Hénon映射的合成根

Dynamics of Newton maps

In this paper, we study the dynamics of Newton maps for arbitrary polynomials. Let $p$ be an arbitrary polynomial with at least three distinct roots, and $f$ be its Newton map. It is shown that the boundary $\partial B$ of any immediate root basin $B$ of $f$ is locally connected. Moreover, $\partial B$ is a Jordan curve if and only if ${\rm deg}(f|_B)=2$. This implies that the boundaries of all components of root basins, for all polynomials' Newton maps, from the viewpoint of topology, are tame.

Newton映射的虚直接盆及其渐近值

Virtual Immediate Basins of Newton Maps and Asymptotic Values

牛顿求根法应用于（超越）整函数f：C->C是亚纯函数N的迭代。众所周知，如果对于某个起始值z，牛顿法收敛于C中的一个点x，则f在x处有根。我们表明，在许多情况下，如果一个轨道收敛到无穷大的牛顿法，那么f有一个“虚拟根”在无穷远。更精确地说，我们证明了，如果N有一个不变的贝克域，满足一些温和的假设，那么0是一个渐近值的f。相反，我们表明，如果f有一个渐近值的对数型在0，然后在0上的奇异性是包含在不变的贝克域N，我们称之为一个虚拟的即时盆地。我们通过反例表明，这是不正确的更一般类型的奇点。

Iterative square roots of functions

An iterative square root of a function $f$ is a function $g$ such that $g(g(\cdot))=f(\cdot)$. We obtain new characterizations for detecting the non-existence of such square roots for self-maps on arbitrary sets. This is used to prove that continuous self-maps with no square roots are dense in the space of all continuous self-maps for various topological spaces. The spaces studied include those that are homeomorphic to the unit cube in ${\mathbb R}^m$ and to the whole of $\mathbb{R}^m$ for every positive integer $m.$ On the other hand, we also prove that every continuous self-map of a space ho…

一元多项式与某些切映射的秩

Polynomials in one variable and ranks of certain tangent maps

研究了一个映射，它将一个无重根的单次n阶复多项式f（x）送到f（x）根上的n阶导数的集合.我们回答了Ju.S.伊利亚申科

放松牛顿法作为求根方法族：动力学与收敛性

Relaxed Newton's Method as a Family of Root-finding Methods: Dynamics and Convergence

本文研究了放松牛顿法作为求根方法族的动力学与收敛性，揭示了其在不同多项式下的收敛行为及复杂动力学特性。

An augmented phase plane approach for discrete planar maps: Introducing next-iterate operators

The next-iterate operators and corresponding next-iterate root-sets and root-curves associated with the nullclines of a planar discrete map are introduced. How to augment standard phase portraits that include the nullclines and the direction field, by including the signs of the root-operators associated with their nullclines, thus producing an augmented phase portrait, is described. The sign of a next-iterate operator associated with a nullcline determines whether a point is mapped above or below the corresponding nullcline and can, for example, identify positively invariant regions. Using a L…