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Zalcman functions and similarity between the Mandelbrot set, Julia sets, and the tricorn

Zalcman functions and similarity between the Mandelbrot set, Julia sets, and the tricorn We present a simple proof of Tan’s theorem on asymptotic similarity between the Mandelbrot set and Julia sets at Misiurewicz parameters. Then we give a new perspective on this phenomenon in terms of Zalcman functions, that is, entire functions generated by applying Zalcman’s lemma to complex dynamics. We also show asymptotic similarity between the tricorn and Julia sets at Misiurewicz parameters, which is an antiholomorphic counterpart of Tan’s theorem. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Analysis and Mathematical Physics Springer Journals

Zalcman functions and similarity between the Mandelbrot set, Julia sets, and the tricorn

Analysis and Mathematical Physics , Volume 10 (2) – Mar 6, 2020

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Publisher
Springer Journals
Copyright
Copyright © Springer Nature Switzerland AG 2020
ISSN
1664-2368
eISSN
1664-235X
DOI
10.1007/s13324-020-00357-4
Publisher site
See Article on Publisher Site

Abstract

We present a simple proof of Tan’s theorem on asymptotic similarity between the Mandelbrot set and Julia sets at Misiurewicz parameters. Then we give a new perspective on this phenomenon in terms of Zalcman functions, that is, entire functions generated by applying Zalcman’s lemma to complex dynamics. We also show asymptotic similarity between the tricorn and Julia sets at Misiurewicz parameters, which is an antiholomorphic counterpart of Tan’s theorem.

Journal

Analysis and Mathematical PhysicsSpringer Journals

Published: Mar 6, 2020

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