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

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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.

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Authors and Affiliations

  1. Faculty of Mathematics and Mechanics, University of Science Pyongyang, D.P.R. of Korea

    Hye Gyong Jang

  2. Fakultät für Mathematik, Technische Universität Dortmund, Germany

    Norbert Steinmetz

Authors
  1. Hye Gyong Jang
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  2. Norbert Steinmetz
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Correspondence to Hye Gyong Jang.

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Jang, H.G., Steinmetz, N. On the Dynamics of the Rational Family \(f_{t}(z)=- t/4{(z^{2}- 2)^{2}/(z^{2}- 1)}\) . Comput. Methods Funct. Theory 12, 1–17 (2012). https://doi.org/10.1007/BF03321809

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  • DOI: https://doi.org/10.1007/BF03321809

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