Generation of Antifractals via Hybrid Picard-Mann Iteration

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 <inline-formula> <tex-math notation="LaTeX">$z\rightarrow \overline {z}^{k}+c$ </tex-math></inline-formula> of the complex polynomial <inline-formula> <tex-math notation="LaTeX">$z^{k}+c$ </tex-math></inline-formula>, for <inline-formula> <tex-math notation="LaTeX">$k\geq 2$ </tex-math></inline-formula>. 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.


I. INTRODUCTION
The branch of mathematics known as fixed point theory is a powerful tool to study natural phenomena that are usually nonlinear and has many applications in almost every area of research including biology, computer sciences, image generations, complex graphics, etc [1], [2]. Fractal and antifractal images can be generated by using iterative algorithms for finding fixed points of particular mappings [1]. Very first time, Julia in the year 1918 [3] studied graphics of the following complex function (1) and lead the foundation of Fractal geometry: where z ∈ C and c is a fixed complex number. Later this complex set named as Julia set, which is the classical example of Fractal. Definition 1 ( [4]): Let us consider a complex valued polynomial f : C −→ C with degree ≥ 2. Then the set of complex numbers F f whose orbits does not converge to infinity is known as filled Julia set of f . Mathematically the filled Julia set is: The associate editor coordinating the review of this manuscript and approving it for publication was Haiyong Zheng .
The Julia set is denoted by J (f c ) and is the boundary of filled Julia set F f and the complement of Julia set is called Fatou set [5].
The word Fractal was introduced by Mandelbrot and introduced Mandelbrot set [6]. He took c as a complex variable in (1) and utilized the idea of Gostan Julia to observe the behaviour of Julia sets that are connected sets [6]. Fractals don't have a conventional definition, anyway they are distinguished through their irregular structure that can't be found in Euclidean geometry.
Definition 2 ( [7]): For the complex polynomial Q c (z) = z 2 + c, the Mandelbrot set is the collection of all c for which the orbit of 0 is bounded and is denoted by M . Mathematical, we can write The 0 has been taken as initial point because it is the only critical point for Q c . The Julia sets and Mandelbrot sets are the fundamental sets in fractal geometry and got special place in fractal art [8]. These sets are most complex sets till the date and can not be seen without computer [9]. There are many ways to generate Fractals and fixed point theory is one of them in which iterative algorithms are used to find fixed point of complex polynomials [10], [11]. Different researcher used different iterative algorithms to generate Fractals, for example, see [10]- [14]. Generation of Fractals is an aesthetic endeavor, a sooting diversion or only a numerical model and fractal art is totally different from other computer activities [15].
In [16], Crowe et al., first time studied the connected locus and dynamics of the quadratic antiholomorphic polynomials z 2 + c and this connected locus is called ''tricorn'' by Milnor in [17].
Tricorn, being a complex subset of complex numbers, plays an important role in quadratic and cubic polynomials and very much similar with Mandelbrot set. The nature of tricorn is three-cornered and the style of its self similarity exactly same as that of Mandelbrot set. In the year 1983, Crowe with his coauthors [16] studied the relationship between tricorn and Mandelbrot set, named it "Mandelbar sets" and proved that the features bifurcations of tricorn is along arcs rather than at points. Later in [18], Winters proved that the boundary of tricorn contains a smooth arc and in [19], Lau and Schleicher investigated symmetries of tricorn and multicorn. In continuation of these works, the tricorn set is generalized and multicorn set has been introduced [20]. In fact multicorns is the generalization of tericorns or multicorn set is a tricorn set of higher order. Moreover they proved that the Julia set of mapping A c (z) = z k + c for k ≥ 2 is either connected or disconnected. Now we define the multicorn sets.
Definition 3 ( [7]): Let us consider the mapping A c (z) = z k + c for k ≥ 2. The multicorn det is denoted by M * c which is the collection of all complex numbers c for which the orbit of 0 is bounded. Mathematical, we can write as: does not tend to ∞}, where A n c is the nth iterate of the function A c (z). In the above definition, if we take k = 2, multicorn set become tricorn set. In literature, one can find many ways to generate antifractals and one way is to use iterative algorithms for finding fixed point of mappings. The aim of this paper is to generate antifractals by using Picard-Mann iterative algorithm. For a complex valued mapping T : C → C, the Picard orbit (PO) is defined as [21]: where n ≥ 0. The Mann orbit (MO) is defined as [22]: where θ ∈ (0, 1]. This is one step iterative algorithm and antifractals via MO process were studied by Rani in [23], [24]. Many authors studied the dynamical behaviour of antiholomorphic complex mappings and various fixed point iterative algorithms were utilized [25]- [27].
Kwun et al. [29] and Chen et al. [30] explored tricorns and multicorns via Noor iteration with s-convexity and modified S-iteration respectively. Kang et al. [31] visualized tricorns and multicorns by using following S-iteration process [32]: where θ and δ ∈ (0, 1]. Recently, Li et al. [33] introduced the antifractals by utilizing CR-iteration with s-convexity. It is seen that for each iterative process the behaviour and dynamics of the tricorn and multicorns differ. Now we define the Picard-Mann orbit (PMO) [34].
Definition 4: Consider the mapping f : C → C, where C is a subset of complex plane. Then the PMO is the following sequence of iterates with initial guess where 0 < θ ≤ 1.
The PMO is a function of three variables (f , x 0 , θ) which can be written as PMO(f , x 0 , θ n ). Khan in [34] proved that the PMO converges faster than PO, MO and Ishikawa iteration process. Now, for polynomial Q c (z n ) of any degree, we have following PM scheme: where θ ∈ (0, 1]. In this paper we consider the iteration process of unicritical antiholomorphic polynomials f c (z) = z k + c, for any degree k ≥ 2 and c ∈ C via PMO and attained diverse graphical patterns of tricorn, multicorn and anti-Julia sets that are totally different from those generated by using MO.

