Julia Set Research Articles

The Julia and Mandelbrot sets for the function [formula omitted] exhibit Mann and Picard–Mann orbits along with [formula omitted]-convexity

This research paper introduces a novel approach to visualize Julia and Mandelbrot sets by employing iterative techniques, which play a crucial role in creating fractals. The primary focus is on complex functions of the form F(z)=zp−qz2+rz+sincw for all z∈ℂ, where p∈N∖{1}, q∈ℂ, r,c∈ℂ∖{0} and w∈[1,∞). The Mann and Picard–Mann iteration schemes with s-convexity are utilized throughout the study. Innovative escape criteria are developed to generate Julia and Mandelbrot sets using these iterative methods. These criteria serve as guidelines for determining when the iterative process should terminate, leading to the creation of captivating fractal patterns. The research investigates the impact of parameter variations within the iteration schemes on the resulting fractal’s shape, size, and color. By manipulating these parameters, a wide range of captivating fractal patterns can be generated and visualized, encompassing various aesthetic possibilities. Additionally, we discuss the numerical examples related to Julia and Mandelbrot sets generated through the proposed iteration. We also delve into discussions concerning execution time and the average number of iterations.

On the stochastic bifurcations regarding random iterations of polynomials of the form

Abstract In this paper, we consider random iterations of polynomial maps $z^{2} + c_{n}$ , where $c_{n}$ are complex-valued independent random variables following the uniform distribution on the closed disk with center c and radius r. The aim of this paper is twofold. First, we study the (dis)connectedness of random Julia sets. Here, we reveal the relationships between the bifurcation radius and connectedness of random Julia sets. Second, we investigate the bifurcation of our random iterations and give quantitative estimates of bifurcation parameters. In particular, we prove that for the central parameter $c = -1$ , almost every random Julia set is totally disconnected with much smaller radial parameters r than expected. We also introduce several open questions worth discussing.

ON THE ITERATIONS AND THE ARGUMENT DISTRIBUTION OF MEROMORPHIC FUNCTIONS

AbstractThis paper consists of two parts. The first is to study the existence of a point a at the intersection of the Julia set and the escaping set such that a goes to infinity under iterates along Julia directions or Borel directions. Additionally, we find such points that approximate all Borel directions to escape if the meromorphic functions have positive lower order. We confirm the existence of such slowly escaping points under a weaker growth condition. The second is to study the connection between the Fatou set and argument distribution. In view of the filling disks, we show nonexistence of multiply connected Fatou components if an entire function satisfies a weaker growth condition. We prove that the absence of singular directions implies the nonexistence of large annuli in the Fatou set.

Lower order and limiting directions of Julia sets of solutions to second order differential equations

In this paper, we consider the properties of entire solutions to second order differential equation(⁎)f″+Af′+Bf=0, where A(z) and B(z)≢0 are entire functions. Under certain assumptions on A(z) and B(z), we prove that every non-trivial solution f of equation (*) is of infinite lower order, and then obtain the measure estimation of the limiting directions of Julia sets for those infinite lower order entire solutions. The existence of Baker domain for f(n) is also discussed.

On hyperbolic dimension gap for entire functions

Polynomials and entire functions whose hyperbolic dimension is strictly smaller than the Hausdorff dimension of their Julia set are known to exist but in all these examples the latter dimension is maximal, i.e. equal to two. In this paper we show that there exist hyperbolic entire functions f f having Hausdorff dimension of the Julia set HD ⁡ ( J f ) > 2 \operatorname {HD} (\mathcal {J}_f)>2 and hyperbolic dimension H y p D i m ( f ) > H D ( J f ) \mathrm {HypDim}(f)>\mathrm {HD}(\mathcal {J}_f) .

Slowly recurrent Collet–Eckmann maps on the Riemann sphere

In this paper we study perturbations of rational Collet–Eckmann maps for which the Julia set is the whole sphere, and for which the critical set is allowed to be slowly recurrent. Generically, if each critical point is simple, we show that each such Collet–Eckmann map is a Lebesgue point of Collet–Eckmann maps in the space of rational maps of the same degree d≥2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d \\ge 2$$\\end{document}. The same result holds in each subspace, where we fix the multiplicities of the critical points.

