Newton's method and 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.
- Publication:
-
arXiv e-prints
- Pub Date:
- October 2007
- DOI:
- 10.48550/arXiv.0710.1147
- arXiv:
- arXiv:0710.1147
- Bibcode:
- 2007arXiv0710.1147B
- Keywords:
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- Mathematics - Complex Variables;
- Mathematics - Dynamical Systems;
- 37F10;
- 30D05;
- 49M15;
- 65H05
- E-Print:
- 6 pages