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