Topological properties of the immediate basins of attraction for the secant method

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 $\alpha$ 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 ${\mathcal{A}}^{\ast }(\alpha )$ 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 ${\mathcal{A}}^{\ast }(\alpha )$, when $\alpha$ is an internal real root of $p$. More precisely, we show the existence of a 4-cycle in $\partial {\mathcal{A}}^{\ast }(\alpha )$ and we give conditions on $p$ to guarantee the simple connectivity of ${\mathcal{A}}^{\ast }(\alpha )$.

24 pages, 19 figures