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.

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 |