On Newton's Method for Entire Functions
Dynamical Systems
2007-08-21 v2
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.
Cite
@article{arxiv.math/0505652,
title = {On Newton's Method for Entire Functions},
author = {Johannes Rueckert and Dierk Schleicher},
journal= {arXiv preprint arXiv:math/0505652},
year = {2007}
}
Comments
19 pages, 4 figures. Changes in Version 2: Sharpened the result in Section 4