On Newton's Method for Entire Functions

Details On Newton's Method for Entire Functions On Newton's Method for Entire Functions

2005-05-30

arxiv.org/abs/math/0505652v2

J London Math Soc (2) 75 (2007) 659-676

Mathematics

Dynamical Systems

19 pages, 4 figures. Changes in Version 2: Sharpened the result in Section 4

Scientific paper

10.1112/jlms/jdm046

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.

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