# Tricorns and Multicorns ofS-Iteration Scheme

Authors: Shin Min Kang; Arif Rafiq; Abdul Latif; Abdul Aziz Shahid; Young Chel Kwun
Journal: Journal of Function Spaces
 (2015)

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## Abstract

Complex graphics of dynamical system have striking features of fractals and become a wide area of research due to their beauty and complexity of their nature. The aim of this paper is to study dynamics of relative superior tricorns and multicorns usingS-iteration schemes. Several examples are presented to explore the geometry of relative superior tricorns and multicorns for antipolynomial z→z¯n+c of complex polynomial zn+c for n≥2.


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## Body

## 1. Introduction

In 1918, French mathematician Julia [1] investigated the iteration process of complex function and attained a Julia set. On the other hand, the object Mandelbrot set was given by Mandelbrot [2]. In 1989, Crowe et al. [3] considered formal analogy with Mandelbrot set and named it “Mandelbrot sets” and showed its feature bifurcations along arcs rather than at points. The word “tricorn” was coined by Milnor for the connectedness locus for antiholomorphic polynomials z¯2+c, which plays an intermediate role between quadratic and cubic polynomials. Tricorn has many similarities with the Mandelbrot set due to a compact subset of C.Milnor [4] found it in a real slice of the cubic connectedness locus. Winters [5] explained that boundary of the tricorn contains a smooth arc. The symmetries of tricorns and multicorns have been analyzed by Lau and Schleicher [6]. Nakane and Schleicher [7] presented various properties of tricorns and multicorns and quoted that the multicorns are the generalized tricorns or the tricorns of higher order. They also investigate that the Julia set of a polynomial of the form pc(z)=z¯n+c for n≥2 is either connected or disconnected. The set of parameters c such that the Julia set of pc is connected is called the multicorn. Tricorn prints, such as tricorn mugs and tricorn shirts, are being used for commercial purpose.The dynamics of antiholomorphic complex polynomialsz→z¯n+c for n≥2 was studied and explored using Mann iteration by Rani [8, 9]. Relative superior tricorns and relative superior multicorns were introduced using Ishikawa iterates by Chauhan et al. [10]. Also, they studied their corresponding relative superior Julia sets.In this paper we introduce and visualize a new class of relative superior tricorns and relative superior multicorns usingS-iteration scheme.This paper is organized as follows. In Section2, some basic definitions are presented. Section 3 contains the escape criterion for relative superior tricorns and multicorns. In Section 4, we generate relative superior tricorns and multicorns of S-iteration scheme for quadratic, cubic, and biquadratic functions. At last, paper has been concluded in Section 5.


## 2. Preliminaries

Definition 1 (see [11], multicorn).
The multicornAc for the quadratic function Ac(z)=z¯n+c is defined as the collection of all c∈C for which the orbit of the point 0 is bounded; that is,
(1)Ac=c∈C:Acn0doesnottendto∞,
where C is a complex space and Acn is the nth iterate of the function Ac(z). An equivalent formulation is that the connectedness of loci for higher degree antiholomorphic polynomials Ac(z)=z¯n+c is called multicorns.Notice that, atn=2, multicorns reduce to tricorn. Moreover, the tricorn naturally lives in the real slice d=c¯ in the two-dimensional parameter space of maps z→(z2+d)2+c. They have (n+1)-fold rotational symmetries. Also, by dividing these symmetries, the resulting multicorns are called unicorns [7].Definition 2 (see [12], S-iteration scheme for relative superior tricorns and multicorns).
LetX be a subset of real or complex numbers and f:X→X. For x0∈X, one constructs the sequences {xn} and {yn} in X in the following manner:
(2)xn=1-sn-1fxn-1+sn-1fyn-1,yn=1-sn′xn+sn′fxn,
where 0<sn<1, 0<sn′<1, and sn, sn′ both are convergent to nonzero number.
The sequences{xn} and {yn} constructed above are called S-iteration scheme sequences of iterations or relative superior sequences of iterates. We denote it by RSO(x0,sn,sn′,t).Definition 3 (Mandelbrot set).
The Mandelbrot setM consists of all parameters c for which the filled Julia set of Qc is connected; that is
(3)M={c∈C:KQcisconnected}.
In fact, M contains an enormous amount of information about the structure of Julia sets. The Mandelbrot set M for the quadratic Qc(z)=z2+c is defined as the collection of all c∈C for which the orbit of the point 0 is bounded; that is,
(4)M=c∈C:Qcn0n=0,1,2,…isbounded.
We choose the initial point 0, as 0 is the only critical point of Qc [11].


