Simple Mandelpinski necklaces for z2 + λ/z2

Daniel Cuzzocreo; Robert L. Devaney

doi:10.1007/978-3-662-52927-0_5

Simple Mandelpinski necklaces for z² + λ/z²

Daniel Cuzzocreo, Robert L. Devaney^*

^*Corresponding author for this work

Mathematics

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

2 Scopus citations

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 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)
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	https://doi.org/10.1007/978-3-662-52927-0_5
State	Published - 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
Volume	180
ISSN (Print)	2194-1009
ISSN (Electronic)	2194-1017

Other

Other	18th International Conference on Difference Equations and Applications, ICDEA 2012
Country/Territory	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

General Mathematics

Access to Document

10.1007/978-3-662-52927-0_5

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

@inproceedings{1ea483233d594d4fb8c6043549438191,

title = "Simple Mandelpinski necklaces for z2 + λ/z2",

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.",

keywords = "Complex dynamics, Julia set, Mandelbrot set, Mandelpinski necklace, Rational map, Sierpinski hole",

author = "Daniel Cuzzocreo and Devaney, {Robert L.}",

note = "Publisher Copyright: {\textcopyright} Springer-Verlag Berlin Heidelberg 2016.; 18th International Conference on Difference Equations and Applications, ICDEA 2012 ; Conference date: 23-07-2012 Through 27-07-2012",

year = "2016",

doi = "10.1007/978-3-662-52927-0_5",

language = "English (US)",

isbn = "9783662529263",

series = "Springer Proceedings in Mathematics and Statistics",

publisher = "Springer New York LLC",

pages = "63--72",

editor = "Cushing, {Jim M.} and Pinto, {Alberto A.} and Saber Elaydi and {i Soler}, {Lluis Alseda}",

booktitle = "Difference Equations, Discrete Dynamical Systems and Applications - ICDEA 2012",

}

Cuzzocreo, D & Devaney, RL 2016, Simple Mandelpinski necklaces for z² + λ/z². in JM Cushing, AA Pinto, S Elaydi & LA i Soler (eds), Difference Equations, Discrete Dynamical Systems and Applications - ICDEA 2012. Springer Proceedings in Mathematics and Statistics, vol. 180, Springer New York LLC, pp. 63-72, 18th International Conference on Difference Equations and Applications, ICDEA 2012, Barcelona, Spain, 7/23/12. https://doi.org/10.1007/978-3-662-52927-0_5

Simple Mandelpinski necklaces for z² + λ/z². / Cuzzocreo, Daniel; Devaney, Robert L.
Difference Equations, Discrete Dynamical Systems and Applications - ICDEA 2012. ed. / Jim M. Cushing; Alberto A. Pinto; Saber Elaydi; Lluis Alseda i Soler. Springer New York LLC, 2016. p. 63-72 (Springer Proceedings in Mathematics and Statistics; Vol. 180).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

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N2 - 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.

AB - 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.

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KW - Julia set

KW - Mandelbrot set

KW - Mandelpinski necklace

KW - Rational map

KW - Sierpinski hole

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