II. ESCAPE CRITERION FOR ANTIFRACTALS
Many techniques are used to generate and analyze fractals, such as iterated function systems, random fractals, escape time fractals etc. The escape time algorithm depends on the maximum number of iterations necessary to measure if the orbit sequence tends to infinity or not. This algorithm gives a useful mechanism applied to demonstrate some features of dynamic system under iterative procedure. Usually, the escape criterion for fractal sets is: then Q n c (z) → ∞ as n → ∞ Here max{|c| , 2} is called escape radius threshold which may be different for different iterative processes. Now we obtain a general escape criterion that is necessary to construct VOLUME 8, 2020 the antifractals for antipolynomials of the form Q c (z) = z k + c, k ≥ 2 in PMO.
Now, we can get following escape criterion immediatly.

IV. CONCLUSION
In the present paper we introduced and visualized antifractals in hybrid PMO and compare it with antifractals generated in MO. Escape criterion for antifractals has been established corresponding to PMO and visualized the pattern of tricorns, multicorns and anti-Julia sets. In the dynamics of antipolynomials z → z k + c for k ≥ 2, we obtained various patterns of tricorns and multicorns for the same value of k and choosing different values of θ in PMO. We observed that the number of branches attached to the main body of the tricorns and multicorns are k + 1 and many branches have subbranches. We also found that the symmetry of multicorn is about both x-axis and y-axis when k is odd but for k is even the symmetry is preserved only along x-axis. A few examples of connected anti-Julia sets have been presented for quadratic and cubic functions. Interesting changes are seen in the figures for different values of parameter θ. Tricorn prints are utilized commercially such as tricorn mugs and tricorn dresses like tricorn T-shirts. We think that results of this paper will impress those who are interesting in creating automatically aesthetic patterns.
ABDUL AZIZ SHAHID received the M.Phil. degree in mathematics from Lahore Leads University, Lahore, Pakistan, in 2014. He is currently a Ph.D. Research Scholar with The University of Lahore, Lahore. He has published over 15 research articles in different international journals. His research interests include fixed point theory and fractal generation via different fixed point iterative schemes.

MINGYE WANG was born in China, in 1992.
He is currently pursuing the Ph.D. degree with Beihang University, with major of pattern recognition and intelligent systems. His main research interests include deep learning and computer vision. VOLUME 8, 2020