论文标题
割线方法的直接吸引力盆地的拓扑特性
Topological properties of the immediate basins of attraction for the secant method
论文作者
论文摘要
我们研究了应用于真实多项式$ p $的SECANT方法给出的$ r^2 $子集定义的离散动力系统。 $ p $的每个简单的真实根$α$都将其吸引力的盆地$ \ MATHCAL A（α）$相关联，该点由$ s $的固定点$（α，α）$组成。我们用$ \ Mathcal a^*（α）$表示其吸引的直接盆地，即$ \ Mathcal a（α）$的连接组件，其中包含$（α，α）$。当$α$是$ p $的内部真实根时，我们专注于$ \ mathcal a^*（α）$的某些拓扑属性。更确切地说，我们在$ \ partial \ mathcal a^*（α）$中显示了4个周期的存在，并在$ p $上提供条件，以保证$ \ Mathcal a^*（α）$的简单连接。
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 $α$ of $p$ has associated its basin of attraction $\mathcal A(α)$ formed by the set of points converging towards the fixed point $(α,α)$ of $S$. We denote by $\mathcal A^*(α)$ its immediate basin of attraction, that is, the connected component of $\mathcal A(α)$ which contains $(α,α)$. We focus on some topological properties of $\mathcal A^*(α)$, when $α$ is an internal real root of $p$. More precisely, we show the existence of a 4-cycle in $\partial \mathcal A^*(α)$ and we give conditions on $p$ to guarantee the simple connectivity of $\mathcal A^*(α)$.
