areas

  • Mathematics(all)

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  • Cite this

    Cuzzocreo, D., & Devaney, R. L. (2016). Simple Mandelpinski necklaces for z2 + λ/z2. 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