On a topological closeness of perturbed Mandelbrot sets

In the present work we propose a numerical and visual tool for the study of the deformation of the Mandelbrot sets of perturbed Mandelbrot maps by noise in comparison with the original Mandelbrot set. Further, by employing these numerical tools, we support the invariance of the Mandelbrot set of a noise-perturbed Mandelbrot map under different noise realizations. Finally, we provide evidence for the non-fractal structure of the Mandelbrot set of a noise-perturbed Mandelbrot map.     

On the Julia set of the perturbed Mandelbrot map

In this work, we present numerical results which support the smooth decomposition method of the generalized Julia set by Peintge et al., in the case of other perturbations of the Mandelbrot map studied in our previous work (Argyris J, Andreadis I, Karakasidis T. Chaos, Solitons & Fractals 1999). We also calculate the generalized Julia set of a Mandelbrot map subject to noise. Hence, we are in a position to examining numerically the stability of this set under small noise.     

Reverse bifurcation and fractal of the compound logistic map

The nature of the fixed points of the compound logistic map is researched and the boundary equation of the first bifurcation of the map in the parameter space is given out. Using the quantitative criterion and rule of chaotic system, the paper reveal the general features of the compound logistic map transforming from regularity to chaos, the following conclusions are shown: (1) chaotic patterns of the map may emerge out of double-periodic bifurcation and (2) the chaotic crisis phenomena and the reverse bifurcation are found. At the same time, we analyze the orbit of critical point of the compound logistic map and put forward the definition of Mandelbrot–Julia set of compound logistic map. We generalize the Welstead and Cromer’s periodic scanning technology and using this technology construct a series of Mandelbrot–Julia sets of compound logistic map. We investigate the symmetry of Mandelbrot–Julia set and study the topological inflexibility of distributing of period region in the Mandelbrot set, and finds that Mandelbrot set contain abundant information of structure of Julia sets by founding the whole portray of Julia sets based on Mandelbrot set qualitatively.     

ON THE DISTRIBUTION OF JULIA SETS OF HOLOMORPHIC MAPS

In 1965 Baker first considered the distribution of Julia sets of transcendental entire maps and proved that the Julia set of an entire map cannot be contained in any finite set of straight lines. In this paper we shall consider the distribution problem of Julia sets of meromorphic maps. We shall show that the Julia set of a transcendental meromorphic map with at most finitely many poles cannot be contained in any finite set of straight lines.Meanwhile, examples show that the Julia sets of meromorphic maps with infinitely many poles may indeed be contained in straight lines. Moreover, we shall show that the Julia set of a transcendental analytic self-map of C* can neither contain a free Jordan arc nor be contained in any finite set of straight lines.     

Quatre applications du lemme de Zalcman à la dynamique complexe

We give four applications of Zalcman’s lemma to the dynamics of rational maps on the Riemann sphere: a parameter analogue of a proof of the density of repelling cycles in the Julia sets; similarity between the Mandelbrot set and the Julia sets; a construction of the Lyubich-Minsky lamination and its variant; and a unified characterization of conical points by Lyubich and Minsky and those by Martin and Mayer.     

Multipliers of periodic orbits of quadratic polynomials and the parameter plane

We prove a result about an extension of the multiplier of an attracting periodic orbit of a quadratic map as a function of the parameter. This has applications to the problem of geometry of the Mandelbrot and Julia sets. In particular, we prove that the size of p/q-limb of a hyperbolic component of the Mandelbrot set of period n is O(4 ⁿ /p), and give an explicit condition on internal arguments under which the Julia set of corresponding (unique) infinitely renormalizable quadratic polynomial is not locally connected. In memory of my grandmother Esfir Garbuz     

On perturbations of the Mandelbrot map

In this work, we propose new applications of analytic and non-analytic perturbations of the Mandelbrot map as expressed in a two-parameter deformation family of it. The influence of alternative applications of noise for specific choices of a Mandelbrot set is also provided. Hence, we are in a position of examining the stability of this set under stochastic perturbations.     

Generating Fractals Using Geometric Algebra

In this paper we investigate how, using the language of Geometric Algebra [7, 4], the common escape-time Julia and Mandelbrot set fractals can be extended to arbitrary dimension and, uniquely, non-Eulidean geometries. We develop a geometric analog of complex numbers and show how existing ray-tracing techniques [2] can be extended. In addition, via the use of the Conformal Model for Geometric Algebra, we develop an analog of complex arithmetic for the Poincaré disc and show that, in non-Euclidean geometries, there are two related but distinct variants of the Julia and Mandelbrot sets.     

