Julia Set Research Articles

On the quaternion Julia sets via Picard–Mann iteration

In recent years, an extensive study on the use of various iteration schemes from fixed point theory for the generation of Mandelbrot and Julia sets in complex space has been carried out. In this work, inspired by these progresses, we study the use of the Picard–Mann iteration scheme for the Julia sets in the quaternion space. Specifically, in our study, we prove the escape criterion of the Picard–Mann orbit and examine the symmetry of the Julia set for the quadratic function. Moreover, we present and discuss some 2D and 3D graphical examples of the sets generated using the Picard–Mann iteration scheme. We further analyse the influence of a parameter of interest used in the Picard–Mann iteration scheme on the average number of iterations for 2D cross sections of quaternion Julia sets of different degrees.

A new perspective on the Sullivan dictionary via Assouad type dimensions and spectra

The Sullivan dictionary provides a beautiful correspondence between Kleinian groups acting on hyperbolic space and rational maps of the extended complex plane. We focus on the setting of geometrically finite Kleinian groups with parabolic elements and parabolic rational maps. In this context an especially direct correspondence exists concerning the dimension theory of the associated limit sets and Julia sets. In recent work we established formulae for the Assouad type dimensions and spectra for these fractal sets and certain conformal measures they support. This allows a rather more nuanced comparison of the two families in the context of dimension. In this expository article we discuss how these results provide new entries in the Sullivan dictionary, as well as revealing striking differences between the two families.

A Shape Modulus for Fractal Geometry Generation

AbstractWe present an efficient new method for computing Mandelbrot‐like fractals (Julia sets) that approximate a user‐defined shape. Our algorithm is orders of magnitude faster than previous methods, as it entirely sidesteps the need for a time‐consuming numerical optimization. It is also more robust, succeeding on shapes where previous approaches failed. The key to our approach is a versor‐modulus analysis of fractals that allows us to formulate a novel shape modulus function that directly controls the broad shape of a Julia set, while keeping fine‐grained fractal details intact. Our formulation contains flexible artistic controls that allow users to seamlessly add fractal detail to desired spatial regions, while transitioning back to the original shape in others. No previous approach allows Mandelbrot‐like details to be “painted” onto meshes.

The long-term behavior of the p-adic Sigmoid Beverton-Holt dynamical systems in the projective line [formula omitted

In this paper we study the long-term behavior of the p-adic dynamical systems associated with the Sigmoid Beverton-Holt model that arises in population dynamics. The corresponding fixed points, maximal Siegel disks, attractors, and periodic trajectories are analyzed in the projective line P1(Qp). Moreover, the Julia and Fatou sets associated with these dynamical systems are also examined.

Fractal transformations of recursive walks in the Julia fern

One of the most studied fractals corresponds to the Julia Set and one of its elements can be obtained with the following recursive formula in the complex plane: Zi+1 = Zi2 - 1, the result depends on the chosen starting complex number. We can define the Julia set of a complex variable polynomial as the boundary of the set of points that escape to infinity when iterating this polynomial. This means that the orbit of an element of the Julia set does not escape infinity.

On dynamics of asymptotically minimal polynomials

We study dynamical properties of asymptotically extremal polynomials associated with a non-polar planar compact set E. In particular, we prove that if the zeros of such polynomials are uniformly bounded then their Brolin measures converge weakly to the equilibrium measure of E. In addition, if E is regular and the zeros of such polynomials are sufficiently close to E then we show that the filled Julia sets converge to polynomial convex hull of E in the Klimek topology.

The Hausdorff Dimension of Escaping Sets of Meromorphic Functions in the Speiser Class

Abstract Bergweiler and Kotus gave sharp upper bounds for the Hausdorff dimension of the escaping set of a meromorphic function in the Eremenko–Lyubich class, in terms of the order of the function and the maximal multiplicity of the poles. We show that these bounds are also sharp in the Speiser class. We apply this method also to construct meromorphic functions in the Speiser class with preassigned dimensions of the Julia set and the escaping set.

Symmetric cubic laminations

To investigate the degree d d connectedness locus, Thurston [On the geometry and dynamics of iterated rational maps, Complex Dynamics, A K Peters, Wellesley, MA, 2009, pp. 3–137] studied σ d \sigma _d -invariant laminations, where σ d \sigma _d is the d d -tupling map on the unit circle, and built a topological model for the space of quadratic polynomials f ( z ) = z 2 + c f(z) = z^2 +c . In the spirit of Thurston’s work, we consider the space of all cubic symmetric polynomials f λ ( z ) = z 3 + λ 2 z f_\lambda (z)=z^3+\lambda ^2 z in a series of three articles. In the present paper, the first in the series, we construct a lamination C s C L C_sCL together with the induced factor space S / C s C L \mathbb {S}/C_sCL of the unit circle S \mathbb {S} . As will be verified in the third paper of the series, S / C s C L \mathbb {S}/C_sCL is a monotone model of the cubic symmetric connectedness locus, i.e. the space of all cubic symmetric polynomials with connected Julia sets.

