Sebastian Mayer, Dierk Schleicher
We investigate the well known Newton method to find roots of entire holomorphic functions. Our main result is that the immediate basin of attraction for every root is simply connected and unbounded. We also introduce ``virtual immediate basins'' in which the dynamics converges to infinity; we prove that these are simply connected as well.
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We give sufficient conditions under which the set of eventually periodic points in the intersection of immediate attracting basins boundaries is non-empty. We give other conditions under which this set is dense in the intersection.
Dynamics on parabolic immediate basins for rational Newton maps of entire functions have been studied. It is proved that every parabolic immediate basin contains invariant accesses to the parabolic fixed point at infinity. Moreover, among these accesses there exists a unique dynamically defined access where dynamics are attracted towards the parabolic fixed point, whereas for other accesses, if there is any, the dynamics are repelled.
In this paper, we give a formula of the Hausdorff dimension of the boundary of the immediate basin of infinity of McMullen maps $f_p(z)=z^Q+p/z^Q$, where $Q\geq 3$ and $p$ is small. This gives a lower bound of the Hausdorff dimension of the Julia sets of McMullen maps in the special cases.
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…
The Newton map N_f of an entire function f turns the roots of f into attracting fixed points. Let U be the immediate attracting basin for such a fixed point of N_f. We study the behavior of N_f in a component V of C\U. If V can be surrounded by an invariant curve within U and satisfies the condition that each point in the extended plane has at most finitely many preimages in V, we show that V contains another immediate basin of N_f or a virtual immediate basin. A virtual immediate basin is an unbounded invariant Fatou component in which the dynamics converges to infty through an absorbing set.
Boundedness and simple connectivity of the basins of attraction for some numerical methods
该文研究哈雷与特劳布求根算法在高次对称多项式族上的动力学，证明存在多项式在哈雷法下根的直接吸引域有界，探讨了吸引域的有界性与单连通性。
Let f_n be a sequence of analytic functions in a domain U with a common attracting fixed point z_0. Suppose that f_n converges to f_0 uniformly on each compact subset of U and that z_0 is a Siegel point of f_0. We establish a sufficient condition for the immediate basins of attraction $\mathcal A^*(z_0,f_n,U)$ to form a sequence that converges to the Siegel disk of f_0 as to the kernel with respect to z_0. The same condition is shown to imply convergence of the Koenings functions associated with f_n to that of f_0. Our method allows us also to obtain a kind of quantitative result for analytic …
Newton's root finding method applied to a (transcendental) entire function f:C->C is the iteration of a meromorphic function N. It is well known that if for some starting value z, Newton's method converges to a point x in C, then f has a root at x. We show that in many cases, if an orbit converges to infinity for Newton's method, then f has a `virtual root' at infinity. More precisely, we show that if N has an invariant Baker domain that satisfies some mild assumptions, then 0 is an asymptotic value for f. Conversely, we show that if f has an asymptotic value of logarithmic type at 0, then the…
We prove that if A is the basin of immediate attraction to a periodic attracting or parabolic point for a rational map f on the Riemann sphere, then periodic points in the boundary of A are dense in this boundary. To prove this in the non simply- connected or parabolic situations we prove a more abstract, geometric coding trees version.
On the dimension of the boundaries of attracting basins of entire maps
本文研究了超越整个映射吸引盆地边界的维数，证明了其超几何维数大于1，并展示了其边界不可能是光滑或可测曲线。
Let $H^d$ be the set of all rational maps of degree $d\ge 2$ on the Riemann sphere which are expanding on Julia set. We prove that if $f\in H^d$ and all or all but one critical points (or values) are in the immediate basin of attraction to an attracting fixed point then there exists a polynomial in the component $H(f)$ of $H^d$ containing $f$. If all critical points are in the immediate basin of attraction to an attracting fixed point or parabolic fixed point then $f$ restricted to Julia set is conjugate to the shift on the one-sided shift space of $d$ symbols. We give exotic examples of maps …
Newton polygons and Böttcher coordinates near infinity for polynomial skew products
本文研究次数至少为2的多项式斜积，利用 $p$ 的次数与 $q$ 的 Newton 多边形构造无穷附近的不变区域，并证明其在该区域上共轭于单项式映射。
On the boundary of an immediate attracting basin of a hyperbolic entire function
本文研究了有限阶超越整函数的吸引周期点，在附加假设下证明其 Fatou 集分量边界与逃逸集交集的 Hausdorff 维数为 1，推广了已有结果。
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.
