On Newton's Method for Entire Functions
Abstract
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. 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 Nf 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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