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Journal of the London Mathematical Society Advance Access originally published online on July 11, 2007
Journal of the London Mathematical Society 2007 75(3):659-676; doi:10.1112/jlms/jdm046
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© 2007 London Mathematical Society

On Newton's method for entire functions

Johannes Rückert and Dierk Schleicher

School of Engineering and Science
International University Bremen
Postfach 750 561
28725 Bremen
Germany
jrueckert{at}jacobs-alumni.de

The Newton map Nf of an entire function f turns the roots of f into attracting fixed points. Let U be the immediate attracting basin for such a fixed point of Nf. We study the behavior of Nf in a component V of C\U. If V can be surrounded by an invariant curve within U and satisfies the condition that for all z isin Formula, Nf–1({z}) {cap} V is a finite set, then it is shown that V contains another immediate basin of Nf or a virtual immediate basin.

dierk{at}iu-bremen.de

2000 Mathematics Subject Classification 30D05, 37F10, 37F20, 49M15.

Received May 30, 2005; revised May 17, 2006; published online July 11, 2007.


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