On the basins of attraction of a one-dimensional family of root finding algorithms: from Newton to Traub
该研究分析阻尼 Traub 求根方法族的动力学与吸引盆拓扑，给出保证收敛到多项式全部根的通用初始条件，并数值揭示 Newton 法与 Traub 法的联系。
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。
We consider the family of entire transcendental maps given by $F_{λ,m}= λz^m exp(z)$ where m>=2. All functions $F_{λ,m}$ have a superattracting fixed point at z=0, and a critical point at z=-m. In the dynamical plane we study the topology of the basin of attraction of z=0. In the parameter plane we focus on the capture behaviour, i.e., λvalues such that the critical point belongs to the basin of attraction of z=0. In particular, we find a capture zone for which this basin has a unique connected component, whose boundary is then non-locally connected. However, there are parameter values for whi…
Chebyshev's method applied to polynomials with two distinct roots
该研究探讨了Chebyshev方法在两个不同根的多项式中的应用，证明了Julia集的连通性及Fatou集的结构，同时分析了根重数对盆地边界的影响。
We investigate boundedness of hyperbolic components in the moduli space of Newton maps. For quartic maps, (i) we prove hyperbolic components possessing two distinct attracting cycles each of period at least two are bounded, and (ii) we characterize the possible points on the boundary at infinity for some other types of hyperbolic components. For general maps, we prove hyperbolic components whose elements have fixed superattracting basins mapping by degree at least three are unbounded.
Horn maps of holomorphic functions locally pseudo-conjugate on their parabolic basins
本文引入了直接抛物盆地上不作连续性假设的局部伪共轭概念，并证明了两个全纯函数局部伪共轭当且仅当其提升角映射满足覆盖等价条件，为理解抛物重整化的不变类奠定了基础。
The purpose of this work is to give a first approach to the dynamical behavior of Schröder's method, a well known iterative process for solving nonlinear equations. In this context we consider equations defined in the complex plane. By using topological conjugations, we characterize the basins of attraction of Schröder's method applied to polynomials with two roots and different multiplicities. Actually, we show that these basins are half-planes or circles, depending on the multiplicities of the roots. We finish our study with a graphical gallery that allow us to compare the basins of attracti…
We give a combinatorial classification for the class of postcritically fixed Newton maps of polynomials as dynamical systems. This lays the foundation for classification results of more general classes of Newton maps. A fundamental ingredient is the proof that for every Newton map (postcritically finite or not) every connected component of the basin of an attracting fixed point can be connected to $\infty$ through a finite chain of such components.
