Julia Set Research Articles

On invariant measures of ‘satellite’ infinitely renormalizable quadratic polynomials

Abstract Let $f(z)=z^2+c$ be an infinitely renormalizable quadratic polynomial and $J_\infty $ be the intersection of forward orbits of ‘small’ Julia sets of its simple renormalizations. We prove that if f admits an infinite sequence of satellite renormalizations, then every invariant measure of $f: J_\infty \to J_\infty $ is supported on the postcritical set and has zero Lyapunov exponent. Coupled with [13], this implies that the Lyapunov exponent of such f at c is equal to zero, which partly answers a question posed by Weixiao Shen.

Equilibrium measures on Julia sets of random quadratic polynomials

Abstract We consider sequences of compositions of quadratic polynomials f c n ( z ) = z 2 + c n . For such sequences one can naturally generalise the definitions of the Julia set and basin of infinity from the autonomous case. In this setting the Julia set depends on a sequence ω = ( c 0 , c 1 , … ) . We study the equilibrium (harmonic) measure on such Julia sets. In particular, we calculate the Hausdorff dimension of the equilibrium measure and study its dependence on the ‘scale of randomnes’.

On the area rate of perturbed Siegel disks

Abstract In the famous work by Buff and Chéritat constructing quadratic Julia sets with positive area, the control of the shape of perturbed Siegel disks is a key technique. To do it, Buff and Chéritat added a restrictive condition on rotation numbers. In this paper, we show that this restrictive condition is not necessary. As a consequence, more quadratic Julia sets with positive area can be constructed.

Disconnected Julia sets as buried Julia components

Disconnected Julia sets as buried Julia components

On the Simulation of Artificial Cracks in Brittle Materials Using Julia Set Fractals

On the Simulation of Artificial Cracks in Brittle Materials Using Julia Set Fractals

Transcendental Julia sets of positive finite Lebesgue measure

We show that there exists a transcendental entire function whose Julia set has positive finite Lebesgue measure.

Asymptotic behaviors of Hausdorff dimensions of radial Julia sets of exponential and cosine functions

In this paper, we obtained a four terms asymptotic formula of the Hausdorff dimension of the radial Julia set Jr(fλ) for the exponential function fλ(z) = λez as λ → 0. Moreover, we also discussed the asymptotic behavior of the Hausdorff dimension of the radial Julia set of cosine functions fa,b(z) = aez + be−z as a, b → 0 in various ways.

Exploring the Complex Dynamics of Julia and Mandelbrot Sets for the Complex-valued Mapping sin(z<sup>k</sup>) + az + c Using Four-Step Iterative Scheme with s-Convexity

In this paper, we investigate unique variations of the Julia and Mandelbrot sets, establishing criteria for the escape of a function Tcz = sin(zk) + az + c for all z ∈ C, where k ∈ N \ {1} and a, c ∈ C with a̸ = 0 by utilizing a four-step iterative scheme extended with s-convexity, we analyze the dynamics of these fractals by formulating and implementing the algorithms. Our exploration focuses on understanding how distinct parameters impact these fractals’ color, dynamics, and overall visual characteristics. Our findings reveal that specific fractal patterns resemble exquisite natural objects like flowers and spiders. These fractal designs are also employed as visually striking artistic motifs on garments and textile materials. The captivating and complex properties of fractal patterns within dynamic systems make them an enticing and visually appealing area of research.

Adaptive synchronization and anti-synchronization of Julia sets generated by the competitive model

In this paper, we study the fractal behaviour of a competitive model that describes the interaction of plankton allelopathy. This paper aims to establish synchronization and anti-synchronization of Julia sets of two competitive systems with some different parameters by using an adaptive control strategy. Firstly, a discrete version of the competitive model is obtained, and then the Julia set of the discrete version is generated by using the escape-time algorithm. Adaptive controllers and parameter update laws for unknown parameters are designed to achieve synchronization and anti-synchronization of Julia sets. Furthermore, we can determine unknown parameters of the competitive system by using this adaptive control technique. Here, the adaptive synchronization and anti-synchronization of Julia sets are accomplished by its trajectories synchronization and anti-synchronization due to the close relation of trajectories of the system with the Julia set of the system. Numerical simulations are carried out to validate several key theoretical results as well as the efficacy and accuracy of the applied methodologies. Moreover, with the help of this analysis, we can study other models of a similar type.

