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On Newton's Method for Entire Functions

Johannes Rueckert·Dierk Schleicher
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Abstract

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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Johannes Rueckert, Dierk Schleicher. 2006-05-17. On Newton's Method for Entire Functions. https://doi.org/10.1112/jlms%2Fjdm046

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