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

# metadata.dc.description.abstract[].value
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 $\mathcal{A}^*(\alpha)$.

# metadata.dc.relation.ispartof[].value
Mediterranean Journal of Mathematics, 2021, vol. 18, num. 221

# metadata.dc.source[].value
Articles publicats en revistes (Matemàtiques i Informàtica)

# metadata.dc.title[].value
Topological properties of the immediate basins of attraction for the secant method