On Newton's Method for Entire Functions

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.

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

Cited by in corpus (5)

• Accesses to infinity from Fatou components
• A Combinatorial classification of postcritically fixed Newton maps
• Newton's method and Baker domains
• A characterization of postcritically minimal Newton maps of complex exponential functions
• Dynamics of projectable functions: Towards an atlas of wandering domains for a family of Newton maps