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

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 $$\alpha $$ of p has associated its basin of attraction $${\mathcal {A}}(\alpha )$$ formed by the set of points converging towards the fixed point $$(\alpha ,\alpha )$$ of S. We denote by $${\mathcal {A}}^*(\alpha )$$ its immediate basin of attraction, that is, the connected component of $${\mathcal {A}}(\alpha )$$ which contains $$(\alpha ,\alpha )$$ . We focus on some topological properties of $${\mathcal {A}}^*(\alpha )$$ , when $$\alpha $$ is an internal real root of p. More precisely, we show the existence of a 4-cycle in $$\partial {\mathcal {A}}^*(\alpha )$$ and we give conditions on p to guarantee the simple connectivity of $${\mathcal {A}}^*(\alpha )$$ .