Julia Set Research Articles

Julia sets are quasicircles for rational maps of degree N

This research aims to study complex dynamics, where the complex dynamic is considered a branch of dynamical systems, which takes maps from the Riemann sphere to the Riemann sphere. The Julia sets are also examples of fractals and the most important of them, thus being a fractal set. This work aims to study the Julia sets of a family of rational maps are quasicircles and offer the boundaries of the immediate basin of attraction of zero and the infinity are quasi-circles. Now, we introduce the family of maps are defined as: Gb (z) = 2bn+1 z–n + z2n – b3n+1/zn (z2n – bn–1), (1) where n ≥ 2 and b ϵ ℂ\{0} also bn+1 ≠ 1 and b2n+2 ≠ 1.

On Julia Set of the Functions Which Have Parabolic Fixed Points

In the present article, functions with parabolic fixed points are examined and Julia sets have been investigated, including rotational symmetry revealing. Collections of functions which have lines and rays as Julia sets, have been adduced. Construction algorithms for Julia sets of some functions and a number of results obtained by analytical investigations which are visualised by means of computer programs, have been described. Comparison of holomorphic dynamics of a collection of functions which have parabolic fixed point, with holomorphic dynamics of a collection of functions which have attracting fixed point, has been conducted.

CALCULATING JULIA FRACTAL SETS IN ANY EMBEDDING DIMENSION

In this paper, we compute and display hyperdimensional Julia sets using a multiplication operator that can be applied to any embedding dimension. Special attention is given to five-dimensional (5D) Julia sets, which are visualized in 3D through a voxel-based representation and volumetric ray casting rendering.

The lower bound on the measure of sets consisting of Julia limiting directions of solutions to some complex equations associated with Petrenko's deviation

<abstract><p>In the value distribution theory of complex analysis, Petrenko's deviation is to describe more precisely the quantitative relationship between $ T (r, f) $ and $ \log M (r, f) $ when the modulus of variable $ |z| = r $ is sufficiently large. In this paper we introduce Petrenko's deviations to the coefficients of three types of complex equations, which include difference equations, differential equations and differential-difference equations. Under different assumptions we study the lower bound of limiting directions of Julia sets of solutions of these equations, where Julia set is an important concept in complex dynamical systems. The results of this article show that the lower bound of limiting directions mentioned above is closely related to Petrenko's deviation, and our conclusions improve some known results.</p></abstract>

A bounded sequence of bitransitive and capture Sierpiński curve Julia sets for 3-circle inversions

A bounded sequence of bitransitive and capture Sierpiński curve Julia sets for 3-circle inversions

Symmetries and Dynamics of Generalized Biquaternionic Julia Sets Defined by Various Polynomials

Higher-dimensional hypercomplex fractal sets are getting more and more attention because of the discovery of more and more interesting properties and visual aesthetics. In this study, the attention was focused on generalized biquaternionic Julia sets and a generalization of classical Julia sets, defined by power and monic higher-order polynomials. Despite complex and quaternionic Julia sets, their biquaternionic analogues are still not well investigated. The performed morphological analysis of 3D projections of these sets allowed for definition of symmetries, limit shapes, and similarities with other fractal sets of this class. Visual observations were confirmed by stability analysis for initial cycles, which confirm similarities with the complex, bicomplex, and quaternionic Julia sets, as well as manifested differences between the considered formulations of representing polynomials.

Mandelbrot and Julia Sets of Complex Polynomials Involving Sine and Cosine Functions via Picard–Mann Orbit

Mandelbrot and Julia Sets of Complex Polynomials Involving Sine and Cosine Functions via Picard–Mann Orbit

Hausdorff Dimension of Julia Sets in the Logistic Family

A closed interval and circle are the only smooth Julia sets in polynomial dynamics. D. Ruelle proved that the Hausdorff dimension of unicritical Julia sets close to the circle depends analytically on the parameter. Near the tip of the Mandelbrot set \({{\mathcal {M}}}\), the Hausdorff dimension is generally discontinuous. Answering a question of J-C. Yoccoz in the conformal setting, we observe that the Hausdorff dimension of quadratic Julia sets depends continuously on c and find explicit bounds at the tip of \({{\mathcal {M}}}\) for most real parameters in the the sense of 1-dimensional Lebesgue measure.

