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In this paper, we visualise and analyse the dynamics of fractals (Julia and Mandelbrot sets) for complex polynomials of the form T (z) = z n + m z + r , where n ≥ 2 and m , r ∈ C , by adopting the viscosity approximation type iteration process which is most widely used iterative method for finding fixed points of non-linear operators. We establish a convergence condition in the form of escape criterion which allows to adapt the escape-time algorithm to the considered iteration scheme. We also present some graphical examples of the Mandelbrot and Julia fractals showing the dependency of Julia and Mandelbrot sets on complex polynomials, contraction mappings, and iteration parameters. Moreover, we propose two numerical measures that allow the study of the dependency of the set shape change on the values of the iteration parameters. Using these two measures, we show that the dependency for the considered iteration method is non-linear. [ABSTRACT FROM AUTHOR]

This paper presents a detailed study of the dynamics of transcendental entire maps which are symmetric with respect to rays of the form Λ (a , θ) = { t e i θ + a | t ≥ 0 } , with a ∈ C and θ ∈ [ 0 , 2 π) . The symmetric families considered in the paper contain many well-known entire maps. Let f be such a symmetric entire map. Then (i) there exists a point z 0 = t 0 e i θ + a , (t 0 ≥ 0) , such that for every point z on the ray Λ 0 (a , θ) = { t e i θ + a | t ≥ t 0 } , and any point z n in its forward orbit, any connected compact set containing z and z n must intersect the Julia set of f; (ii) The map f either has a multiply-connected wandering domain, or its Julia set contains the ray Λ 0 (a , θ) ; (iii) If t 0 > 0 , then the Fatou set of f has an invariant immediate attractive basin, or an invariant Leau domain containing a specific disc { z : | z - a | < t 0 } , with z 0 being a repelling fixed point and f ′ (z 0) > 1 , or a rationally indifferent fixed point and f ′ (z 0) = 1 , respectively; As applications, new dynamical properties of parametric families of the most well-known maps, including exponential maps of the forms a e z n + b e - z n , hyperbolic maps of the forms λ sinh z , and cosh z + λ , trigonometric maps of the forms λ sin z , and λ cos z , etc, are obtained. Moreover, a new map with wandering domains is constructed. [ABSTRACT FROM AUTHOR]

In this paper, we present an application of the viscosity approximation type iterative method introduced by Nandal et al. (Iteration Process for Fixed Point Problems and Zeros of Maximal Monotone Operators, Symmetry, 2019) to visualize and analyse the Julia and Mandelbrot sets for a complex polynomial of the type T (z) = z n + p z + r , where p , r ∈ C , and n ≥ 2 . This iterative method has many applications in solving various fixed point problems. We derive an escape criterion to visualize Julia and Mandelbrot sets via the proposed viscosity approximation type method. Moreover, we present several graphical examples of the fractals generated with the proposed iteration method. [ABSTRACT FROM AUTHOR]

In this paper, a novel reversible blind dual-color image watermarking algorithm is proposed by using singular value decomposition (SVD), Hachimoji Deoxyribonucleic Acid (HDNA) biogenetic encryption, coupled map lattice-based Tent–Sine system (TSS-CML) and mathematical Julia set. For watermark embedding, the watermark image is firstly encrypted using 8-bases HDNA sequences, TSS-CML and Julia set image. Then the encrypted HDNA watermark is obtained. Next, decompose the host image into equal non-overlapping blocks and utilize SVD on the randomly selected blocks. Further, embed the HDNA watermark through modifying the relation between the elements in the first column of the matrix U or V. The watermarked image can be eventually attained by carrying out the inverse SVD on all selected blocks. Also a reliable extraction algorithm is designed to recover the watermark from the possibly attacked watermarked images without resorting to the original image. Experimental and analysis results demonstrate that the proposed watermarking scheme has not only an excellent imperceptibility but a strong robustness against the common image processing attacks, geometric attacks and some composite attacks. In addition, the running time taken for hiding and exacting is about 1 s, which is suitable for real-time network transmission and application. In conclusion, the proposed method outperforms the related dual images watermarking algorithms in terms of time performance, extraction effect and robustness. [ABSTRACT FROM AUTHOR]

The authors study a family of transcendental entire functions which lie outside the Eremenko-Lyubich class in general and are of infinity growth order. Most importantly, the authors show that the intersection of Julia set and escaping set of these entire functions has full Hausdor. dimension. As a by-product of the result, the authors also obtain the Hausdor. measure of their escaping set is infinity. [ABSTRACT FROM AUTHOR]

