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Julia Sets of Cubic Rational Maps with Escaping Critical Points

Julia Sets of Cubic Rational Maps with Escaping Critical Points It is known that the Julia set of a quadratic rational map is either connected or a Cantor set. In this paper, we explore this dichotomy for the maps in a type of three-dimensional space of cubic rational maps. We show that for a cubic rational map f, if f has an attracting fixed point p and all critical points are attracted to p under the iteration of f, then the Julia set J(f) is either a Cantor set or a connected set (and locally connected) with one possible exception; we also give a necessary and sufficient condition for J(f) to be a Sierpinski curve when it is connected. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Arnold Mathematical Journal Springer Journals

Julia Sets of Cubic Rational Maps with Escaping Critical Points

Arnold Mathematical Journal , Volume 6 (3-4) – Jul 7, 2020

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Publisher
Springer Journals
Copyright
Copyright © Institute for Mathematical Sciences (IMS), Stony Brook University, NY 2020
ISSN
2199-6792
eISSN
2199-6806
DOI
10.1007/s40598-020-00146-8
Publisher site
See Article on Publisher Site

Abstract

It is known that the Julia set of a quadratic rational map is either connected or a Cantor set. In this paper, we explore this dichotomy for the maps in a type of three-dimensional space of cubic rational maps. We show that for a cubic rational map f, if f has an attracting fixed point p and all critical points are attracted to p under the iteration of f, then the Julia set J(f) is either a Cantor set or a connected set (and locally connected) with one possible exception; we also give a necessary and sufficient condition for J(f) to be a Sierpinski curve when it is connected.

Journal

Arnold Mathematical JournalSpringer Journals

Published: Jul 7, 2020

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