Newton's method and Baker domains

TL;DR

This paper constructs specific entire functions to explore the dynamics of their associated Newton functions, demonstrating the existence of functions without Baker domains and analyzing the structure of their attracting basins.

Contribution

It provides new examples of entire functions with particular Newton dynamics, answering open questions about Baker domains and basin structures.

Findings

Existence of entire functions with Newton functions lacking Baker domains.

Construction of functions with a single zero and complex basin structures.

Answers to open questions in complex dynamics regarding Baker domains.

Abstract

We show that there exists an entire function f without zeros for which the associated Newton function N(z)=z-f(z)/f'(z) is a transcendental meromorphic functions without Baker domains. We also show that there exists an entire function f with exactly one zero for which the complement of the immediate attracting basin has at least two components and contains no invariant Baker domains of N. The second result answers a question of J. Rueckert and D. Schleicher while the first one gives a partial answer to a question of X. Buff.