We prove that Collet-Eckmann rational maps have poly-time computable Julia sets. As a consequence, almost all real quadratic Julia sets are poly-time. [ABSTRACT FROM AUTHOR]

In this paper, we investigate some analytic properties for a class of holomorphic matrixvalued functions. In particular, we give a Picard type theorem which depicts the characterization of Picard omitting value in these functions. We also study the relation between asymptotic values and Picard omitting values, and the relation between periodic orbits of the canonical extension on ℂ and Julia set of one dimensional complex dynamic system. [ABSTRACT FROM AUTHOR]

In this note, we give fixed point results in fractal generation (Julia sets and Mandelbrot sets) by using Noor iteration scheme with s-convexity. Researchers have already presented fixed point results in Mann and Ishikawa orbits that are examples of one-step and two-step feedback processes respectively. In this paper we present fixed point results in Noor orbit, which is a three-step iterative procedure. [ABSTRACT FROM AUTHOR]

In this paper, the dynamics of the functions $$ f_{\mu }(z)=z\exp (z+\mu ) $$ with the real parameter is studied. We say that a real parameter $$ \mu $$ belongs to the set $$ B_n $$ for a positive integer n if $$ f_\mu $$ has an attracting cycle of n-order. We prove that the Fatou set $$ F(f_\mu ) $$ is a completely invariant attracting basin for every parameter $$ \mu <0 $$ . Further, regarding the set $$ B_n $$ for $$ n>1 $$ , we prove the following results: (1) There exists $$ \mu _{*}\ne +\infty $$ such that $$ B_2 = (2,\mu _{*}) $$ . (2) For every positive integer $$ n>2 $$ , the set $$ B_n $$ is non-empty. (3) For every prime number $$ p>3 $$ , the set $$ B_p $$ has at least two components. [ABSTRACT FROM AUTHOR]

We study the interplay between infinitesimal deformations of conformal mappings, quasiconformal distortion estimates and integral means spectra. By the work of McMullen, the second derivative of the Hausdorff dimension of the boundary of the image domain is naturally related to asymptotic variance of the Beurling transform. In view of a theorem of Smirnov which states that the dimension of a k-quasicircle is at most $${1 + k^2}$$ , it is natural to expect that the maximum asymptotic variance $${\Sigma^2 = 1}$$ . In this paper, we prove $${0.87913 \le \Sigma^2 \le 1}$$ . For the lower bound, we give examples of polynomial Julia sets which are k-quasicircles with dimensions $${1 + 0.87913 \, k^2}$$ for k small, thereby showing that $${\Sigma^2 \ge 0.87913}$$ . The key ingredient in this construction is a good estimate for the distortion k, which is better than the one given by a straightforward use of the $${\lambda}$$ -lemma in the appropriate parameter space. Finally, we develop a new fractal approximation scheme for evaluating $${\Sigma^2}$$ in terms of nearly circular polynomial Julia sets. [ABSTRACT FROM AUTHOR]

The aim of this paper is to establish some fixed point results in the generation of Julia and Mandelbrot sets by using Jungck Mann and Jungck Ishikawa iterations with s-convexity. [ABSTRACT FROM AUTHOR]

This paper presents an improvement of the sixth-order method of Chun and Neta as a class of three-step iterations with optimal efficiency index, in the sense of Kung-Traub conjecture. Each member of the presented class reaches the highest possible order using four functional evaluations. Error analysis will be studied and numerical examples are also made to support the theoretical results. We then present results which describe the dynamics of the presented optimal methods for complex polynomials. The basins of attraction of the existing optimal methods and our methods are presented and compared to illustrate their performances. [ABSTRACT FROM AUTHOR]

We study the dynamics of commuting rational maps with coefficients in ℂ. By lifting the dynamics from ℙ(ℂ) to Berkovich projective space ℙ, we prove that two nonlinear commuting maps have the same Berkovich Julia set and the same canonical measure. As a consequence, two nonlinear commuting maps with coefficient in ℂ have the same classical Julia set. We also prove that they have the same pre-periodic Berkovich Fatou components. [ABSTRACT FROM AUTHOR]

In this paper, we consider the dynamics of the map z ↦ exp( z)/ z on the punctured plane ℂ* = ℂ\{0}. We show that for almost every point z ∈ ℂ*, the ω-limit set of z is equal to {0,∞}. In particular, the map is not recurrent. [ABSTRACT FROM AUTHOR]

