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We show that the Hausdorff dimension of the set of points of bounded orbit in the Julia set of a meromorphic map with a simply connected direct tract and a certain restriction on the singular values is strictly greater than one. This result is obtained by proving new results related to Wiman–Valiron theory.

SIRS model is one of the most basic models in the dynamic warehouse model of infectious diseases, which describes the temporary immunity after cure. The discrete SIRS models with the Caputo deltas sense and the theories of fractional calculus and fractal theory provide a reasonable and sensible perspective of studying infectious disease phenomenon. After discussing the fixed point of the fractional order system, controllers of Julia sets are designed by utilizing fixed point, which are introduced as a whole and a part in the models. Then, two totally different coupled controllers are introduced to achieve the synchronization of Julia sets of the discrete fractional order systems with different parameters but with the same structure. And new proofs about the synchronization of Julia sets are given. The complexity and irregularity of Julia sets can be seen from the figures, and the correctness of the theoretical analysis is exhibited by the simulation results.

In this article, the fractal behavior in generalized alternated system is discussed. Take the classical fractal set as an example, the anti-synchronization of the Julia set between two different generalized alternated systems is discussed using the adaptive control method in this article. The anti-synchronization of the Julia set between different generalized alternated systems is also accomplished by synchronizing their iterative tracks. The simulations illustrate the effectiveness and correctness of this control method.

The purpose of this paper is to introduce the Mandelbrot and Julia sets by using Picard-Mann iteration procedure. Escape criteria is established which plays an important role to generate Mandelbrot and Julia sets. Also, numerous graphical pictures of these sets have been visualized and certain examples have been recognized. Presented results shows that fractal images generated by Picard-Mann iteration procedure are entirely different from those generated in Mann orbit.

The aim of this paper is to generate antifractals using fixed point iterative algorithms, i.e., we aim to generate anti Julia sets, tricorns and multicorns for the anti-polynomial z → z̅k +c of the complex polynomial zk +c, for k ≥ 2. A hybrid Picard-Mann iterative procedure used to establish escape criterion and explore the geometry of antifractals. A visualization of the antifractals for certain complex antipolynomials is presented and their graphical behavior is compared with antifratals generated via Mann iteration. We also explain the effects of parameters on shape of antifractals.

We prove that for a polynomial diffeomorphism of C^2, uniform hyperbolicity on the set of saddle periodic points implies that saddle points are dense in the Julia set. In particular f satisfies Smale's Axiom A on C^2 .

The aim of this paper is to introduce a topological model of fractals. Self similar fractals will be approached as inverse limit of finite one dimensional topological spaces with alpha continuous bonding functions. The second approach is to investigate topological graphs in terms nano topological spaces for Lellis Thivagar. From these approximations, the dynamics of Julia sets as a special type of self similar fractals will be studied and some physical properties of fractals through their nano topological graphs will be applied.

We prove that every quasisymmetric homeomorphism of a standard square Sierpinski carpet $$S_p$$ , $$p\ge 3$$ odd, is an isometry. This strengthens and completes earlier work by the authors (Bonk and Merenkov in Ann Math (2) 177:591–643, 2013, Theorem 1.2). We also show that a similar conclusion holds for quasisymmetries of the double of $$S_p$$ across the outer peripheral circle. Finally, as an application of the techniques developed in this paper, we prove that no standard square carpet $$S_p$$ is quasisymmetrically equivalent to the Julia set of a postcritically-finite rational map.

In this paper, we propose a dynamics-based approach to develop efficient matrix iterations for computing the matrix sign function. Considering an iterative method with unknown weight functions, our approach constructs a double sequence of weight functions so that for a quadratic polynomial with distinct roots, the Julia set of the method associated with each term of the sequence is the perpendicular bisector of the line segment from one root to another. Based on the sequence-generated methods, we introduce a double sequence of matrix iterations for computing the matrix sign function, and perform precise convergence and stability analyses of these iterations. We show that our matrix iterations contain the well-known globally convergent principal Pade family and its reciprocal as special cases. Based on the result of the numerical experiments, we found several of our methods that do not belong to the principal Pade family and its reciprocal, which are more efficient in computing the matrix sign function.