Halvorsen chaotic system based microwave absorber modelling for fighter jet stealth technologies

This study focuses on the development and detailed analysis of a broadband microwave absorber utilizing Halvorsen chaotic dynamics, that focuses at enhancing stealth capabilities for fighter jets. The investigation begins by exploring the mathematical formulation of the Halvorsen chaotic system and conducting a parametric sweep of its control parameters to generate distinct two-dimensional and three-dimensional chaotic attractor plots. These plots are then post-processed using Julia set theory to develop intricate fractal patterns, which serve as the foundation for the absorber design. Image processing techniques, including filtering and thresholding, are employed to refine the patterns by removing artifacts and noise, ensuring they are suitable for practical implementation. The refined fractal patterns are then imported into a computational electromagnetic simulation environment where they are patterned onto a 0.035 mm thick copper sheet. Moreover, the Magtrex 555 substrate with a thickness of 1.52 mm, is selected for its high permittivity and low-loss characteristics. A comprehensive series of parametric studies are conducted to evaluate the influence of various design parameters such as side length, unit cell geometry, and substrate thickness on the absorber’s electromagnetic performance. Important parameters include the effects of chaotic control parameter optimization and the electromagnetic boundary conditions applied during the simulations. Extensive simulations are performed across the 2–20 GHz frequency range to evaluate absorption efficiency, focusing on key metrics like absorptivity, surface current distribution, and electric field distribution. The final design achieves over 90 % absorption efficiency within the target frequency band when the chaotic control parameters are optimized. Finally, comparative analysis using different commercially available substrates, including FR-4 and Rogers RO3003, reveals that Magtrex 555 offers superior absorption performance. The study concludes with a detailed presentation of the final absorber design, alongside an indepth discussion of the frequency range, parametric variations, and the impact of chaotic system dynamics on the absorption properties. This research provides crucial insights into the design and optimization of chaotic-system-based microwave absorbers, that advances the development of stealth technology in military applications.

Generation of Mandelbrot and Julia sets by using M-iteration process

In this manuscript, we introduce the M-iteration process for generating Mandelbrot and Julia sets. We establish an escape criterion for a polynomial of the form xk+1+c in the complex plane corresponding to the M-iteration process. Next, we present some graphical examples of Mandelbrot and Julia sets generated using the proven escape criterion and the escape-time algorithm. We also compare the images generated with the M, Mann, and Picard–Mann iterations. Moreover, we study the dependency between the iterations’ parameters and three numerical measures (the average escape time, non-escaping area index, and box-counting dimension) used in the literature. The results show that fractal images generated using the M-iteration are entirely different from those generated using the other two analysed iteration schemes. Moreover, the dependencies are highly non-linear and vary between the iterations.

A Mañé-Manning formula for expanding measures for endomorphisms of ℙ^{𝕜}

Let k ≥ 1 k \ge 1 be an integer and f f a holomorphic endomorphism of P k ( C ) \mathbb {P}^k (\mathbb {C}) of algebraic degree d ≥ 2 d\geq 2 . We introduce a dynamical volume dimension for ergodic f f -invariant probability measures with strictly positive Lyapunov exponents. In particular, this class of measures includes all ergodic measures whose measure-theoretic entropy is strictly larger than ( k − 1 ) log ⁡ d (k-1)\log d , a natural generalization of the class of measures of positive measure-theoretic entropy in dimension 1. The volume dimension is equivalent to the Hausdorff dimension when k = 1 k=1 , but depends on the dynamics of f f to incorporate the absence of an analogue of Koebe’s theorem and the non-conformality of holomorphic endomorphisms for k ≥ 2 k\geq 2 . If ν \nu is an ergodic f f -invariant probability measure with strictly positive Lyapunov exponents, we prove a generalization of the Mañé-Manning formula relating the volume dimension, the measure-theoretic entropy, and the sum of the Lyapunov exponents of ν \nu . As a consequence, we give a characterization of the first zero of a natural pressure function for such expanding measures in terms of their volume dimensions. For hyperbolic maps, such zero also coincides with the volume dimension of the Julia set, and with the exponent of a natural (volume-)conformal measure. This generalizes results by Denker-Urbański [Nonlinearity 4 (1991), pp. 365–384; Trans. Amer. Math. Soc. 328 (1991), pp. 563–587] and McMullen [Comment. Math. Helv. 75 (2000), pp. 535–593] in dimension 1 to any dimension k ≥ 1 k\geq 1 . Our methods mainly rely on a theorem by Berteloot-Dupont-Molino [Ann. Inst. Fourier (Grenoble) 58 (2008), pp. 2137–2168], which gives a precise control on the distortion of inverse branches of endomorphisms along generic inverse orbits with respect to measures with strictly positive Lyapunov exponents.

Exploring Weight Functions: A Family of One-Point Methods for Multiple Roots

In this paper we introduce a family of one-point methods for multiple roots using weight functions. We analyze the necessary conditions for the weight function to achieve third-order accuracy, and then present a sequence of rational weight functions for a complex quadratic function. The resulting members of the family exhibit Julia sets that align with the perpendicular bisector of the roots. We compare selected members with existing methods based on basin of attraction analysis, demonstrating their competitive performance. Additionally, we investigate the influence of root multiplicity on the fractal structures of basin of attraction for equations with both simple and multiple roots.

