## Publications

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**ABSTRACT:**We consider the dynamics as a special class of rational functions that are obtained from Newton's method when applied to a polynomial equation. Finding solutions of these equations leads to some beautiful images in complex functions. These images represent the basins of attraction of roots of complex functions. We seek the answer "What is the dynamics near the chosen parabolic fixed points?". In addition, we will provide a detailed history of Fractal and Dynamical System Theory.08/2012; DOI:10.1063/1.4747660 - [Show abstract] [Hide abstract]

**ABSTRACT:**It is known that if we apply Newton's method to the complex function , with , then the immediate basin of attraction of the roots of P has finite area. In this paper, we show that under certain conditions on the polynomial P, if , then there is at least one immediate basin of attraction having infinite area.Journal of Difference Equations and Applications 06/2012; 18(6):1067-1076. DOI:10.1080/10236198.2010.547493 · 0.86 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**In this paper, we consider the family of rational maps $$\F(z) = z^n + \frac{\la}{z^d},$$ where $n \geq 2$, $d\geq 1$, and$\la \in \bbC$. We consider the case where $\la$ lies in the main cardioid of one of the $n-1$ principal Mandelbrot sets in these families. We show that the Julia sets of these maps are always homeomorphic. However, two such maps $\F$ and $F_\mu$ are conjugate on these Julia sets only if the parameters at the centers of the given cardioids satisfy $\mu = \nu^{j(d+1)}\la$ or $\mu = \nu^{j(d+1)}\bar{\la}$ where $j \in \bbZ$ and $\nu$ is an $n-1^{\rm st}$ root of unity. We define a dynamical invariant, which we call the minimal rotation number. It determines which of these maps are are conjugate on their Julia sets, and we obtain an exact count of the number of distinct conjugacy classes of maps drawn from these main cardioids.International Journal of Bifurcation and Chaos 03/2011; 23(2). DOI:10.1142/S0218127413300048 · 1.02 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**For polynomial maps in the complex plane, the notion of external rays plays an important role in determining the structure of and the dynamics on the Julia set. In this paper we consider an extension of these rays in the case of rational maps of the form Fλ(z) = z n + λ/z n where n > 1. As in the case of polynomials, there is an immediate basin of ∞, so we have similar external rays. We show how to extend these rays throughout the Julia set in three specific examples. Our extended rays are simple closed curves in the Riemann sphere that meet the Julia set in a Cantor set of points and also pass through countably many Fatou components. Unlike the external rays, these extended rays cross infinitely many other extended rays in a manner that helps determine the topology of the Julia set. Mathematics Subject Classification (2010)Primary 37F10-Secondary 37F45 KeywordsJulia set-external rays-Mandelbrot set-symbolic dynamicsJournal of Fixed Point Theory and Applications 06/2010; 7(1):223-240. DOI:10.1007/s11784-010-0003-2 · 0.57 Impact Factor -
##### Article: Infinite Basins of Julia Sets.

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**ABSTRACT:**The goal of this paper is to investigate the iterative behavior of a particular class of rational functions which arise from Newton’s method applied to the entire function (z 2 +c)e Q(z) wherec is a complex parameter and Q is a nonconstant polynomial with deg(Q)≤2. In particular, the basins of attracting fixed points will be described.International Journal of Bifurcation and Chaos 10/2008; 18:3169-3173. DOI:10.1142/S0218127408022330 · 1.02 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**Newton's iterative method for finding solutions to nonlinear equations leads to some beautiful images when applied to complex functions. These images represent the basins of attraction of roots of complex functions. The interesting point about the basins of attraction for the roots of a complex function F approximated by relaxed Newton's method is that they are fractals called Julia sets of a rational function N-F,N-h. The main aim of the present paper is to describe such fractal sets. (c) 2006 Elsevier Ltd. All rights reserved.Chaos Solitons & Fractals 04/2007; 32(2):471-479. DOI:10.1016/j.chaos.2006.06.057 · 1.50 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**This paper shows via a reduced family of examples, the relaxed Newton's method is applied to complex exponential function F(z)=zez and F(z)=zez2, the basin of roots has infinite area. In addition, we examined their computer pictures which are fractals for the relaxed Newton's basin. In fact, computer experiments F(z)=P(z)ez and F(z)=P(z)ez2, indicate this to hold for arbitrary non-constant polynomial P(z).Chaos Solitons & Fractals 12/2004; 22(5-22):1189-1198. DOI:10.1016/j.chaos.2004.04.031 · 1.50 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**For a nonconstant function F and a real number h∈]0,1] the relaxed Newton’s method N F,h of F is an iterative algorithm for finding the zeroes of F. We show that when the relaxed Newton’s method is applied to the complex function F(z)=P(z)e Q(z) , where P and Q are polynomials, the basin of attraction of a root of F has a finite area if the degree of Q exceeds or equals 3. The key point is that N F,h is a rational map with a parabolic fixed-point at infinity.International Journal of Bifurcation and Chaos 12/2004; 14(12):4177-4190. DOI:10.1142/S0218127404011879 · 1.02 Impact Factor

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