分形几何和分维数简介

本文摘要介绍分形几何的发展历史,分形几何的特征以及分维数的概念。简单介绍由复迭代用计算机产生的分形几何-Julia集和Mandelbrot集。     

Hausdorff dimension 2 for Julia sets of quadratic polynomials

We consider families of quadratic polynomials which admit parameterisations in a neighbourhood of the boundary of the Mandelbrot set. We show how to find parameters such that the associated Julia sets are of Hausdorff dimension 2. Received October 11, 1999 / Published online April 12, 2001     

Determination of Mandelbrot Sets Hyperbolic Component Centres

In this work we present a very fast and parsimonious method to calculate the centre coordinates of hyperbolic components in the Mandelbrot set. The method we use constitutes an extension for the complex domain of the one developed by Myrberg for the real map x ] x²−p, in which, given the symbolic sequence of a superstable orbit, the parameter value originating such a superstable orbit is worked out. We show that, when dealing with complex domain sequences, some of the solutions obtained correspond to the centres of the Mandelbrot sets hyperbolic components, while some others do not exist.     

Dynamics of McMullen maps

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 answer to the question of Devaney. Higher regularity of this boundary is obtained in almost all cases. We show that the boundary is a quasi-circle if it contains neither a parabolic point nor a recurrent critical point. For the whole Julia set, we show that the McMullen maps have locally connected Julia sets except in some special cases.     

An explicit bound for uniform perfectness of the Julia sets of rational maps

A compact set C in the Riemann sphere is called uniformly perfect if there is a uniform upper bound on the moduli of annuli which separate C. Julia sets of rational maps of degree at least two are uniformly perfect. Proofs have been given independently by Ma né and da Rocha and by Hinkkanen, but no explicit bounds are given. In this note, we shall provide such an explicit bound and, as a result, give another proof of uniform perfectness of Julia sets of rational maps of degree at least two. As an application, we provide a lower estimate of the Hausdorff dimension of the Julia sets. We also give a concrete bound for the family of quadratic polynomials in terms of the parameter c. Received: 7 June 1999; in final form: 9 November 1999 / Published online: 17 May 2001     

McMullen's root-finding algorithm for cubic polynomials

We show that a generally convergent root-finding algorithm for cubic polynomials defined by C. McMullen is of order 3, and we give generally convergent algorithms of order 5 and higher for cubic polynomials. We study the Julia sets for these algorithms and give a universal rational map and Julia set to explain the dynamics.

     

An effective algorithm to compute Mandelbrot sets in parameter planes

In McMullen (2000) it was proven that copies of generalized Mandelbrot set are dense in the bifurcation locus for generic families of rational maps. We develop an algorithm to an effective computation of the location and size of these generalized Mandelbrot sets in parameter space. We illustrate the effectiveness of the algorithm by applying it to concrete families of rational and entire maps.     

Ergodic Properties and Julia Sets of Weierstrass Elliptic Functions

 We discuss properties of the Julia and Fatou sets of Weierstrass elliptic ℘ functions arising from real lattices. We give sufficient conditions for the Julia sets to be the whole sphere and for the maps to be ergodic, exact, and conservative. We also give examples for which the Julia set is not the whole sphere.     

Foundations for an iteration theory of entire quasiregular maps

The Fatou-Julia iteration theory of rational functions has been extended to uniformly quasiregular mappings in higher dimension by various authors, and recently some results of Fatou-Julia type have also been obtained for non-uniformly quasiregular maps. The purpose of this paper is to extend the iteration theory of transcendental entire functions to the quasiregular setting. As no examples of uniformly quasiregular maps of transcendental type are known, we work without the assumption of uniform quasiregularity. Here the Julia set is defined as the set of all points such that the complement of the forward orbit of any neighbourhood has capacity zero. It is shown that for maps which are not of polynomial type, the Julia set is non-empty and has many properties of the classical Julia set of transcendental entire functions.     

填充Jnlia集与Mandelbrot集的面积与直径

本文用单叶函数中的面积定理及Ｇａｒａｂｅｄｉａｎ－Ｓｃｈｉｆｆｅｒ不等式的有关推论．给出了求多项式的填充Ｊｕｌｉａ集及Ｍａｎｄｅｌｂｒｏｔ集面积的方法及直径的上界估计，从而给Ａ．Ｄｏｕａｄｙ所提的有关问题一个回答．     

Ergodic Properties and Julia Sets of Weierstrass Elliptic Functions

 We discuss properties of the Julia and Fatou sets of Weierstrass elliptic ℘ functions arising from real lattices. We give sufficient conditions for the Julia sets to be the whole sphere and for the maps to be ergodic, exact, and conservative. We also give examples for which the Julia set is not the whole sphere. Received September 4, 2001; in revised form March 26, 2002     

Dynamical systems on lattices with decaying interaction II: Hyperbolic sets and their invariant manifolds

This is the second part of the work devoted to the study of maps with decay in lattices. Here we apply the general theory developed in Fontich et al. (2011) [3] to the study of hyperbolic sets. In particular, we establish that any close enough perturbation with decay of an uncoupled lattice map with a hyperbolic set has also a hyperbolic set, with dynamics on the hyperbolic set conjugated to the corresponding of the uncoupled map. We also describe how the decay properties of the maps are inherited by the corresponding invariant manifolds.