Fractals as Julia and Mandelbrot Sets via S-iteration

To understand the phenomena of expanding symmetries Fractals patterns are an important tool which exhibit similar patterns for different scales. In the present paper, establishing an escape criteria by using S-iteration process to visualize fractals namely Julia and Mandelbrot sets for the function F(w)=aewp+c where c,a∈ C and p≥2. The result obtain is a generalization of the existing algorithm and technique providing fractals for different parameter values. Also, the time taken to obtain fractals for different parameters by using computer software MATLAB is computed in seconds.

Fractal as Julia sets of complex functions via a new generalized viscosity approximation type iterative method

In this article, we study and explore novel variants of Julia set patterns that are linked to the complex exponential function $W(z)=pe^{z^n}+qz+r$, and complex cosine function $T(z)=\cos({z^n})+dz+c$, where $n\geq 2$ and $c,d,p,q,r\in \mathbb{C}$ by employing a generalized viscosity approximation type iterative method introduced by Nandal et al. (Iteration process for fixed point problems and zero of maximal monotone operators, Symmetry, 2019) to visualize these sets. We utilize a generalized viscosity approximation type iterative method to derive an escape criterion for visualizing Julia sets. This is achieved by generalizing the existing algorithms, which led to visualization of beautiful fractals as Julia sets. Additionally, we present graphical illustrations of Julia sets to demonstrate their dependence on the iteration parameters. Our study concludes with an analysis of variations in the images and the influence of parameters on the color and appearance of the fractal patterns. Finally, we observe intriguing behaviors of Julia sets with fixed input parameters and varying values of $n$ via proposed algorithms.

LIMITING DIRECTIONS FOR ENTIRE SOLUTIONS OF A

Let us consider f as being an entire solution of the differential-difference equation G(z, f)+h(z)fm(z) = 0, (m ∈ N), where h(z) is a transcendental entire function and G(z, f) is a differential-difference polynomial in f with entire coefficients. By considering the order and deficiency of h(z) and such coefficients, we mainly study the radial distribution of f, and establish a lower bound of measure for the set of common limiting directions of the Julia sets of derivatives and primitives of its shifts.

From hyperbolic to parabolic parameters along internal rays

For the quadratic family f c ( z ) = z 2 + c f_{c}(z) = z^2+c with c c in a hyperbolic component of the Mandelbrot set, it is known that every point in the Julia set moves holomorphically. In this paper we give a uniform derivative estimate of such a motion when the parameter c c converges to a parabolic parameter c ^ {\hat {c}} radially; in other words, it stays within a bounded Poincaré distance from the internal ray that lands on c ^ {\hat {c}} . We also show that the motion of each point in the Julia set is uniformly one-sided Hölder continuous at c ^ {\hat {c}} with exponent depending only on the petal number. This paper is a parabolic counterpart of the authors’ paper “From Cantor to semi-hyperbolic parameters along external rays” [Trans. Amer. Math. Soc. 372 (2019), pp. 7959–7992].

Hausdorff limits of external rays: The topological picture

AbstractWe study Hausdorff limits of the external rays of a given periodic angle along a convergent sequence of polynomials of degree with connected Julia sets.

Strong Probabilistic Stability in Holomorphic Families of Endomorphisms of ℙk (ℂ) and Polynomial-Like Maps

Abstract We prove that, in stable families of endomorphisms of $\mathbb{P}^{k} (\mathbb C)$, the measurable holomorphic motion of the Julia sets introduced by Berteloot, Dupont, and the first author is unbranched at almost every point with respect to all measures on the Julia set with strictly positive Lyapunov exponents and not charging the post-critical set. This provides a parallel in this setting to the probabilistic stability of Hénon maps by Berger–Dujardin–Lyubich. An analogous result holds in families of polynomial-like maps of large topological degree. In this case, we also give a sufficient condition for the positivity of the Lyapunov exponents of an ergodic measure in terms of its measure-theoretic entropy, generalizing to this setting an analogous result by de Thélin and Dupont valid on $\mathbb{P}^{k} (\mathbb C)$.

Generating Geometric Patterns Using Complex Polynomials and Iterative Schemes

Iterative procedures have been proved as a milestone in the generation of fractals. This paper presents a novel approach for generating and visualizing fractals, specifically Mandelbrot and Julia sets, by utilizing complex polynomials of the form QC(p)=apn+mp+c, where n≥2. It establishes escape criteria that play a vital role in generating these sets and provides escape time results using different iterative schemes. In addition, the study includes the visualization of graphical images of Julia and Mandelbrot sets, revealing distinct patterns. Furthermore, the study also explores the impact of parameters on the deviation of dynamics, color, and appearance of fractals.