On Newton's method for entire functions

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 for all z ∈ ℂ̂, N f −1 ({ z }) ∩ V is a finite set, then it is shown that V contains another immediate basin of N f or a virtual immediate basin.