Exploring the Julia and Mandelbrot sets of [formula omitted] using a four-step iteration scheme extended with [formula omitted]-convexity

Iterative approaches have been established to be fundamental for the creation of fractals. This paper introduces an approach to visualize Julia and Mandelbrot sets for a complex function of the form Q(z)=zp+logct for all z∈ℂ, where p∈N∖{1},t∈[1,∞),c∈ℂ∖{0}, using a four-step iteration scheme extended with s-convexity. The study introduces an escape criteria for generating Julia and Mandelbrot sets using a four-step iterative method. It investigates how changes in the iteration parameters influence the shape and color of the resulting Julia and Mandelbrot sets. This approach can generate a wide range of captivating fractals and analyze them through numerical experiments.

Mandelbrot Set as a Particular Julia Set of Fractional Order, Equipotential Lines and External Rays of Mandelbrot and Julia Sets of Fractional Order

This paper deepens some results on a Mandelbrot set and Julia sets of Caputo’s fractional order. It is shown analytically and computationally that the classical Mandelbrot set of integer order is a particular case of Julia sets of Caputo-like fractional order. Additionally, the differences between the fractional-order Mandelbrot set and Julia sets from their integer-order variants are revealed. Equipotential lines and external rays of a Mandelbrot set and Julia sets of fractional order are determined.

On an uncountable family of graphs whose spectrum is a Cantor set

For each p\geq 1 , the star automaton group \mathcal{G}_{S_p} is an automaton group which can be defined starting from a star graph on p+1 vertices. We study Schreier graphs associated with the action of the group \mathcal{G}_{S_p} on the regular rooted tree T_{p+1} of degree p+1 and on its boundary \partial T_{p+1} . With the transitive action on the n -th level of T_{p+1} is associated a finite Schreier graph \Gamma^{p}_{n} , whereas there exist uncountably many orbits of the action on the boundary, represented by infinite Schreier graphs which are obtained as the limits of the sequence \{\Gamma_{n}^{p}\}_{n\geq 1} in the Gromov–Hausdorff topology. We obtain an explicit description of the spectrum of the graphs \{\Gamma_{n}^{p}\}_{n\geq 1} . Then, by using amenability of \mathcal{G}_{S_p} , we prove that the spectrum of each infinite Schreier graph is the union of a Cantor set of zero Lebesgue measure, which is the Julia set of the quadratic map f_{p}(z) = z^{2}-2(p-1)z -2p , and a countable collection of isolated points supporting the Kesten–Neumann–Serre spectral measure. We also give a complete classification of the infinite Schreier graphs up to isomorphism of unrooted graphs, showing that they may have 1 , 2 or 2p ends, and that the case of 1 end is generic with respect to the uniform measure on \partial T_{p+1} .

Generation of Mandelbrot and Julia sets for generalized rational maps using SP-iteration process equipped with [formula omitted]-convexity

In this paper, we introduce a generalized rational map to develop a theory of escape criterion via the SP-iteration process equipped with s-convexity. Furthermore, we develop algorithms for the exploration of unique kinds of Mandelbrot as well as Julia sets. We demonstrate graphically the change in colour, size, and shape of images with the change in values of the considered iteration’s parameters. The new fractals thus obtained are visually very pleasing and attractive. Most of these newly generated fractals resemble natural objects around us. Moreover, we numerically study the dependence between the iteration’s parameters and the set size. The experiments show that this dependency is non-linear. We believe that the obtained conclusions will motivate researchers who are interested in fractal geometry.

