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We study the discrete dynamical system defined on a subset of $R^2$ given bythe iterates of the secant method applied to a real polynomial $p$. Each simplereal root $\alpha$ of $p$ has associated its basin of attraction $\mathcalA(\alpha)$ formed by the set of points converging towards the fixed point$(\alpha,\alpha)$ of $S$. We denote by $\mathcal A^*(\alpha)$ its immediatebasin 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 $p$. Moreprecisely, we show the existence of a 4-cycle in $\partial \mathcalA^*(\alpha)$ and we give conditions on $p$ to guarantee the simple connectivityof $\mathcal A^*(\alpha)$.

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