The Multibrots for Multicorns is a modification of the classic Mandelbrot and Julia sets and it is given by the complex function where and is a constant. The Multibrot fractal type is particularly interesting, with beautiful shapes and lots of spirals. In this paper we have presented different characteristics of Multibrot function for Multicorns using superior iterates. Further, different prope...

The core entropy of polynomials, recently introduced by W. Thurston, is a dynamical invariant which can be defined purely in combinatorial terms, and provides a useful tool to study parameter spaces of polynomials. The theory of core entropy extends the entropy theory for real unimodal maps to complex polynomials: the real segment is replaced by an invariant tree, known as the Hubbard tree, whi...

By means of a nested sequence of some critical pieces constructed by Kozlovski, Shen, and van Strien, and by using a covering lemma recently proved by Kahn and Lyubich, we prove that the Julia set of a polynomial is a Cantor set if and only if each component of the filledin Julia set containing critical points is aperiodic. This result was a conjecture raised by Branner and Hubbard in 1992.

Introduction to Fractal Geometry .................................................................. 1 1. A few Random Quotes ......................................................................... 1 2. Introduction ....................................................................................... 1 3. Types of fractals ........................................................................

A decoration of the Mandelbrot set M is a part of M cut off by two external rays landing at some tip of a satellite copy of M attached to the main cardioid. In this paper we consider infinitely renormalizable quadratic polynomi-als satisfying the decoration condition, which means that the combinatorics of the renormalization operators involved is selected from a finite family of decorations. Fo...

Let G be a semigroup of complex polynomials (under the operation of composition of functions) such that there exists a bounded set in the plane which contains any finite critical value of any map g ∈ G. We discuss the dynamics of such polynomial semigroups as well the structure of the Julia set of G. In general, the Julia set of such a semigroup G may be disconnected, and each Fatou component o...

In recent literature, researchers have generated Julia sets and Mandelbrot sets in Mann and Ishikawa orbits that are examples of two-step and three-step feedback processes respectively. This paper presents further generalization of Julia and Mandelbrot sets for complex-valued polynomials such as quadratic, cubic and higher degree polynomials using a Noor orbit, which is a four-step iterative pr...

This paper is concerned with a generalisation of the classical theory of the dynamics associated to the iteration of a rational mapping of the Riemann sphere, to the more general setting of the dynamics associated to an arbitrary semigroup of rational mappings. We are partly motivated by results of Gehring and Martin which show that certain parameter spaces for KJeinian groups are essentially t...

We investigate when a rational endomorphism of the Riemann sphere C extends to a mapping of the upper half-space H3 which is rational with respect to some measurable conformal structure. Such an extension has the property that it and all its iterates have uniformly bounded distortion. Such maps are called uniformly quasiregular. We show that, in the space of rational mappings of degree d, such ...

For a polynomial with one critical point (maybe multiple), which does not have attracting or neutral periodic orbits, we prove that the backward dynamics is stable provided the Julia set is locally connected. The latter is proved to be equivalent to the non-existence of a wandering continuum in the Julia set or to the shrinking of Yoccoz puzzle-pieces to points.