# Generation of Julia and Mandelbrot Sets via Fixed Points

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## Abstract

**:**

## 1. Introduction

**Definition**

**1**

**.**The set of points in $\mathbb{C}$ whose orbits do not converge to a point at infinity is known as filled Julia set, ${K}_{T}$, that is,

**Theorem**

**1**

**.**If T is a complex polynomial, then ${J}_{T}$ is the closure of the repelling periodic points of T.

**Definition**

**2**

**.**Let T be any complex polynomial of degree $n\ge 2$. A Mandelbrot set M is the set consisting of all parameters r for which the Julia set, ${J}_{{Q}_{r}}$, is connected, that is,

**Theorem**

**2**

**.**For ${Q}_{r}\left(x\right)={x}^{2}+r$, $x,r\in \mathbb{C}$, if there exists $i\ge 0$ such that

## 2. Main Results

#### 2.1. Escape Criterion for Quadratic Complex Polynomials in a Picard Ishikawa Type Orbit

**Theorem**

**3.**

**Proof.**

**Corollary**

**1.**

#### 2.2. Escape Criterion for Cubic Complex Polynomials in a Picard Ishikawa Type Orbit

**Theorem**

**4.**

**Proof.**

**Corollary**

**2.**

#### 2.3. Escape Criterion for General Complex Polynomials in a Picard Ishikawa Type Orbit

**Theorem**

**5.**

**Proof.**

**Corollary**

**3.**

**Theorem**

**6.**

**Proof.**

## 3. Visualization of Fractals

#### 3.1. Generation of Julia sets

Algorithm 1: Generation of Julia Set. |

- For Figure 1, we consider the polynomial $T\left(x\right)={x}^{2}+(-0.5+0.7i)x+(-0.01+0.18i)$ and $A=[-2.5,2.5]\times [-2.1,2.1]$. It is easy to see that T has one attracting fixed point, $p=-0.1427+0.1019i$. Observe that for $\alpha =0.2$, $\beta =0.097$ and $\alpha =0.11$,$\beta =0.18$ we obtain different images due to color variation caused by parameters. It is interesting to note that for $\alpha =1$, $\beta =1$ and $\alpha ={10}^{-10}$, $\beta ={10}^{-10}$ we have similar shapes but there is clear variation of colors.
- For Figure 2, we consider the polynomial $T\left(x\right)={x}^{3}+(-0.275+0.5i)x+(-0.559+0.35i)$ and $A=[-1.5,1.5]\times [-1.8,1.8]$. The polynomial T has attracting fixed point $p\sim -0.6434+0.2687i$ in A. Note that the cubic Julia sets for $\alpha =0.08$ and $\beta =0.09$ have more color variation as compared to the Julia sets for $\alpha =0.1$, and $\beta =0.2$. Again, for $\alpha =1$, $\beta =1$ and $\alpha ={10}^{-10}$, $\beta ={10}^{-10}$ the shapes are same but there is variability in colors.
- For Figure 3, we input $T\left(x\right)={x}^{7}+(0.23+1.2i)x+(0.5+0.7i)$ and $A={[-1.3,1.3]}^{2}$. The attracting fixed point of the polynomial is $p\sim -0.2391+0.5835i$. We can see that for $\alpha =0.01$ and $\beta =0.08$ the shape is spread and stretched while the shape is dense and neatly packed for $\alpha =0.1$ and $\beta =0.05$. Note the variation of colors in figures (C) and (D) as well.

#### 3.2. Generation of Mandelbrot Sets

Algorithm 2: Generation of Mandelbrot set. |

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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Abbas, M.; Iqbal, H.; De la Sen, M.
Generation of Julia and Mandelbrot Sets via Fixed Points. *Symmetry* **2020**, *12*, 86.
https://doi.org/10.3390/sym12010086

**AMA Style**

Abbas M, Iqbal H, De la Sen M.
Generation of Julia and Mandelbrot Sets via Fixed Points. *Symmetry*. 2020; 12(1):86.
https://doi.org/10.3390/sym12010086

**Chicago/Turabian Style**

Abbas, Mujahid, Hira Iqbal, and Manuel De la Sen.
2020. "Generation of Julia and Mandelbrot Sets via Fixed Points" *Symmetry* 12, no. 1: 86.
https://doi.org/10.3390/sym12010086