Newton's method and Baker domains

Abstract

We show that there exists an entire function f without zeros for which the associated Newton function 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 f . The second result answers a question of Rückert and Schleicher, while the first one gives a partial answer to a question of Buff.