Some transcendental entire functions with irrationally indifferent fixed points

Let $S$ be the set of all transcendental entire functions of the form $P(z) \exp (Q(z))$, where $P$ and $Q$ are polynomials. In this paper, by using the theory of polynomial-like mappings, we construct various kinds of functions in $S$ with irrationally indifferent fixed points as follows: (1) We construct functions in $S$ with bounded type Siegel disks centered at points other than the origin bounded by quasicircles containing critical points. This is an extension of Zakeri's result in [24] for $f \in S$. (2) We construct functions in $S$ with Cremer points whose multipliers satisfy some Cremer's condition in [6] only for rational functions. Our method shows that this condition can be applicable even in some transcendental cases. (3) For any integer $d \geq 2$ and some $c \in {\mathbf C} \setminus \{0\}$, we show that the function of the form $e^{2\pi i \theta}z(1 + cz)^{d-1}e^z\,(\theta \in {\mathbf R} \setminus {\mathbf Q})$ has a Siegel point at the origin if and only if $\theta$ is a Brjuno number. This is an extension of Geyer's result in [11]. (4) For the function of the form $(e^{2\pi i\theta}z+\alpha z^2)e^z \,(\theta \in {\mathbf R} \setminus {\mathbf Q}, \alpha \in {\mathbf C} \setminus \{0\})$, we show that if $\alpha$ and $\theta$ satisfy some condition, then the Siegel disk centered at the origin is bounded by a Jordan curve containing a critical point, which is not a quasicircle. Moreover, we can choose $\alpha$ and $\theta$ so that the Lebesgue measure of the Julia set is positive and can also choose them so that it is zero. This is an extension of Keen and Zhang's result in [13].

Asymptotically holomorphic methods for infinitely renormalizable unimodal maps

AbstractThe purpose of this paper is to initiate a theory concerning the dynamics of asymptotically holomorphic polynomial-like maps. Our maps arise naturally as deep renormalizations of asymptotically holomorphic extensions of $C^r$ ( $r>3$ ) unimodal maps that are infinitely renormalizable of bounded type. Here we prove a version of the Fatou–Julia–Sullivan theorem and a topological straightening theorem in this setting. In particular, these maps do not have wandering domains and their Julia sets are locally connected.

A Four Step Feedback Iteration and Its Applications in Fractals

Fractals play a vital role in modeling the natural environment. The present aim is to investigate the escape criterion to generate specific fractals such as Julia sets, Mandelbrot sets and Multi-corns via F-iteration using complex functions h(z)=zn+c, h(z)=sin(zn)+c and h(z)=ezn+c, n≥2,c∈C. We observed some beautiful Julia sets, Mandelbrot sets and Multi-corns for n = 2, 3 and 4. We generalize the algorithms of the Julia set and Mandelbrot set to visualize some Julia sets, Mandelbrot sets and Multi-corns. Moreover, we calculate image generation time in seconds at different values of input parameters.

Multi-directional annular multi-wing chaotic system based on Julia fractals

Fractals and chaos are inextricably linked, and relevant study has been done on both. However, the studies that combine fractals and chaotic systems to produce multi- scroll or multi-wing chaotic attractors are all extend multi-scroll or multi-wing in the even direction. To this end, by improving the traditional fractal mapping, a chaotic system that can extend multiple wings in any direction of odd and even is designed and analyzed. The main point of this method is as follows: Firstly, a new family of even-symmetric segmented composite functions is proposed and applied to a family of generalized Lorenz systems, which can generate multi-wing chaotic systems in two directions and the number of wings in each direction can also be increased with the parameter N. Secondly, the improved Julia fractal is applied to two-direction multi-wing chaotic systems to generate a class of single-direction multi-wing chaotic systems. Finally, the combination of this class of single-direction multi-wing chaotic systems and higher-order Julia fractals is used to realize chaotic systems with extended multi-wing in odd and even arbitrary directions. Computer simulations confirm the method's viability. In addition, the maximum Lyapunov exponent, bifurcation diagram, Poincare diagram, and PE complexity chaos diagram are also analyzed as examples of multi-directional multi-wing Lü chaotic systems, demonstrating the system's superiority in odd directions over even directions and the existence and effective scalability of multi-directional multi-wing chaotic systems.