In a previous paper we produced a complex iteration of a holomorphic function $ϕ$ in the immediate basin of a fixed point whose multiplier is a real number and in between zero and one. We further explore this problem, allowing the multiplier to be a complex number and its modulus to be in between zero and one. We find expansions of results derived before. We will continue to use Ramanujan's master theorem and the process of \emph{factoring} appropriately bounded exponential functions by their values on the natural numbers. We will obtain \emph{all} complex iterations of entire $ϕ$ in the immed…
In this paper, we investigate the behavior of orbits inside attracting basins in higher dimensions. Suppose $F(z, w)=(P(z), Q(w))$, where $P(z), Q(w)$ are two polynomials of degree $m_1, m_2\geq2$ on $\mathbb{C}$, $P(0)=Q(0)=0,$ and $0<|P'(0)|, |Q'(0)|<1.$ Let $Ω$ be the immediate attracting basin of $F(z, w)$. Then there is a constant $C$ such that for every point $(z_0, w_0)\in Ω$, there exists a point $(\tilde{z}, \tilde{w})\in \cup_k F^{-k}(0, 0), k\geq0$ so that $d_Ω\big((z_0, w_0), (\tilde{z}, \tilde{w})\big)\leq C, d_Ω$ is the Kobayashi distance on $Ω$. However, for many other cases, th…
Chebyshev's method applied to polynomials with rotational symmetry
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。
Local connectivity of boundaries of tame Fatou components of meromorphic functions
本文研究超越亚纯映射不变单连通吸引盆边界的局部连通性，在“无穷远处排斥花瓣”等条件下证明结论，并应用于 Newton 方法，给出 $\mathcal B$ 类外的非平凡局部连通 Julia 集例子。
Backtracking New Q-Newton's method, Schröder's theorem, and Linear Conjugacy
该研究证明回溯新Q-牛顿法（BNQN）求解二次多项式根的吸引域与牛顿法（对应经典Schröder结果）一致，但边界动力学更平滑，且BNQN在特定线性算子共轭下不变，明确了二者的异同。
Dynamics of Newton's method for odd and even elliptic functions
研究牛顿法在奇偶椭圆函数中的动力学行为，揭示其雅各布集连通性及游动域与吸引盆地共存的条件。
We study the Cebyshev-Halley methods applied to the family of polynomials $f_{n,c}(z)=z^n+c$, for $n\ge 2$ and $c\in \mathbb{C}^{*}$. We prove the existence of parameters such that the immediate basins of attraction corresponding to the roots of unity are infinitely connected. We also prove that, for $n \ge 2$, the corresponding dynamical plane contains a connected component of the Julia set, which is a quasiconformal deformation of the Julia set of the map obtained by applying Newton's method to $f_{n,-1}$.
We show that there exists an entire function f without zeros for which the associated Newton function N(z)=z-f(z)/f'(z) is a transcendental meromorphic functions without Baker domains. We also show that there exists an entire function f with exactly one zero for which the complement of the immediate attracting basin has at least two components and contains no invariant Baker domains of N. The second result answers a question of J. Rueckert and D. Schleicher while the first one gives a partial answer to a question of X. Buff.
Little is known about the global structure of the basins of attraction of Newton's method in two or more complex variables. We make the first steps by focusing on the specific Newton mapping to solve for the common roots of $P(x,y) = x(1-x)$ and $Q(x,y) = y^2+Bxy-y$. There are invariant circles $S_0$ and $S_1$ within the lines $x=0$ and $x=1$ which are superattracting in the $x$-direction and hyperbolically repelling within the vertical line. We show that $S_0$ and $S_1$ have local super-stable manifolds, which when pulled back under iterates of $N$ form global super-stable spaces $W_0$ and $W…
Dynamics of projectable functions: Towards an atlas of wandering domains for a family of Newton maps
本文提出带零点的超越整函数单参数族$F_λ$，其牛顿法产生游荡域并与根吸引盆共存，通过研究可投影函数类中$f$与$g$的关系及对数提升方法扩展，刻画了相关整函数并分析其参数空间特性。
A small probabilistic universal set of starting points for finding roots of complex polynomials by Newton's method
本文构造了一个大小为 O(d(log log d)^2) 的随机起始点集，以高概率与所有固定次数 d 的（适当归一化）复多项式的所有根的牛顿法吸引域相交，为牛顿法求根提供高效通用的概率起始点。
On dynamics of the Chebyshev's method for quartic polynomials
本文研究了四次多项式 Chebyshev 方法的动力学性质，证明所有额外不动点均为排斥点且不存在不变 Siegel 盘和 Herman 环，并分析了实参数下根的直接吸引域的无界性与单连通性。
Backtracking New Q-Newton's method, Newton's flow, Voronoi's diagram and Stochastic root finding
本文深入探索回溯新Q-牛顿方法（BNQN）寻找多项式和亚纯函数根时的平滑吸引盆地现象，将其与牛顿流和Voronoi图相联系，并证明其对随机扰动比牛顿方法更具鲁棒性。
The Riemann hypothesis and dynamics of Backtracking New Q-Newton's method
本文研究Backtracking New Q-Newton方法在Riemann xi函数上的应用，证明Riemann假设等价于该方法的所有吸引子均位于临界线上，并探讨其吸引盆与Voronoi图的关系以辅助验证假设或寻找反例。
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