The generation of fractals and study of the dynamics of polynomials is one of the emerging and interesting field of research nowadays. We introduce in this paper the dynamics of polynomials z n z + c = 0 for n 2 and applied Jungck Ishikawa Iteration to generate new Relative Superior Mandelbrot sets and Relative Superior Julia sets. In order to solve this function by Jungck –type iterative sche...

Introduction 1. Definitions and Elementary Results 2. When Disconnected Filled-in Julia Sets are Cantor Sets 3. Smooth Julia Sets 4. Fixed Points 5. Remarks on the Topology of the Connectedness Locus Acknowledgements References We consider the rational maps given by z jzj z c, for z and c complex and fixed and real. The case corresponds to quadratic polynomials: some of the well-known results f...

Let T be a finite subset of the complex unit circle S, and define f : S 7→ S by f(z) = z. Let CH(T ) denote the convex hull of T. If card(T ) = N ≥ 3, then CH(T ) defines a polygon with N sides. The N -gon CH(T ) is called a wandering N -gon if for every two non-negative integers i 6= j, CH(f (T )) and CH(f (T )) are disjoint N -gons. A non-degenerate chord of S is said to be critical if its tw...

Julia sets are examined as examples of strange objects which arise in the study of long time properties of simple dynamical systems. Technically they are the closure of the set of unstable cycles of analytic maps. Physically, they are sets of points which lead to chaotic behavior. The map f ( z ) = z2+ p is analyzed for smallp where the Julia set is a closed curve, and for largep where the Juli...

This note gives a simplified proof of the similarity between the Mandelbrot set and the quadratic Julia sets at the Misiurewicz parameters, originally due to Tan Lei [TL]. We also give an alternative proof of the global linearization theorem of repelling fixed points. 1 Simlarity between M and J We first give a proof of Tan’s theorem, in a way inspired by Schwick [Sch]. The Julia sets and the M...

In this paper, we give a proof of Sullivan’s complex bounds for the Feigenbaum quadratic polynomial and show that the Julia set of the Feigenbaum quadratic polynomial is connected and locally connected.

A difference analogue of the logistic equation, which preserves integrability, is derived from Hirota’s bilinear difference equation. The integrability of the map is shown to result from the large symmetry associated with the Bäcklund transformation of the KP hierarchy. We introduce a scheme which interpolates between this map and the standard logistic map and enables us to study integrable and...