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arXiv:math/0505652 [math.DS]AbstractReferencesReviewsResources

On Newton's Method for Entire Functions

Johannes Rueckert, Dierk Schleicher

Published 2005-05-30, updated 2006-05-17Version 2

The Newton map N_f 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 N_f. We study the behavior of N_f in a component V of C\U. If V can be surrounded by an invariant curve within U and satisfies the condition that each point in the extended plane has at most finitely many preimages in V, we show that V contains another immediate basin of N_f or a virtual immediate basin. A virtual immediate basin is an unbounded invariant Fatou component in which the dynamics converges to infty through an absorbing set.

Comments: 19 pages, 4 figures. Changes in Version 2: Sharpened the result in Section 4
Journal: J London Math Soc (2) 75 (2007) 659-676
Categories: math.DS
Subjects: 30D05, 37F10, 37F20, 49M15
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