## 3. Escape Criterion for Relative Superior Tricorns and Multicorns

The escape criterion plays an important role in the generation and analysis of relative superior tricorns and multicorns. We now obtain a general escape criterion for polynomials of the formGc(z)=zn+c.Theorem 4.
For general functionGc(z)=zn+c, n=1,2,3,…, suppose that |z|≥|c|>(2/s)1/n-1 and |z|≥|c|>(2/s′)1/n-1, where 0<s, s′<1, and c is a complex number. Define
(5)z1=1-szn+c+sGcz,⋮zn=(1-s)(zn-1n+c)+sGc(zn-1).
Then |zn|→∞ as n→∞. Thus the general escape criterion is max⁡{c,(2/s)1/n-1,(2/s′)1/n-1}.Proof.
We will use induction. Forn=1, we get Gc(z)=z+c, so the escape criterion is c, which is obvious; that is, z>max⁡{c,0,0}. For n=2, we get Gc(z)=z2+c so the escape criterion is z>max⁡{c,2/s,2/s′}. For n=3, we get Gc(z)=z3+c so the escape criterion is z>max⁡{c,(2/s)1/2,(2/s′)1/2}.
Now suppose that theorem is true for anyn. Let Gc(z)=zn+1+c and z≥c>(2/s)1/n and z≥c>(2/s′)1/n exist. Then for Gc′(z)=zn+1+c, consider
(6)Gcz=1-s′z+s′Gc′z=1-s′z+s′zn+1+c=1-s′z+s′zn+1+s′c≥s′zn+1+1-s′z-s′c≥s′zn+1+1-s′z-s′z≥s′zn+1-1-s′z-s′z=s′zn+1-z+s′z-s′z≥z(s′zn-1).
Also, for
(7)zn=(1-s)f(zn-1)+sGc(z),
we obtain
(8)z1=1-szn+1+c+szs′zn-1=1-szn+1+1-sc+szs′zn-sz=ss′z·zn-sz+1-szn+1+1-sc≥ss′z·zn-sz-s-1zn+1-1-sc≥ss′zn+1-sz-s-1zn+1-1-sz≥ss′zn+1-sz-s-1zn+1-z+sz≥ss′-s+1zn+1-z≥z((ss′-s+1)zn-1).
Since |z|>(2/s)1/n and |z|>(2/s′)1/n it follows that |zn|>2/s and |zn|>2/s′. It can be easily seen that
(9)zn>2(ss′)>2(ss′-s+1),
which implies that
(10)ss′-s+1zn-1>1.
Hence there exists λ>0 such that (ss′-s+1)|zn|-1>1+λ. Consequently
(11)z1>(1+λ)z,⋮zn>1+λnz.
Hence zn→∞ as n→∞. So |z|>max⁡{c,(2/s)1/n,(2/s′)1/n} is the escape criterion. This completes the proof.Corollary 5.
Suppose that|c|>(2/s)1/n-1 and |c|>(2/s′)1/n-1 exist. Then the relative superior orbit of S-iteration scheme RSO(Gc,0,s,s′) escapes to infinity.Corollary 6.
Assume thatzk>max⁡{c,(2/s)1/k-1,(2/s′)1/k-1} for some k≥0. Then zk+1>γnzk and zn→∞ as n→∞.This corollary provides an algorithm for computing the relative superior Mandelbrot sets for the functions of the formGc(z)=zn+c and n=2,3,… also gives escape criterion to generate relative superior tricorns and multicorns.


## 4. Generation of Relative Superior Tricorns and Multicorns

We generate relative superior tricorns and multicorns ofS-iteration scheme for quadratic, cubic, and biquadratic functions using software MAPLE.


### 4.1. Relative Superior Tricorns for Quadratic Functions

In case of quadratic antipolynomial, relative superior tricorns maintain the symmetry alongx-axis (Figures 1–6).Figure 1
Relative superior tricorn fors=1.0 and s′=1.0.Figure 2
Relative superior tricorn fors=0.2 and s′=0.4.Figure 3
Relative superior tricorn fors=0.4 and s′=0.3.Figure 4
Relative superior tricorn fors=0.6 and s′=0.5.Figure 5
Relative superior tricorn fors=0.4 and s′=0.7.Figure 6
Relative superior tricorn fors=0.3 and s′=0.7.


### 4.2. Relative Superior Multicorns for Cubic Functions

In case of cubic antipolynomial, relative superior multicorns maintain the symmetry alongx-axis and y-axis (Figures 7–12).Figure 7
Relative superior multicorn fors=1.0 and s′=1.0.Figure 8
Relative superior multicorn fors=0.3 and s′=0.5.Figure 9
Relative superior multicorn fors=0.2 and s′=0.6.Figure 10
Relative superior multicorn fors=0.6 and s′=0.5.Figure 11
Relative superior multicorn fors=0.4 and s′=0.3.Figure 12
Relative superior multicorn fors=0.3 and s′=0.8.


