  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 $\mathcal A^*(\alpha)$.
