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Proceedings of the London Mathematical Society Advance Access published online on April 15, 2008
Proceedings of the London Mathematical Society, doi:10.1112/plms/pdn012
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© 2008 London Mathematical Society

On connectivity of Julia sets of transcendental meromorphic maps and weakly repelling fixed points I

Núria Fagella, Xavier Jarque and Jordi Taixés

Departament de Matemàtica Aplicada i Anàlisi
Universitat de Barcelona
Gran Via, 585
08007 Barcelona
Spain
fagella@maia.ub.es
xavier.jarque@ub.edu

Received 22 November 2006. Revision received 20 December 2007.

It is known that the Julia set of the Newton method of a non-constant polynomial is connected (Mitsuhiro Shishikura, Preprint, 1990, M/90/37, Inst. Hautes Études Sci.). This is, in fact, a consequence of a much more general result that establishes the relationship between simple connectivity of Fatou components of rational maps and fixed points which are repelling or parabolic with multiplier 1. In this paper we study Fatou components of transcendental meromorphic functions; that is, we show the existence of such fixed points, provided that immediate attractive basins or preperiodic components are multiply connected.

2000 Mathematics Subject Classification 30D05 (primary), 37F10, 30D30, 37F30, 37F50 (secondary).

All authors were supported by MTM2005–02139. Also,the first and the second authors were supported by MTM2006–05849/Consolider (including a FEDER contribution), the second author by CIRIT no. 2005SGR-00550 and the first and the third authors by 2005SGR–01028.


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