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Newton's method and a class of meromorphic functions without wandering domains

Published online by Cambridge University Press:  19 September 2008

Walter Bergweiler
Affiliation:
Lehrstuhl II für Mathematik, RWTH Aachen, Templergraben 55, D-5100 Aachen, Germany†

Abstract

Let N be the class of meromorphic functions f with the following properties: f has finitely many poles;f′ has finitely many multiple zeros; the superattracting fixed points of f are zeros of f′ and vice versa, with finitely many exceptions; f has finite order. It is proved that if fN, then f does not have wandering domains. Moreover, if fN and if ∞ is among the limit functions of fn in a cycle of periodic domains, then this cycle contains a singularity of f−1. (Here fn denotes the nth iterate of f) These results are applied to study Newton's method for entire functions g of the form where p and q are polynomials and where c is a constant. In this case, the Newton iteration function f(z) = zg(z)/g′(z) is in N. It follows that fn(z) converges to zeros of g for all z in the Fatou set of f, if this is the case for all zeros z of g″. Some of the results can be extended to the relaxed Newton method.

Type
Research Article
Copyright
Copyright © Cambridge University Press 1993

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