标题： 哈雷方法的动力学 显示英文标题

标题： On the dynamics of Halley's method

摘要： 在本文中，我们研究了将Halley方法应用于复多项式时的全局动力学。 具体来说，我们分析了该方法的Julia集的结构和连通性。 对于各种多项式类，包括具有非平凡对称群的单临界多项式、三次多项式和四次多项式，研究了相应的Fatou集和Julia集的收敛行为、对称性质和拓扑特征。 特别是，我们证明了当$p$属于上述类别之一时，Halley方法$H_p$是收敛的，其Julia集是连通的，瞬时盆地是无界的，并且其对称群与多项式的对称群一致。 我们进一步将结果扩展到更广泛的多项式类。 结果显示，对应于$p$的一个根的Halley方法$H_p$的瞬时盆地可以是有界的。 我们还对一般三次多项式应用Halley方法的动力学做出了一些备注。 显示英文摘要

摘要： In this article, we study the global dynamics of Halley's method applied to complex polynomials. Specifically, we analyze the structure and connectivity of the Julia set of this method. The convergence behavior, symmetry properties, and topological features of the corresponding Fatou and Julia sets are studied for various classes of polynomials, including unicritical, cubic, and quartic polynomials with non-trivial symmetry groups. In particular, we prove that the Halley's method $H_p$ is convergent, its Julia set is connected, the immediate basins are unbounded and the symmetry group of it coincides with that of the polynomial whenever $p$ belongs to one of the above classes. We further extend our results to a broader class of polynomials. It is shown that the immediate basin of the Halley's method $H_p$ corresponding to a root of $p$ can be bounded. We also make some remarks on the dynamics of the Halley's method applied to a cubic polynomial in general.