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We show that for each d ∈ (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. [ABSTRACT FROM AUTHOR]

Let g (z) = ∫ 0 z p (t) exp (q (t)) d t + c where p, q are polynomials and c ∈ C , and let f be the function from Newton's method for g. We show that under suitable assumptions on the zeros of g ′ ′ the Julia set of f has Lebesgue measure zero. Together with a theorem by Bergweiler, our result implies that f n (z) converges to zeros of g almost everywhere in C if this is the case for each zero of g ′ ′ that is not a zero of g or g ′ . In order to prove our result, we establish general conditions ensuring that Julia sets have Lebesgue measure zero. [ABSTRACT FROM AUTHOR]

We study the dynamics of a generic endomorphism f of an Oka–Stein manifold X. Such manifolds include all connected linear algebraic groups and, more generally, all Stein homogeneous spaces of complex Lie groups. We give several descriptions of the Fatou set and the Julia set of f. In particular, we show that the Julia set is the derived set of the set of attracting periodic points of f and that it is also the closure of the set of repelling periodic points of f. Among other results, we prove that f is chaotic on the Julia set and that every periodic point of f is hyperbolic. We also give an explicit description of the "Conley decomposition" of X induced by f into chain-recurrence classes and basins of attractors. For X = C , we prove that every Fatou component is a disc and that every point in the Fatou set is attracted to an attracting cycle or lies in a dynamically bounded wandering domain (whether such domains exist is an open question). [ABSTRACT FROM AUTHOR]

In recent years, researchers have studied the use of different iteration processes from fixed point theory for the generation of complex fractals. Examples are the Mann, the Ishikawa, the Noor, the Jungck-Mann and the Jungck-Ishikawa iterations. In this paper, we present a generalisation of complex fractals, namely Mandelbrot, Julia and multicorn sets, using the Jungck-CR implicit iteration scheme. This type of iteration does not reduce to any of the other iterations previously used in the study of complex fractals; thus, this generalisation gives rise to new fractal forms. We prove a new escape criterion for a polynomial of the following form zm − az + c, where a, c ∈ ℂ, and present some graphical examples of the obtained complex fractals. [ABSTRACT FROM AUTHOR]

We prove the existence of rational maps whose Julia sets are Sierpiński 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 Sierpiński carpet Julia sets with zero area but with Hausdorff dimension two. Moreover, for any given number s ∈ (1 , 2) , we prove the existence of Sierpiński carpet Julia sets having Hausdorff dimension exactly s. [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 the Julia set of the family λ exp( z)/ z with real parameters λ. We look for what values of real parameters λ such that the Julia set of λ exp( z)/ z does not coincide with the whole plane, and thus gives a complete classification for real parameters, which is similar to Jang's result of a family of transcendental entire functions. Moreover, We also discuss the shape and size of Fatou sets and Julia sets of λ exp( z)/ z with real parameters λ when the Julia sets are not the whole plane. [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]

Devaney and Krych showed that for $$0<\lambda <1/e$$ the Julia set of $$\lambda e^z$$ consists of pairwise disjoint curves, called hairs, which connect finite points, called the endpoints of the hairs, with $$\infty $$ . McMullen showed that the Julia set has Hausdorff dimension 2 and Karpińska showed that the set of hairs without endpoints has Hausdorff dimension 1. We study for which gauge functions the Hausdorff measure of the set of hairs without endpoints is finite. [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]

complex exponential map is said to be Misiurewicz if the forward trajectory of the asymptotic value 0 lies in the Julia set and is bounded. We prove that the set of Misiurewicz parameters in the exponential family $${\lambda\exp(z),\lambda\in\mathbb{C}{\setminus}\{0\}}$$, has the Lebesgue measure zero. [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]

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

In order to further enrich the form of 3D Mandelbrot and Julia sets, this paper first presents two methods of generating 3D fractal sets by utilizing discrete modifications of the standard quaternion algebra and analyzes the limitations in them. To overcome these limitations, a novel method for generating 3D fractal sets based on a 3D number system named ternary algebra is proposed. Both theoretical analyses and experimental results demonstrate that the ternary-algebra-based method is superior to any one of the quad-algebra-based methods, including the first two methods presented in this paper, because it is more intuitive, less time consuming and can completely control the geometric structure of the resulting sets. A ray-casting algorithm based on period checking is developed with the goal of obtaining high-quality fractal images and is used to render all the fractal sets generated in our experiments. It is hoped that the investigations conducted in this paper would result in new perspectives for the generalization of 3D Mandelbrot and Julia sets and for the generation of other deterministic 3D fractals as well. [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]