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Computational Methods and Function Theory 12 (2012), No. 1, 1--17
Copyright Heldermann Verlag 2012

On the Dynamics of the Rational Family ft(z)=-t/4 (z2-2)2/(z2-1)

Hye Gyong Jang
pptayang@co.chesin.com , University of Science Pyongyang, Faculty of Mathematics and Mechanics, D.P.R. of Korea.

Norbert Steinmetz
stein@math.tu-dortmund.de , Technische Universität Dortmund, Fakultät für Mathematik, Germany.

[Abstract-pdf] [Abstract-ps]

In this paper we discuss the dynamics as well as the structure of the parameter space of the one-parameter family of rational maps
ft(z)=-t/4 (z2-2)2/(z2-1)
with free critical orbit
±√2–(2)→0–(4)→t–(1)→...
In particular we show that for any escape parameter t, the boundary of the basin at infinity At 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.

Keywords: Julia set, Mandelbrot set, hyperbolic component, escape component, Sierpiński curve, bifurcation locus, Misiurewicz point.

MSC 2000: 37F10, 37F15, 37F45.

[FullText-pdf (16240 K)] [FullText-ps (15400 K)]