Newton's method and Baker domains

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.

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