Topological Properties of the Immediate Basins of Attraction for the Secant Method

Identifier:  imarina:9228573

Handle: https://hdl.handle.net/20.500.11797/imarina9228573

Authors:  Gardini, Laura; Garijo, Antonio; Jarque, Xavier

Abstract:
We study the discrete dynamical system defined on a subset of R-2 given by the iterates of the secant method applied to a real polynomial p. Each simple real root a of p has associated its basin of attraction A(alpha) formed by the set of points converging towards the fixed point (alpha, alpha) of S. We denote by A* (alpha) its immediate basin of attraction, that is, the connected component of A( a) which contains (alpha, alpha). We focus on some topological properties of A* (alpha), when a is an internal real root of p. More precisely, we show the existence of a 4-cycle in. A* (alpha) and we give conditions on p to guarantee the simple connectivity of A* (alpha).