Estimations and Control of Julia Sets of the SIS Model Perturbed by Noise.

The estimations and control of Julia sets of the SIS(susceptible-infectious-susceptible) model under noise perturbation are studied. At first, a discrete SIS model is introduced, and the effects of additive and multiplicative noises on the fractal characteristics of the SIS model are discussed. Then, estimations of the Julia sets of the SIS model under additive and multiplicative noise perturbations are given, respectively. At last, the feedback control method is used to set appropriate controllers to realize control of the Julia set, and the influence of noise on the Julia set of the SIS model is reduced. The reason why this method is effective is also explained.

Hausdorff dimension of Julia sets of unicritical correspondences

We show that if β > 1 \beta >1 is a rational number and the Julia set J J of the holomorphic correspondence z β + c z^{\beta }+c is a locally eventually onto hyperbolic repeller, then the Hausdorff dimension of J J is bounded from above by the zero of the associated pressure function. As a consequence, we conclude that the Julia set of the correspondence has zero Lebesgue measure for parameters close to zero, whenever q 2 > p q^2>p and β = p / q \beta =p/q in lowest terms.

On Petrenko's deviations and the Julia limiting directions of solutions of complex differential equations

The Julia set J(f) of a transcendental meromorphic function f has chaotic behavior in the complex dynamics. The ray arg⁡z=θ∈[0,2π) is said to be a limiting direction of J(f) if there is an unbounded sequence {zn}⊆J(f) such that limn→∞⁡arg⁡zn=θ. In this paper, we mainly investigated the Julia limiting directions of entire solutions of complex differential equations, of which coefficients are associated with Petrenko's deviations. In fact, we proved the lower bounds of the sets of Julia limiting directions of these solutions have closed relations with Petrenko's deviations of the coefficients.

A Note on the Julia Sets of Entire Solutions to Delay Differential Equations

A Note on the Julia Sets of Entire Solutions to Delay Differential Equations

When do two rational functions have locally biholomorphic Julia sets?

In this article we address the following question, whose interest was recently renewed by problems arising in arithmetic dynamics: under which conditions does there exist a local biholomorphism between the Julia sets of two given one-dimensional rational maps? In particular we find criteria ensuring that such a local isomorphism is induced by an algebraic correspondence. This extends and unifies classical results due to Baker, Beardon, Eremenko, Levin, Przytycki and others. The proof involves entire curves and positive currents.

Topological dynamics of cosine maps

AbstractThe set of points that escape to infinity under iteration of a cosine map, that is, of the form $C_{a,\,b} \colon z \mapsto ae^z+be^{-z}$ for $a,\,b\in \mathbb{C}^\ast$ , consists of a collection of injective curves, called dynamic rays. If a critical value of $C_{a,\,b}$ escapes to infinity, then some of its dynamic rays overlap pairwise and split at critical points. We consider a large subclass of cosine maps with escaping critical values, including the map $z\mapsto \cosh(z)$ . We provide an explicit topological model for their dynamics on their Julia sets. We do so by first providing a model for the dynamics near infinity of any cosine map, and then modifying it to reflect the splitting of rays for functions of the subclass we study. As an application, we give an explicit combinatorial description of the overlap occurring between the dynamic rays of $z\mapsto \cosh(z)$ , and conclude that no two of its dynamic rays land together.

The Hausdorff dimension of Julia sets of meromorphic functions in the Speiser class

We show that for each din (0,2] there exists a meromorphic function f such that the inverse function of f has three singularities and the Julia set of f has Hausdorff dimension d.

On the lambda function and a quantification of Torhorst Theorem

For any compact K⊂Cˆ we define a map λK:Cˆ→N∪{∞}, called the lambda function of K. The lambda function of a continuum K has the property that λK(x)=0 for all x∈Cˆ if and only if K is locally connected. We establish several inequalities and a gluing lemma for the lambda functions. These inequalities reflect the relationship between the topology of K and the complexity that may be involved in the boundary of a component of Cˆ∖K. One of them, called the lambda inequality, generalizes and quantifies the Torhorst Theorem. We also find three sufficient conditions under each of which the lambda inequality becomes an equality. These results cover Whyburn's Theorem on the partial converse to the Torhorst Theorem. The gluing lemma for the lambda functions is of research interest in the study of point set topology and of polynomial Julia sets as well.