Computational Methods and Function Theory 1 (2001), No. 1, 61--79
Copyright Heldermann Verlag 2001
Franz Peherstorfer
Franz.Peherstorfer@jk.uni-linz.ac.at, Johannes Kepler Universität Linz, Institut für Mathematik, A-4040 Linz, Austria.
Christoph Stroh
Christoph.Stroh@jk.uni-linz.ac.at, Johannes Kepler Universität Linz, Institut für Mathematik, A-4040 Linz, Austria.
For a polynomial P it is well known that its Julia set $\mathcal{J}_P$ is connected if and only if the orbits of the finite critical points are bounded. But there is no such simple criteria for the connectedness of the Julia set of a rational function. Indeed, up to the very nice result of Shishikura that any rational function which has one repelling fixed point only has a connected Julia set almost nothing is known on the connectivity. In the first part of the paper we give constructive sufficient conditions for a basin of attraction to be completely invariant and the Julia set to be connected. Then it is shown that the connectedness of a basin of attraction depends heavily on the fact whether the critical points from the basin tend to the attracting fixed point $z_0$ via a preimage of $z_0$ or not. As a consequence we obtain for instance that rational functions with a finite postcritical set or with a Fatou set which contains no Herman rings and each component of which contains at most one critical point, counted without multiplicity, have a connected Julia set.
Keywords: Julia set, Fatou set, basin of attraction, connectedness.
MSC 2000: 37F10.
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