The article describes an algorithm for generating fractals using polar coordinates. The classic Julia and Mandelbrot polynomial iteration applied to a complex number is replaced by an iteration with separate functions for distance and angle. A polynomial function is used for an angle and a power function for a distance. Varying the functions parameters allows to create a wide range of attractive pictures. Distance values ​​are used for coloring fractal images.

This work reports on a study of the Mandelbrot set and Julia set for a generalization of the well-explored function η(z)=z2+λ. The generalization consists of composing with a fixed Möbius transformation at each iteration step. In particular, affine and inverse Möbius transformations are explored. This work offers a new way of visualizing the Mandelbrot and filled-in Julia sets. An interesting and unexpected appearance of hyperbolic triangles occurs in the structure of the Mandelbrot sets for the case of inverse Möbius transforms. Several lemmas and theorems associated with these types of fractal sets are presented.

In this paper, a novel escape-time algorithm is proposed to calculate the connectivity’s degree of Julia sets generated from polynomial maps. The proposed algorithm contains both quantitative analysis and visual display to measure the connectivity of Julia sets. For the quantitative part, a connectivity criterion method is designed by exploring the distribution rule of the connected regions, with an output value Co in the range of [0,1]. The smaller the Co value outputs, the better the connectivity is. For the visual part, we modify the classical escape-time algorithm by highlighting and separating the initial point of each connected area. Finally, the Julia set is drawn into different brightnesses according to different Co values. The darker the color, the better the connectivity of the Julia set. Numerical results are included to assess the efficiency of the algorithm.

We study the dynamics induced by homogeneous polynomials on Banach spaces. It is known that no homogeneous polynomial defined on a Banach space can have a dense orbit. We show a simple and natural example of a homogeneous polynomial with an orbit that is at the same time $\unicode[STIX]{x1D6FF}$-dense (the orbit meets every ball of radius $\unicode[STIX]{x1D6FF}$), weakly dense and such that $\unicode[STIX]{x1D6E4}\cdot \text{Orb}_{P}(x)$ is dense for every $\unicode[STIX]{x1D6E4}\subset \mathbb{C}$ that either is unbounded or has 0 as an accumulation point. Moreover, we generalize the construction to arbitrary infinite-dimensional separable Banach spaces. To prove this, we study Julia sets of homogeneous polynomials on Banach spaces.

We prove the existence of rational maps whose Julia sets are Sierpinski carpets having positive area. Such rational maps can be constructed such that they either contain a Cremer fixed point, a Siegel disk or are infinitely renormalizable. We also construct some Sierpinski carpet Julia sets with zero area but with Hausdorff dimension two. Moreover, for any given number $$s\in (1,2)$$, we prove the existence of Sierpinski carpet Julia sets having Hausdorff dimension exactly s.

This article gives a precise description of the Fatou sets and Julia sets of matrix-valued polynomials in M ( 2 , C ) in terms of the corresponding polynomials in C . A normality criterion given by Aladro–Krantz ( [1] ) plays a crucial role in proving this result. Further, we construct Green functions and Bottcher-type functions for these matrix-valued polynomials.

We prove that a nonempty, proper subset $S$ of the complex plane can be approximated in a strong sense by polynomial filled Julia sets if and only if $S$ is bounded and $\hat{\mathbb{C}} \setminus \textrm{int}(S)$ is connected. The proof that such a set is approximable by filled Julia sets is constructive and relies on Fekete polynomials. Illustrative examples are presented. We also prove an estimate for the rate of approximation in terms of geometric and potential theoretic quantities.

By investigating which level of universality composition operators $C_f$ can have, where the symbol $f$ is given by the restriction of a transcendental entire function to suitable parts of the Fatou set of $f$, this work combines the theory of dynamics of continuous linear operators on spaces of holomorphic functions with the theory of non-linear complex dynamics on the complex plane.