Generation of Julia and Mandelbrot fractals for a generalized rational type mapping via viscosity approximation type iterative method extended with $ s $-convexity

<abstract><p>A dynamic visualization of Julia and Mandelbrot fractals involves creating animated representations of these fractals that change over time or in response to user interaction which allows users to gain deeper insights into the intricate structures and properties of these fractals. This paper explored the dynamic visualization of fractals within Julia and Mandelbrot sets, focusing on a generalized rational type complex polynomial of the form $ S_{c}(z) = a z^{n}+\frac{b}{z^{m}}+c $, where $ a, b, c \in \mathbb{C} $ with $ |a| > 1 $ and $ n, m \in \mathbb{N} $ with $ n > 1 $. By applying viscosity approximation-type iteration processes extended with $ s $-convexity, we unveiled the intricate dynamics inherent in these fractals. Novel escape criteria was derived to facilitate the generation of Julia and Mandelbrot sets via the proposed iteration process. We also presented graphical illustrations of Mandelbrot and Julia fractals, highlighting the change in the structure of the generated sets with respect to the variations in parameters.</p></abstract>

Analytic Studies of a Class of Langevin Differential Equations Dominated by a Class of Julia Fractal Functions

In this investigation, we study a class of analytic functions of type Carathéodory style in the open unit disk connected with some fractal domains. This class of analytic functions is formulated based on a kind of Langevin differential equations (LDEs). We aim to study the analytic solvability of LDEs in the advantage of geometric function theory consuming the geometric properties of the Julia fractal (JF) and other fractal connected with the logarithmic function. The analytic solutions of the LDEs are obtainable by employing the subordination theory.

Public Art Design Practice under Visual Communication Design

Abstract The expansion of visual communication design in public art makes the scope of visual communication wide. This new art form reflects the cross and integration of disciplines but also makes the form of public art in our life richer, and people can get more beautiful enjoyment. Research on the fractal algorithm in the escape time algorithm studied the Julia set in different function conditions of the fractal graph of the change and the function of different indices of the joint. The fractal graph obtained a colorful, more tense structure. Subsequently, it is applied to public art design, and after testing its performance, the fractal design of visual communication is combined with the IPA model to explore the practical effect of the fractal design in public art design. The results show that the improved fractal algorithm proposed in this paper increases the pattern generation rate from 61.5% to 92.6%. The fractal dimension measurement of 15 typical batik patterns shows that more than 85% of the batik patterns have an average value of more than 15000.

FRACTAL ORACLE NUMBERS

Consider orbits [Formula: see text] of the fractal iterator [Formula: see text], [Formula: see text], that start at initial points [Formula: see text], where [Formula: see text] is the set of all rational complex numbers (their real and imaginary parts are rational) and [Formula: see text] consists of all such [Formula: see text] whose complexity does not exceed some complexity parameter value [Formula: see text] (the complexity of [Formula: see text] is defined as the number of bits that suffice to describe the real and imaginary parts of [Formula: see text] in lowest form). The set [Formula: see text] is a bounded-complexity approximation of the filled Julia set [Formula: see text]. We present a new perspective on fractals based on an analogy with Chaitin’s algorithmic information theory, where a rational complex number [Formula: see text] is the analog of a program [Formula: see text], an iterator [Formula: see text] is analogous to a universal Turing machine [Formula: see text] which executes program [Formula: see text], and an unbounded orbit [Formula: see text] is analogous to an execution of a program [Formula: see text] on [Formula: see text] that halts. We define a real number [Formula: see text] which resembles Chaitin’s [Formula: see text] number, where, instead of being based on all programs [Formula: see text] whose execution on [Formula: see text] halts, it is based on all rational complex numbers [Formula: see text] whose orbits under [Formula: see text] are unbounded. Hence, similar to Chaitin’s [Formula: see text] number, [Formula: see text] acts as a theoretical limit or a “fractal oracle number” that provides an arbitrarily accurate complexity-based approximation of the filled Julia set [Formula: see text]. We present a procedure that, when given [Formula: see text] and [Formula: see text], it uses [Formula: see text] to generate [Formula: see text]. Several numerical examples of sets that estimate [Formula: see text] are presented.

