Transcendental Picard-Mann hybrid Julia and Mandelbrot sets

In this paper, the fascinating Julia and Mandelbrot sets for the complex-valued transcendental functions z → sin(z)+ c, (m ≥ 2) ∈ N have been obtained in Picard, Ishikawa and Noor orbits. The purpose of the paper is to visualize transcendental Julia and Mandelbrot sets in Picard-Mann hybrid orbit.


Introduction and Preliminaries
The science of fractal graphics has spread in almost all branches of science and engineering. Fractals are either natural or generated using a mathematical recipe [1]. Gaston Julia iterated complex polynomials and introduced Julia set as a classical example of fractals. Benoit B. Mandelbrot (1977) extended the work of Gaston Julia and created Mandelbrot set using computer graphics. Since then, the study of Julia and Mandelbrot sets has become an area of intense research. Julia and Mandelbrot sets have been studied for quadratic, cubic and higher degree polynomials in Picard orbit (see [7] and references therein). Generally, Mandelbrot and Julia sets lie in a complex plane. Rochon [17] studied a more generalized form of Mandelbrot sets in bi-complex plane (see also [12,19,20] and references therein). For a detailed study of Julia and Mandelbrot sets, one may refer to [2]. Transcendental function forms a rich dynamics for well-known Julia and Mandelbrot sets. The study of dynamical behaviour of the transcendental functions was initiated by Fatou [6]. For a transcendental function, points with unbounded orbits are not in Fatou sets but they must lay in Julia sets. For the entire transcendental functions, the point 1 is an essential singularity. Erneko (see [12] and references therein) studied that for every transcendental function, the set of escaping points is always non-empty. For the special cases of exponential functions, every escaping point may be connected to 1 along with unique curve running entirely through the escaping points. The process of generating fractals from z → sin(z m ) + c and self-squared function is similar [11]. In 2004, Rani and Kumar [16] introduced superior iterates (Mann iterates) in the study of fractal theory, and created superior Julia and Mandelbrot sets. Later on, in a series of papers Rani jointly with other researchers generated and analyzed superior Julia and superior Mandelbrot sets for quadratic, cubic and higher degree complex polynomials (see [13,15,16] and references therein). Per-turbed superior Julia and superior Mandelbrot sets due to noise have also been studied by researchers (see [14] and references therein). For a complete literature review of superior fractals until 2010, one may refer to Singh, Mishra and Sinkala [18]. In 2010, Chauhan, Rana and Negi [5] obtained relative Julia and Mandelbrot sets via Ishikawa iterates. Ashish, Rani and Chugh [4] visualized new Julia sets and Mandelbrot sets in Noor orbit [10]. Further, the cubic Julia sets and antifractals [3] were studied in the Noor orbit. Julia and Mandelbrot sets have been studied in Jungck orbit also (see [9] and references therein).
In the series of Julia and Mandelbrot sets via various iterates, Picard-Mann hybrid iterates was introduced by Rani and others (see [13,14] and references therein). She obtained not only hybrid Julia and Mandelbrot sets in Picard-Mann hybrid orbit but used iterates in chaos theory also. The purpose of this paper is to obtain transcendental Picard-Mann hybrid Julia and Mandelbrot sets. Definition 1.1 (Picard iterates). Let Z be a non-empty subset of real or complex numbers and f : Z → Z. We consider a sequence {z n } of iterates for an initial point z 0 ∈ Z such that O(f, z 0 ) = {z n : z n = f (z n−1 ), n ∈ N}.
The sequence constructed above is called as Picard orbit, abbreviated as O [16]. Definition 1.2 (Picard-Mann hybrid iterates). Let Z be a non-empty subset of real or complex numbers and f : Z → Z. We consider a sequence {z n } of iterates for an initial point z 0 ∈ Z such that (f, z 0 , β n ) = z n+1 : z n+1 = f y n ; y n = (1 − β n )z n + β n f z n , n = 0, 1, 2, . . . , where {β n } is the sequence in [0, 1]. We call P-MO as Picard-Mann hybrid orbit [8].
Notice that when β n = 0, then P-MO reduces to O. Also, at β n = 1, P-MO may be considered as double Picard iteration method. Khan [8] claimed that the P-MO converges faster than Ishikawa orbit. Rani [12]

Transcendental Picard-Mann hybrid Julia and Mandelbrot sets
The transcendental function of the form sin(z m ) + c, (m ≥ 2) ∈ N is iterated in P -M O for some initial choice z 0 as follows to obtain the sequence {z n }:

Generation of Transcendental P-MHJ and P-MHM sets
The escape criterion for the complex polynomial z m + c, (m ≥ 2) ∈ N to obtain Julia and Mandelbrot sets in superior orbit or (Mann orbit) is max c, ( 2 β ) 1 n−1 (see [16] and references therein). The same escape criterion has been used obtain Julia and Mandelbrot sets in P -M O [12]. The same escape criterion may be used in the iteration of the transcendental function z → sin(z m ) + c. We generate transcendental Julia and Mandelbrot sets for m ≥ 2 at β n = β.
3.1. Transcendental P-MHJ sets for sin(z m ) + c. Quadratic transcendental P -M HJ sets have been obtained at c = 0.275 for β = 1 and 0.1 in Fig. 1. The quadratic transcendental P -M HJ sets hold a perfect symmetry about both the axes, and have 4 sticks around them. It can be observed that the set for β = 0.1 is fattier than the set at β = 1.  Fig. 6 shows some of the cubic transcendental P -M HM sets at β = 0.8, 0.5, 0.3, 0.1. Some example of biquadratic transcendental P -M HM sets at β = 1, 0.8, 0.5, 0.1 are in Fig. 7. See two transcendental P -M HM sets for sin(z 5 ) + c at β = 1 and 0.5 in Fig. 8. Transcendental P -M HM sets for higher degree of polynomial are shown in Fig. 9. In all the transcendental P -M HM sets, it can be observed that the number of bulbs are (m − 1), and for higher values of m, the sets become circular saw.

Conclusion
In this paper, we have obtained Julia and Mandelbrot sets for the transcendental function sin(z m )+c, where (m ≥ 2) ∈ N, via Picard-Mann hybrid iterates. It is concluded that in Picard-Mann hybrid orbit, the number of bulbs in a transcendental Mandelbrot set is (m−1). Further, transcendental Julia and Mandelbrot sets both become circular saw for higher values of m.