Some Properties of the Mandelbrot Sets M(Q_α)

The aim of this work is to study the concept of the parameter plane, so we are interested in introducing the Mandebrot set,as well as the Mandelbrot set is a fractals ,whereas, if we zoom in on any small piece in the border regions of the shape, we notice that it is similar to the original shape, that is the Mandelbrot set, with special topological specifications, and we introduce of some properties of the Mandelbrot sets for the map in the form , where is a complex constant, and we prove that the conjecture is the Duadi-Hubbard theorem . By studying of the parameter plane that the Mandelbrot set is a component form in one of its details in the dynamical plane it is the Julia set , in other words every small detail of the Mandelbrot set represents the Julia set, so there is more than one characteristic of the Julia set, we will limit ourselves only to presenting the connection of the Julia set for .

Imaginary scators quadratic mapping in 1+2D dynamic space

The elliptic scator algebra quadratic iteration is evaluated in 1+2 dimensions in dynamic space. There exists a non divergent K set in the scator three dimensional space with a highly complex boundary J=∂K. Two and three dimensional renderings of the sets in dynamic space are published for the first time. The sets exhibit a rich fractal boundary in all three directions. Some of the salient features of the sets can be described in terms of square nilpotent iterations. The Julia and filled in Julia sets are identically reproduced at two perpendicular planes where only one non-vanishing hypercomplex director component is present. The fixed points of K in S1+2 with real constant c, can be obtained from the roots of a scator quadratic polynomial equation. In S1+2 there can be, in addition to the usual complex roots, hypercomplex roots that give rise to four additional fixed points. The inverse orbits of the hypercomplex roots reveal a rich complex structure, that allow for the evaluation of an infinite set of points in ∂K. The ∂K set of the origin is equal to the unit magnitude scator surface, named a cusphere. The J set exhibits self similar structures in 3D at different scales, typical of fractal phenomena. The ix set, is the three dimensional equivalent of the M-set in three dimensions. It is conjectured that the ix-set with some restrictions, is the set of parameters where the J set is connected.

Interior dynamics of fatou sets

In this paper, we investigate the precise behavior of orbits inside attracting basins. Let f be a holomorphic polynomial of degree mge 2 in mathbb {C}, mathcal {A}(p) be the basin of attraction of an attracting fixed point p of f, and Omega _i (i=1, 2, cdots ) be the connected components of mathcal {A}(p). Assume Omega _1 contains p and {f^{-1}(p)}cap Omega _1ne {p}. Then there is a constant C so that for every point z_0 inside any Omega _i, there exists a point qin cup _k f^{-k}(p) inside Omega _i such that d_{Omega _i}(z_0, q)le C, where d_{Omega _i} is the Kobayashi distance on Omega _i. In paper Hu (Dynamics inside parabolic basins, 2022), we proved that this result is not valid for parabolic basins.

Multicorn Sets of z¯k+cm via S-Iteration with h-Convexity

Fractals represent important features of our natural environment, and therefore, several scientific fields have recently begun using fractals that employ fixed-point theory. While many researchers are working on fractals (i.e., Mandelbrot and Julia sets), only a very few have focused on multicorn sets and their dynamic nature. In this paper, we study the dynamics of multicorn sets of z¯k+cm, where k≥2, c≠0∈C, and m∈R, by using S-iteration with h-convexity instead of standard S-iteration. We develop escape criterion z¯k+cm for S-iteration with h-convexity. We analyse the dynamical behaviour of the proposed conjugate complex function and discuss the variation of iteration parameters along with function parameter m. Moreover, we discuss the effects of input parameters of the proposed iteration and conjugate complex functions of the behaviour of multicorn sets with numerical simulations.

On critical renormalization of complex polynomials

Holomorphic renormalization plays an important role in complex polynomial dynamics. We consider invariant continua that are not polynomial-like Julia sets because of extra critical points. However, under certain assumptions, these invariant continua can be identified with Julia sets of lower degree polynomials up to a topological conjugacy. Thus we extend the concept of renormalization.