Dynamics of transcendental entire maps with coefficients in an algebraically closed and complete non-Archimedean field $$K$$ is studied. It is shown, among other things, that periodic repelling points are dense in Berkovich Julia set, the forward union of any open set which intersects the Julia set is the whole Berkovich affine line, and a multi-connected Fatou component is wandering, in which all points go to infinity under iteration. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the rational maps with the positive real parameter λ. It is shown that the immediately attracting basin B of ∞ for F is always a Jordan domain if the Julia set of F is not a Cantor set. Furthermore, B is a quasidisk if there is no parabolic fixed point on the boundary of B. It is also shown that if the Julia set of F is connected, then it is locally connected and all Fatou components are Jordan domains. Finally, a complete description to the problem when the Julia set is a Sierpiński curve is given. [ABSTRACT FROM AUTHOR]

Given a complex polynomial p( z) with at least three distinct roots, we first prove that no rational iteration function exists where the basin of attraction of a root coincides with its Voronoi cell. In spite of this negative result, we prove that the Voronoi diagram of the roots can be well approximated through a high order sequence of iteration functions, the Basic Family, B( z), m≥2. Let θ be a simple root of p( z), V( θ) its Voronoi cell, and A( θ) its basin of attraction with respect to B( z). We prove that given any closed subset C of V( θ), including any homothetic copy of V( θ), there exists m such that for all m≥ m, C is also a subset of A( θ). This implies that when all roots of p( z) are simple, the basins of attraction of B( z) uniformly approximate the Voronoi diagram of the roots to within any prescribed tolerance. Equivalently, the Julia set of B( z), and hence the chaotic behavior of its iterations, will uniformly lie to within prescribed strip neighborhood of the boundary of the Voronoi diagram. In a sense, this is the strongest property a rational iteration function can exhibit for polynomials. Next, we use the results to define and prove an infinite layering within each Voronoi cell of a given set of points, whether known implicitly as roots of a polynomial equation, or explicitly via their coordinates. We discuss potential application of our layering in computational geometry. [ABSTRACT FROM AUTHOR]

In 1992, Branner and Hubbard raised a conjecture that the Julia set of a polynomial is a Cantor set if and only if each critical component of its filled-in Julia set is not periodic. This conjecture was solved recently. In this paper, we generalize this result to a class of rational functions. [ABSTRACT FROM AUTHOR]

It is proved that the rational maps in the family { z → z+ λ/z: λ ∈ ℂ\{0}} for integers m, d ≥ 2 have no Herman rings. [ABSTRACT FROM AUTHOR]

In this paper, we study the dynamics of the family of rational maps f λ( z) = z n + λ/ z m, n ≥ 2, m ≥ 1, λ ∈ ℂ. We construct an example of buried Sierpinski curve Julia set in this family. We also give an estimate of the location of bifurcation locus of f λ. [ABSTRACT FROM AUTHOR]

We show that if a polynomial filled Julia set has empty interior, then it is computable. [ABSTRACT FROM AUTHOR]

The dynamics of functions fλ(z)=λezz+1forz∈C,λ>0 is studied showing that there exists λ∗>0 such that the Julia set of fλ is disconnected for 0<λ<λ∗ whereas it is the whole Riemann sphere for λ>λ∗. Further, for 0<λ<λ∗, the Julia set is a disjoint union of two topologically and dynamically distinct completely invariant subsets, one of which is totally disconnected. The union of the escaping set and the backward orbit of ∞ is shown to be disconnected for 0<λ<λ∗ whereas it is connected for λ>λ∗. For complex λ, it is proved that either all multiply connected Fatou components ultimately land on an attracting or parabolic domain containing the omitted value of the function or the Julia set is connected. In the latter case, the Fatou set can be empty or consists of Siegel disks. All these possibilities are shown to occur for suitable parameters. Meromorphic functions En(z)=ez(1+z+z22!+⋯+znn!)-1, which we call exponential-like, are studied as a generalization of f(z)=ezz+1 which is nothing but E1(z). This name is justified by showing that En has an omitted value 0 and there are no other finite singular value. In fact, it is shown that there is only one singularity over 0 as well as over ∞ and both are direct. Non-existence of Herman rings are proved for λEn. [ABSTRACT FROM AUTHOR]