Abstract
For a polynomial P it is well known that its Julia set \({\cal J_P}\) 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.
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The second author has been supported by the Austrian Science Fund (FWF) under the project P12985-TEC.
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Peherstorfer, F., Stroh, C. Connectedness of Julia Sets of Rational Functions. Comput. Methods Funct. Theory 1, 61–79 (2001). https://doi.org/10.1007/BF03320977
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DOI: https://doi.org/10.1007/BF03320977