### 4.3. Relative Superior Multicorns for Biquadratic Functions

In case of biquadratic antipolynomial, relative superior multicorns maintain the symmetry alongx-axis (Figures 13–18).Figure 13
Relative superior multicorn fors=1.0 and s′=1.0.Figure 14
Relative superior multicorn fors=0.5 and s′=0.6.Figure 15
Relative superior multicorn fors=0.1 and s′=0.8.Figure 16
Relative superior multicorn fors=0.2 and s′=0.6.Figure 17
Relative superior multicorn fors=0.7 and s′=0.4.Figure 18
Relative superior multicorn fors=0.3 and s′=0.8.


### 4.4. Generalization of Relative Superior Multicorns

See Figures19–24.Figure 19
Relative superior multicorn fors=0.01, s′=0.4, and n=5.Figure 20
Relative superior multicorn fors=0.3, s′=0.4, and n=7.Figure 21
Relative superior multicorn fors=0.6, s′=0.5, and n=8.Figure 22
Relative superior multicorn fors=0.2, s′=0.7, and n=10.Figure 23
Relative superior multicorn fors=0.3, s′=0.6, and n=12.Figure 24
Relative superior multicorn fors=0.4, s′=0.6, and n=16.


## 4.1. Relative Superior Tricorns for Quadratic Functions

In case of quadratic antipolynomial, relative superior tricorns maintain the symmetry alongx-axis (Figures 1–6).Figure 1
Relative superior tricorn fors=1.0 and s′=1.0.Figure 2
Relative superior tricorn fors=0.2 and s′=0.4.Figure 3
Relative superior tricorn fors=0.4 and s′=0.3.Figure 4
Relative superior tricorn fors=0.6 and s′=0.5.Figure 5
Relative superior tricorn fors=0.4 and s′=0.7.Figure 6
Relative superior tricorn fors=0.3 and s′=0.7.


## 4.2. Relative Superior Multicorns for Cubic Functions

In case of cubic antipolynomial, relative superior multicorns maintain the symmetry alongx-axis and y-axis (Figures 7–12).Figure 7
Relative superior multicorn fors=1.0 and s′=1.0.Figure 8
Relative superior multicorn fors=0.3 and s′=0.5.Figure 9
Relative superior multicorn fors=0.2 and s′=0.6.Figure 10
Relative superior multicorn fors=0.6 and s′=0.5.Figure 11
Relative superior multicorn fors=0.4 and s′=0.3.Figure 12
Relative superior multicorn fors=0.3 and s′=0.8.


## 4.3. Relative Superior Multicorns for Biquadratic Functions

In case of biquadratic antipolynomial, relative superior multicorns maintain the symmetry alongx-axis (Figures 13–18).Figure 13
Relative superior multicorn fors=1.0 and s′=1.0.Figure 14
Relative superior multicorn fors=0.5 and s′=0.6.Figure 15
Relative superior multicorn fors=0.1 and s′=0.8.Figure 16
Relative superior multicorn fors=0.2 and s′=0.6.Figure 17
Relative superior multicorn fors=0.7 and s′=0.4.Figure 18
Relative superior multicorn fors=0.3 and s′=0.8.


## 4.4. Generalization of Relative Superior Multicorns

See Figures19–24.Figure 19
Relative superior multicorn fors=0.01, s′=0.4, and n=5.Figure 20
Relative superior multicorn fors=0.3, s′=0.4, and n=7.Figure 21
Relative superior multicorn fors=0.6, s′=0.5, and n=8.Figure 22
Relative superior multicorn fors=0.2, s′=0.7, and n=10.Figure 23
Relative superior multicorn fors=0.3, s′=0.6, and n=12.Figure 24
Relative superior multicorn fors=0.4, s′=0.6, and n=16.


## 5. Conclusions

In this paper relative superior antifractal has been visualized with respect to relative superior orbit and analyzed the pattern of symmetry among them. In the dynamics of antipolynomialsz→z¯n+c for n≥2, we obtained many relative superior tricorns and multicorns for the same value of n by using different values of s and s′ in S-iteration scheme. We found that the number of branches and main ovoids attached to the branches of the relative superior tricorns and multicorns had been n+1, where n is the power of z¯. We also found that for n is odd the symmetry of relative superior multicorn is about both x-axis and y-axis but for n is even the symmetry is maintained only along x-axis. We believe that results of this paper will inspire those who are interested in generating automatically nicely looking graphics.



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