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Journal of the London Mathematical Society Advance Access published online on May 27, 2008
Journal of the London Mathematical Society, doi:10.1112/jlms/jdn030
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© 2008 London Mathematical Society

Sierpinski and non-Sierpinski curve Julia sets in families of rational maps

Norbert Steinmetz

Technische Universität (TU) Dortmund
Fakultät für Mathematik
D-44221 Dortmund
Germany

We discuss the dynamics as well as the structure of the parameter plane of certain families of rational maps with few critical orbits. Our paradigm is the family Rt(z) = t (1 + (4/27)z3/(1 – z)), with dynamics governed by the behaviour of the postcritical orbit (Rtn(t))nisinN. In particular, it is shown that if t escapes (that is, Rtn(t) tends to infinity), then the Julia set of Rt is a Cantor set, or a Sierpinski curve, or a curve with one or else infinitely many cut-points; each of these cases actually occurs.

2000 Mathematics Subject Classification 37F10, 37F15, 37F45.

Received September 20, 2006; revised July 13, 2007;
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