In a classical work of the 1950s, Lee and Yang proved that for fixed nonnegative temperature, the zeros of the partition functions of a ferromagnetic Ising model always lie on the unit circle in the complex magnetic field. Zeros of the partition function in the complex temperature were then considered by Fisher, when the magnetic field is set to zero. Limiting distributions of Lee–Yang and of Fisher zeros are physically important as they control phase transitions in the model. One can also consider the zeros of the partition function simultaneously in both complex magnetic field and complex temperature. They form an algebraic curve called the Lee–Yang–Fisher (LYF) zeros. In this paper, we continue studying their limiting distribution for the Diamond Hierarchical Lattice (DHL). In this case, it can be described in terms of the dynamics of an explicit rational function $$R$$ in two variables (the Migdal–Kadanoff renormalization transformation). We study properties of the Fatou and Julia sets of this transformation and then we prove that the LYF zeros are equidistributed with respect to a dynamical (1, 1)-current in the projective space. The free energy of the lattice gets interpreted as the pluripotential of this current. We also prove a more general equidistribution theorem which applies to rational mappings having indeterminate points, including the Migdal–Kadanoff renormalization transformation of various other hierarchical lattices.

not available GANIT J. Bangladesh Math. Soc.Vol. 38 (2018) 119-125

In today's world, complex patterns of the dynamical framework have astounding highlights of fractals and become a huge field of research because of their beauty and unpredictability of their structure. The purpose of this paper is to visualize anti-Julia sets, tricorns, and multicorns by means of the Noor iteration with s-convexity. Various patterns are displayed to investigate the geometry of antifractals for antipolynomial z̅k+1 + c of complex polynomial zk+1 + c, for k ≥ 1 in Noor orbit with s-convexity.

The aim of this paper is to modify the Jungck-S iterative scheme by adding the idea of $s$ -convexity. We define and analyze the modified Jungck-S orbit (MJSO) with $s$ -convex combination and derive the escape criterion for MJSO. Moreover, we establish the algorithms to visualize some Julia sets, Mandelbrot sets, and biomorphs in this orbit. In the biomorph generation algorithm, we did not fix the threshold radius of proposed orbit (i.e., MJSO) as fixed in literature earlier. We also discuss the graphical behavior of some complex polynomials in the generation of Julia sets, Mandelbrot sets, and biomorphs in MJSO.

In this paper, we investigate the dynamics of the following family of rational maps \begin{document}$ \begin{equation*} f_{\lambda}(z) = \frac{z^{2n} - \lambda^{3n+1}}{z^n(z^{2n} - \lambda^{n - 1})} \end{equation*} $\end{document} with one parameter \begin{document}$ \lambda \in \mathbb{C}^* - \{\lambda: \lambda^{2n + 2} = 1\} $\end{document} , where \begin{document}$ n\geq 2 $\end{document} . This family of rational maps is viewed as a singular perturbation of the bi-critical map \begin{document}$ P_{-n}(z) = z^{-n} $\end{document} if \begin{document}$ \lambda \neq 0 $\end{document} is small. It is proved that the Julia set \begin{document}$ J(f_\lambda) $\end{document} is either a quasicircle, a Cantor set of circles, a Sierpinski carpet or a degenerate Sierpinski carpet provided the free critical orbits of \begin{document}$ f_\lambda $\end{document} are attracted by the super-attracting cycle \begin{document}$ 0\leftrightarrow\infty $\end{document} . Furthermore, we prove that there exists suitable \begin{document}$ \lambda $\end{document} such that \begin{document}$ J(f_\lambda) $\end{document} is a Cantor set of circles but the dynamics of \begin{document}$ f_{\lambda} $\end{document} on \begin{document}$ J(f_{\lambda}) $\end{document} is not topologically conjugate to that of any known rational maps with only one or two free critical orbits (including McMullen maps and the generalized McMullen maps). The connectivity of \begin{document}$ J(f_{\lambda}) $\end{document} is also proved if the free critical orbits are not attracted by the cycle \begin{document}$ 0\leftrightarrow\infty $\end{document} . Finally we give an estimate of the Hausdorff dimension of the Julia set of \begin{document}$ f_\lambda $\end{document} in some special cases.

In today's world, fractals play an important role in many fields, e.g., image compression or encryption, biology, physics, and so on. One of the earliest studied fractal types was the Mandelbrot and Julia sets. These fractals have been generalized in many different ways. One of such generalizations is the use of various iteration processes from the fixed point theory. In this paper, we study the use of Jungck-CR iteration process, extended further by the use of s -convex combination. The Jungck-CR iteration process with s-convexity is an implicit three-step feedback iteration process. We prove new escape criteria for the generation of Mandelbrot and Julia sets through the proposed iteration process. Moreover, we present some graphical examples obtained by the use of escape time algorithm and the derived criteria.

