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arXiv · 2006.01528

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

Laura Gardini·Antonio Garijo·Xavier Jarque
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Abstract

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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Laura Gardini, Antonio Garijo, Xavier Jarque. 2020-06-02. Topological properties of the immediate basins of attraction for the secant method. https://arxiv.org/abs/2006.01528

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