Fractal control and synchronization of population competition model based on the T–S fuzzy model

It is of great significance to study the law of population change in species protection, resource utilization, ecological stability, and so on. The fractal behavior of the population competition model is discussed from the fractal viewpoint. By using the sector nonlinear method, the Takagi–Sugeno (T–S) fuzzy model of the population competition model is established, and the Julia set of the population competition model based on the T–S fuzzy model is introduced. The parallel distributed compensation (PDC) is used to design the corresponding fuzzy controller, and sufficient conditions for the stability of the system are given in the form of linear matrix inequalities (LMIs). The parameters of controller gain are obtained by solving LMIs, and the control of the Julia set of the population competition model based on the T–S fuzzy model is carried out. Using the exact linearization technique, the synchronization of the Julia sets of the population competition models is realized based on the T–S fuzzy model by the linear control method, and the Julia set diagram of the corresponding population competition model is analyzed.

Fractal Complexity of a New Biparametric Family of Fourth Optimal Order Based on the Ermakov–Kalitkin Scheme

In this paper, we generalize the scheme proposed by Ermakov and Kalitkin and present a class of two-parameter fourth-order optimal methods, which we call Ermakov’s Hyperfamily. It is a substantial improvement of the classical Newton’s method because it optimizes one that extends the regions of convergence and is very stable. Another novelty is that it is a class containing as particular cases some classical methods, such as King’s family. From this class, we generate a new uniparametric family, which we call the KLAM, containing the classical Ostrowski and Chun, whose efficiency, stability, and optimality has been proven but also new methods that in many cases outperform these mentioned, as we prove. We demonstrate that it is of a fourth order of convergence, as well as being computationally efficienct. A dynamical study is performed allowing us to choose methods with good stability properties and to avoid chaotic behavior, implicit in the fractal structure defined by the Julia set in the related dynamic planes. Some numerical tests are presented to confirm the theoretical results and to compare the proposed methods with other known methods.

Stability, Data Dependence, and Convergence Results with Computational Engendering of Fractals via Jungck–DK Iterative Scheme

We have developed a Jungck version of the DK iterative scheme called the Jungck–DK iterative scheme. Our analysis focuses on the convergence and stability of the Jungck–DK scheme for a pair of non-self-mappings using the more general contractive condition. We demonstrate that this iterative scheme converges faster than all other leading Jungck-type iterative schemes. To further illustrate its effectiveness, we provide an example to verify the rate of convergence and prove the data dependence result for the Jungck–DK iterative scheme. Finally, we calculate the escape criteria for generating Mandelbrot and Julia sets for polynomial functions and present visually appealing images of these sets by our modified iteration.

Computing eigenvalues of the Laplacian on rough domains

We prove a general Mosco convergence theorem for bounded Euclidean domains satisfying a set of mild geometric hypotheses. For bounded domains, this notion implies norm-resolvent convergence for the Dirichlet Laplacian which in turn ensures spectral convergence. A key element of the proof is the development of a novel, explicit Poincaré-type inequality. These results allow us to construct a universal algorithm capable of computing the eigenvalues of the Dirichlet Laplacian on a wide class of rough domains. Many domains with fractal boundaries, such as the Koch snowflake and certain filled Julia sets, are included among this class. Conversely, we construct a counterexample showing that there does not exist a universal algorithm of the same type capable of computing the eigenvalues of the Dirichlet Laplacian on an arbitrary bounded domain.

Fractional quantum Julia set

This paper proposes fractional quantum Julia sets based on a fractional q-difference map and preliminarily investigates their fractal dynamic characteristics by numerical methods and graphical explorations. The influence of parameters on the novel fractal sets is performed by dimension analysis. Memory and scale mainly affect the appearance and existence of the sets, respectively. The robustness of the sets to three types of dynamic noise perturbations is discussed by the Julia deviation tools. In particular, the deviation index is fitted concerning noise intensity. Several extensive numerical experiments are done to support the main conclusions.

Dynamics inside Fatou sets in higher dimensions

Dynamics inside Fatou sets in higher dimensions

Lower bounds on the Hausdorff dimension of some Julia sets

We present an algorithm for a rigorous computation of lower bounds on the Hausdorff dimensions of Julia sets for a wide class of holomorphic maps. We apply this algorithm to obtain lower bounds on the Hausdorff dimension of the Julia sets of some infinitely renormalizable real quadratic polynomials, including the Feigenbaum polynomial . In addition to that, we construct a piecewise constant function on that provides rigorous lower bounds for the Hausdorff dimension of the Julia sets of all quadratic polynomials with . Finally, we verify the conjecture of Ludwik Jaksztas and Michel Zinsmeister that the Hausdorff dimension of the Julia set of a quadratic polynomial , is a C 1-smooth function of the real parameter c on the interval .