Newton's method and Baker domains

Abstract

We show that there exists an entire function f without zeros for which the associated Newton function is a transcendental meromorphic functions without Baker domains. We also show that there exists an entire function f with exactly one zero for which the complement of the immediate attracting basin has at least two components and contains no invariant Baker domains of N _f . The second result answers a question of Rückert and Schleicher, while the first one gives a partial answer to a question of Buff.

Keywords:

• transcendental meromorphic functions
• Newton function
• Baker domains
• Newton

Acknowledgements

Support by the German–Israeli Foundation (G.I.F.) for Scientific Research and Development, Grant G-809-234.6/2003, the EU Research Training Network CODY and the ESF Research Networking Programme HCAA is gratefully acknowledged.

 I thank Xavier Buff, Nuria Fagella, Lasse Rempe, Phil Rippon, Johannes Rückert and Dierk Schleicher for useful discussions on the topics of this paper. I am also grateful to the referee for some helpful comments.