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Remarks on the simple connectedness of basins of sinks for iterations of rational maps. (English) Zbl 0703.58033
Dynamical systems and ergodic theory, 28th Sem. St. Banach Int. Math. Cent., Warsaw/Pol. 1986, Banach Cent. Publ. 23, 229-235 (1989).
[For the entire collection see Zbl 0686.00015.]
Consider a polynomial P: \({\mathbb{C}}\to {\mathbb{C}}\). Newton’s method of looking for its roots is to consider iterates of the rational function \(NP(z)=z- P(z)/P'(z)\) on the Riemann sphere \({\hat {\mathbb{C}}}\). Roots of P are fixed points, sinks for NP. For every P, a root of P, the set of points whose trajectories under iteration of NP converge to p splits into components. The component containing p is called the immediate basin of attraction to p.
The author proves the following Theorem A. Immediate basins of attraction to the roots of a complex polynomial, for Newton’s method, are simply connected.
This answers a question of A. Manning [“How to be sure of solving a complex polynomial using Newton’s method”, Preprint, Univ. Warwick, Nov. 1986].
Theorem B. Immediate basins of attraction to periodic sinks of period at least 2 for a rational map of degree 2 are simply connected.
The methods used by the author are classical, going back to the famous memoirs by P. Fatou and G. Julia.
Reviewer: G.M.Rassias

MSC:
37C70 Attractors and repellers of smooth dynamical systems and their topological structure
30D30 Meromorphic functions of one complex variable, general theory
30D50 Blaschke products, etc. (MSC2000)
37B99 Topological dynamics
30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable