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

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