This paper introduces an effective image encryption method based on fractals for colored images. Fractals are now widely used in image security due to its randomness property and infinite boundaries. Since images are traversed over the public channel in day-to-day life confidentiality, integrity and authenticity must be pledged for prosperous transmission. This method capsulizes a larger key space and complex behavior to accomplish a productive confusion and diffusion techniques in a fruitful way. Julia set and Mandelbrot set are constructed and active confusion is exerted till the image pixels are altogether disordered and ensuing diffusion with XOR operation. Experimental results show the ability of the proposed system with respect to various statistical measures.

We prove that for any infinite collection of quadratic Julia sets, there exists a transcendental entire function whose Julia set contains quasiconformal copies of the given quadratic Julia sets. In order to prove the result, we construct a quasiregular map with required dynamics and employ the quasiconformal surgery to obtain the desired transcendental entire function. In addition, the transcendental entire function has order zero.

In today's world, fractals play an important role in many fields, e.g., image compression or encryption, biology, physics, and so on. One of the earliest studied fractal types was the Mandelbrot and Julia sets. These fractals have been generalized in many different ways. One of such generalizations is the use of various iteration processes from the fixed point theory. In this paper, we study the use of Jungck-CR iteration process, extended further by the use of s -convex combination. The Jungck-CR iteration process with s-convexity is an implicit three-step feedback iteration process. We prove new escape criteria for the generation of Mandelbrot and Julia sets through the proposed iteration process. Moreover, we present some graphical examples obtained by the use of escape time algorithm and the derived criteria.

In this paper, the two-parameter family F b ≡ { f λ , μ ( z ) = λ b z + μ b − z for z ∈ C : λ ≥ μ > 0 } of transcendental entire functions is considered. By investigating on the existence and nature of the fixed points of f λ , μ , the dynamics of functions f λ , μ ∈ F b is studied. It is shown that the Julia set of f λ , μ is nowhere dense and not locally connected whenever parameters λ and μ satisfy 1 + 4 λ μ ( ln ⁡ b ) 2 ≤ ln ⁡ ( 1 + 1 + 4 λ μ ( ln ⁡ b ) 2 2 λ ln ⁡ b ) and the Julia set is the whole of extended complex plane for 1 + 4 λ μ ( ln ⁡ b ) 2 > ln ⁡ ( 1 + 1 + 4 λ μ ( ln ⁡ b ) 2 2 λ ln ⁡ b ) .

In this paper, we study the generalized Jungck Mann orbit (GJMO) and prove the converse theorem of results. We develop algorithms for the generation of filled Julia sets and their boundaries in the GJMO. In simple Julia set (i.e., boundary of filled Julia set), we show the internal structure of Julia set and establish the correspondence between Julia points via dark blue lines in graphs. Moreover, we demonstrate the pictorial effects of filled Julia set and its boundary and graphically present the image change with the change of complex parameter c in the GJMO.

In this paper, we study on the local and semi-local convergence and dynamic of an iterative optimal two-step Steffensen's method. We study local convergence by using Lipschitz conditions and continuity of differentiable function F. We also obtain semi-local convergence by using majorizing sequences. By this approach, we can obtain the convergence of the method by minimum of auxiliary sequences and few initial conditions. Moreover, we analyze the stability of the method via Julia sets and parameter planes that is a novelty for Steffensen-type methods. We extend this method for obtaining the sign function of a matrix. Numerical examples are given to confirm our theoretical results.

We study the quasisymmetric geometry of the Julia sets of McMullen maps $f_\lambda(z)=z^m+\lambda/z^\ell$, where $\ell$, $m\geq 2$ are integers satisfying $1/\ell+1/m, Comment: 18 pages, 3 figures

We consider complex dynamics of a critically finite holomorphic map from P^k to P^k, which has symmetries associated with the symmetric group S_{k+2} acting on P^k, for each k \ge 1. The Fatou set of each map of this family consists of attractive basins of superattracting points. Each map of this family satisfies Axiom A., 12 pages

We discuss computability and computational complexity of conformal mappings and their boundary extensions. As applications, we review the state of the art regarding computability and complexity of Julia sets, their invariant measures, and external ray impressions.