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Please use this identifier to cite or link to this item: https://hdl.handle.net/2445/189388
Topological properties of the immediate basins of attraction for the secant method
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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 $\mathcal{A}^*(\alpha)$.
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GARDINI, Laura, GARIJO, Antonio and JARQUE I RIBERA, Xavier. Topological properties of the immediate basins of attraction for the secant method. Mediterranean Journal of Mathematics. 2021. Vol. 18, num. 221. ISSN 1660-5446. [consulted: 17 of June of 2026]. Available at: https://hdl.handle.net/2445/189388