发表机构
Indian Institute of Technology Bhubaneswar（印度技术学院布巴内斯瓦尔分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究探讨切比雪夫方法应用于多项式族$p_n(z)=z(z^n-1)$的动力学，明确其吸引域性质、收敛条件及对称群，证实了Nayak与Pal的相关猜想。
AI 中文摘要
我们研究了将切比雪夫方法应用于多项式族$p_n(z)=z(z^n-1)$（$n>1$）时的动力学特性，所得映射记为$C_n$。研究证明，对应于非零根的直接吸引域是无界且单连通的；还表明$C_n$的朱利亚集是连通的。进一步证明，原点处根的直接吸引域表现出不同行为：当$n\leq16$时为无界，当$n\geq17$时为有界。我们确定$C_n$在$n\leq16$或$n$为奇数时收敛；最后确定$C_n$的对称群与多项式$p_n$的对称群一致，从而针对该多项式族证实了Nayak和Pal提出的一个猜想。
英文摘要
We investigate the dynamics of Chebyshev's method applied to the polynomial family $p_n(z)=z(z^n-1)$ for $n>1$. The resulting map is denoted by $C_n$. It is proved that the immediate basins corresponding to the non-zero roots are unbounded and simply connected. We also show that the Julia set of $C_n$ is connected. It is proved that the immediate basin of the root at the origin exhibits a different behavior: it is unbounded for $n\leq 16$ and bounded for $n\geq 17$. We establish that $C_n$ is convergent whenever $n\leq 16$ or $n$ is odd. Finally, we determine the symmetry group of $C_n$ and prove that it coincides with the symmetry group of the polynomial $p_n$, thereby confirming, for this family, a conjecture proposed by Nayak and Pal.
Comments30 Pages