Slowly recurrent Collet–Eckmann maps with non‐empty Fatou set

AbstractIn this paper, we study rational Collet–Eckmann maps for which the Julia set is not the whole sphere and for which the critical points are recurrent at a slow rate. In families where the orders of the critical points are fixed, we prove that such maps are Lebesgue density points of hyperbolic maps. In particular, if all critical points are simple, these maps are Lebesgue density points of hyperbolic maps in the full space of rational maps of any degree .

Fuzzy Julia Sets and Fuzzy Superior Julia Sets

This article examine the past study of fuzzy Mandelbrot set and fuzzy superior Mandelbrot set, then give the definition of fuzzy Julia sets and fuzzy superior Julia sets with the idea of utilizing membership functions to represent the escape velocity of Julia sets or superior Julia sets of each complex number in the definition of fuzzy Mandelbrot set and fuzzy superior Mandelbrot set inherited while the membership functions are selected to distinguish complex numbers with different escape velocity and same orbit. Then some examples of fuzzy Julia sets and fuzzy superior Julia sets are presented. With some observation of the examples, some analytical and topological properties of fuzzy Julia sets are demonstrated and sketches of the proofs are also presented. In the part of fuzzy superior Julia sets, some trivial conclusions as well as an example is presented where more complicated properties which involves the study into very chaotic behaviors are left open.

Synchronization and Clustering in Complex Quadratic Networks.

Synchronization and clustering are well studied in the context of networks of oscillators, such as neuronal networks. However, this relationship is notoriously difficult to approach mathematically in natural, complex networks. Here, we aim to understand it in a canonical framework, using complex quadratic node dynamics, coupled in networks that we call complex quadratic networks (CQNs). We review previously defined extensions of the Mandelbrot and Julia sets for networks, focusing on the behavior of the node-wise projections of these sets and on describing the phenomena of node clustering and synchronization. One aspect of our work consists of exploring ties between a network's connectivity and its ensemble dynamics by identifying mechanisms that lead to clusters of nodes exhibiting identical or different Mandelbrot sets. Based on our preliminary analytical results (obtained primarily in two-dimensional networks), we propose that clustering is strongly determined by the network connectivity patterns, with the geometry of these clusters further controlled by the connection weights. Here, we first explore this relationship further, using examples of synthetic networks, increasing in size (from 3, to 4, to 20 nodes). We then illustrate the potential practical implications of synchronization in an existing set of whole brain, tractography-based networks obtained from 197 human subjects using diffusion tensor imaging. Understanding the similarities to how these concepts apply to CQNs contributes to our understanding of universal principles in dynamic networks and may help extend theoretical results to natural, complex systems.

Synchronization of Julia Sets in Three-Dimensional Discrete Financial Models

When aiming to achieve consistency in fractal characteristics between different models, it is crucial to consider the synchronization of Julia sets. This paper studies the synchronization of Julia sets in three-dimensional discrete financial models. First, three-dimensional discrete financial models with different model parameters are proposed and their Julia sets are presented. According to the model forms, two kinds of synchronous couplers that can achieve synchronization of Julia sets between different models are designed by changing the synchronization parameters. The proposed synchronization method is theoretically derived and the efficiency of different synchronous couplers are compared. Finally, the effectiveness is verified by Julia sets graphics. This method has reference value for theoretical research into financial models in the field of fractals.

Julia Sets, Jordan Curves and Quasi-circles

In this paper, the classification of rational functions whose Julia sets are Jordan arcs or curves, which started in (Carleson and Gamelin in Complex dynamics, Springer, Berlin, 1993; Steinmetz in Math Ann 307:531–541, 1997), will be completed. The method of proof is based on two quasi-conformal surgery procedures, which enables shifting the critical points in simply connected (super-)attracting and parabolic basins into a single critical point of highest possible multiplicity.

Limit Directions of Julia Sets of Entire Functions and Complex Differential Equations

The limiting directions of Julia sets of infinite order entire functions are studied by combining the theory of complex dynamic system and the theory of complex differential equations, in which the lower bound of the measure of limiting direction of Julia set of entire solutions of complex differential equations is obtained.