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On the Dynamics of the Rational Family $f_{t}(z)=- t/4{(z^{2}- 2)^{2}/(z^{2}- 1)}$

On the Dynamics of the Rational Family $f_{t}(z)=- t/4{(z^{2}- 2)^{2}/(z^{2}- 1)}$ In this paper we discuss the dynamics as well as the structure of the parameter space of the one-parameter family of rational maps $${f_{t}(z)}= - {t\over 4} {(z^{2}- 2)^{2}\over {z^{2}- 1}}$$ with free critical orbit $$\pm\sqrt{2}\mathop \rightarrow \limits^{(2)}0 \mathop \rightarrow \limits^{(4)}t \mathop\rightarrow \limits^{(1)}\cdots.$$ In particular we show that for any escape parameter t, the boundary of the basin at infinity A t is either a Cantor set, a curve with infinitely many complementary components, or else a Jordan curve. In the latter case the Julia set is a Sierpiński curve. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Computational Methods and Function Theory Springer Journals

On the Dynamics of the Rational Family $f_{t}(z)=- t/4{(z^{2}- 2)^{2}/(z^{2}- 1)}$

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Publisher
Springer Journals
Copyright
Copyright © 2012 by Heldermann  Verlag
Subject
Mathematics; Analysis; Computational Mathematics and Numerical Analysis; Functions of a Complex Variable
ISSN
1617-9447
eISSN
2195-3724
DOI
10.1007/BF03321809
Publisher site
See Article on Publisher Site

Abstract

In this paper we discuss the dynamics as well as the structure of the parameter space of the one-parameter family of rational maps $${f_{t}(z)}= - {t\over 4} {(z^{2}- 2)^{2}\over {z^{2}- 1}}$$ with free critical orbit $$\pm\sqrt{2}\mathop \rightarrow \limits^{(2)}0 \mathop \rightarrow \limits^{(4)}t \mathop\rightarrow \limits^{(1)}\cdots.$$ In particular we show that for any escape parameter t, the boundary of the basin at infinity A t is either a Cantor set, a curve with infinitely many complementary components, or else a Jordan curve. In the latter case the Julia set is a Sierpiński curve.

Journal

Computational Methods and Function TheorySpringer Journals

Published: Aug 24, 2011

References