On a topological closeness of perturbed Julia sets

In the present work we expand our previous work in [1] by introducing the Julia Deviation Distance and the Julia Deviation Plot in order to study the stability of the Julia sets of noise-perturbed Mandelbrot maps. We observe a power-law behaviour of the Julia Deviation Distance of the Julia sets of a family of additive dynamic noise Mandelbrot maps from the Julia set of the Mandelbrot map as a function of the noise level. Additionally, using the above tools, we support the invariance of the Julia set of a noise-perturbed Mandelbrot map under different noise realizations.     

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.     

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     

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.     

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.     

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     

On the description of self-similarity for attractors by means of conditional entropy

There exist several sets having similar structure on arbitrarily small scales. Mandelbrot called such sets fractals, and defined a dimension that assigns non-integer numbers to fractals. On the other hand, a dynamical system yielding a fractal set referred to as a strange attractor is a chaotic map. In this paper, a characterization of self-similarity for attractors is attempted by means of conditional entropy.     

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.     

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。     

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.     

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.     

Dynamical properties of discrete Lotka–Volterra equations

A discrete version of the Lotka–Volterra differential equations for competing population species is analyzed in detail in much the same way as the discrete form of the logistic equation has been investigated as a source of bifurcation phenomena and chaotic dynamics. It is found that in addition to the logistic dynamics – ranging from very simple to manifestly chaotic regimes in terms of governing parameters – the discrete Lotka–Volterra equations exhibit their own brands of bifurcation and chaos that are essentially two-dimensional in nature. In particular, it is shown that the system exhibits “twisted horseshoe” dynamics associated with a strange invariant set for certain parameter ranges.     

Topological entropy of quadratic polynomials and dimension of sections of the Mandelbrot set

A fundamental theme in holomorphic dynamics is that the local geometry of parameter space (e.g. the Mandelbrot set) near a parameter reflects the geometry of the Julia set, hence ultimately the dynamical properties, of the corresponding dynamical system. We establish a new instance of this phenomenon in terms of entropy.     

Connectivity of the Mandelbrot set for the family of renormalization transformations

We show that the Mandelbrot set for the family of renormalization transformations of 2-dimensional diamond-like hierachical Potts models in statistical mechanics is connected. We also give an upper bound for the Hausdorff dimension of Julia set when it is a quasi-circle.     

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.

     

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.     

Delayed q-deformed logistic map

The delay logistic map with two types of q-deformations: Tsallis and Quantum-group type are studied. The stability of the logistic map and its bifurcation scheme is analyzed as a function of the deformation and delay parameters. Chaos is suppressed in a certain region of deformation and delay parameter space. By introducing delay, the steady state in one type of deformation is maintained while chaotic behavior is recovered in another type.     

An epicycloidal boundary of the main component in the degree-n bifurcation set

It is known that the parametric boundary equation for the main component in the Mandelbrot set represents a cardioid. We derive an epicycloidal boundary equation of the main component in the degree-n bifurcation set by extending the parameter which describes the cardioid in the Mandelbrot set. Computational results as well as some useful properties are presented together with the programming source codes written inMathematica. Various boundaries are displayed for 2≤n≤7 and show a good agreement with the theory presented here. The known boundary equation enables us to significantly reduce the construction time for the degree-n bifurcation set.     

Asymptotic porosity of planar harmonic measure

We study the distribution of harmonic measure on connected Julia sets of unicritical polynomials. Harmonic measure on a full compact set in ? is always concentrated on a set which is porous for a positive density of scales. We prove that there is a topologically generic set $\mathcal{A}$ in the boundary of the Mandelbrot set such that for every $c\in \mathcal{A}$ , β>0, and λ∈(0,1), the corresponding Julia set is a full compact set with harmonic measure concentrated on a set which is not β-porous in scale λ ⁿ for n from a set with positive density amongst natural numbers.