Related papers: Dynamics of Newton maps

The theory of polynomial-like maps is of fundamental importance in holomorphic dynamics. We study dynamical properties of a larger class of maps. Our main result is that, under some natural conditions, a map of this class has a completely…

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…

We prove: If $f(z)$ is a critically finite rational map which has exactly two critical points and which is not conjugate to a polynomial, then the boundary of every Fatou component of $f$ is a Jordan curve. If $f(z)$ is a hyperbolic…

An intriguing and unexpected result for students learning numerical analysis is that Newton's method, applied to the simple polynomial z^3 - 1 = 0 in the complex plane, leads to intricately interwoven basins of attraction of the roots. As…

In this article, we develop the Yoccoz puzzle technique to study a family of rational maps termed McMullen maps. We show that the boundary of the immediate basin of infinity is always a Jordan curve if it is connected. This gives a positive…

We study the dynamics of polynomial maps on the boundary of the central hyperbolic component $\mathcal H_d$. We prove the local connectivity of Julia sets and a rigidity theorem for maps on the regular part of $\partial\mathcal H_d$. Our…

Let $f(z,w)=(p(z),q(z,w))$ be a polynomial skew product such that the degrees of $p$ and $q$ are grater than or equal to $2$. Under one or two conditions, we prove that $f$ is conjugate to a monomial map on an invariant region near…

We introduce the notion of an approximate jacobian Newton diagram which is the jacobian Newton diagram of the morphism $(f^{(k)},f)$, where $f$ is a branch and $f^{(k)}$ is a characteristic approximate root of $f$. We prove that the set of…

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…

We investigate the dynamics of semigroups generated by a family of polynomial maps on the Riemann sphere such that the postcritical set in the complex plane is bounded. Moreover, we investigate the associated random dynamics of polynomials.…

The purpose of this paper is to initiate a theory concerning the dynamics of asymptotically holomorphic polynomial-like maps. Our maps arise naturally as deep renormalizations of asymptotically holomorphic extensions of $C^r$ ($r>3$)…

For a rational function $R$, let $N_R(z)=z-\frac{R(z)}{R'(z)}.$ Any such $N_R$ is referred to as a Newton map. We determine all the rational functions $R$ for which $N_R$ has exactly two attracting fixed points, one of which is an…

Let $f:\mathbb{C}^2 \to \mathbb{C}$ be a polynomial map. Let $\mathbb{C}^2 \subset X$ be a compactification of $\mathbb{C}^2$ where $X$ is a smooth rational compact surface and such that there exists a morphism of varieties $\Phi :X\to…

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…

We investigate the root finding algorithm given by the secant method applied to a real polynomial $p$ as a discrete dynamical system defined on $\mathbb R^2$. We study the shape and distribution of the basins of attraction associated to the…

We give a combinatorial classification for the class of postcritically fixed Newton maps of polynomials and indicate potential for extensions. As our main tool, we show that for a large class of Newton maps that includes all hyperbolic…

A new variant of Newton's method - named Backtracking New Q-Newton's method (BNQN) - which has strong theoretical guarantee, is easy to implement, and has good experimental performance, was recently introduced by the third author.…

In this paper, we study the structure of Newton polygons for compositions of polynomials over the rationals. We establish sufficient conditions under which the successive vertices of the Newton polygon of the composition $ g(f^n(x)) $ with…

In this paper we study the dynamics of Halley's and Traub's root-finding algorithms applied to a symmetric family of polynomials of degree $d+1\geq 3$. We discuss the (un)boundedness and simple connectivity of the immediate basins of…

We study the $p$-adic absolute value of the roots of the $L$-functions associated to certain twisted character sums, and additive character sums associated to polynomials $P(x^d)$, when $P$ varies among the space of polynomial of fixed…