# Generalized Mandelbrot Sets of a Family of PolynomialsPnz=zn+z+c;n≥2

Authors: Salma M. Farris
Journal: International Journal of Mathematics and Mathematical Sciences
 (2022)

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## Abstract

In this paper, we study the general Mandelbrot set of the family of polynomialsPnz=zn+z+c;n≥2, denoted by GM(Pn). We construct the general Mandelbrot set for these polynomials by the escaping method. We determine the boundaries, areas, fractals, and symmetry of the previous polynomials. On the other hand, we study some topological properties of GMPn. We prove that GMPn is bounded and closed; hence, it is compact. Also, we characterize the general Mandelbrot set as a union of basins of attraction. Finally, we make a comparison between the properties of famous Mandelbrot set Mz2+c and our general Mandelbrot sets.


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## Body

## 1. Introduction

In the complex dynamical systems, a most beautiful and complicated object, with a simple definition, was studied from 1980s, and it is the well-known Mandelbrot set (see Figure1). In fact, this set has many properties, like fractals, chaos, self-similarity, and others [1–3]. The Mandelbrot set is used to study the dynamics of the quadratic polynomial Pz=z2+c [4, 5]. The Mandelbrot set, denoted by M, consists of all values of c for which the orbit of the critical point, 0, of P is bounded, i.e., Mf=c∈⊄:fn0is bounded∀n∈N where fnz=ffn−1z [6]. Note that the critical points are important in studying the complex dynamics because of the following fact: if a function possesses an attracting periodic orbit, then the orbit of the critical points must converge to that periodic orbit [7–10].Figure 1
The Mandelbrot set.There are a number of works that studied the general Mandelbrot sets, like the general Mandelbrot set ofz2+c+Rez, the general Mandelbrot set of z¯+c, the general Mandelbrot set of sinz/c, and others [11–15].The general Mandelbrot set, denoted by GM(f), is defined by GM(f) = c∈⊄:fnzcris bounded,zcris the critical point off [16, 17].Wang et al. emphasized that the generalized Mandelbrot set contains a lot of information to construct the Julia set, which enters the state of chaos through the solution of fractal bifurcation and triple rotational bifurcation [18].The paper [19] studies the structure topological inflexibility and the discontinuity evolution law of the generalized M–J sets generated from the complex mapping . Moreover, it studies the physical meaning of the generalized M–J sets. The researcher has other works in this field that the reader can find through the Internet.In this paper, we construct the general Mandelbrot sets of the family of polynomialsPnz=zn+z+c, (n≥2), using the escaping algorithm. Moreover, we study their areas, their boundaries, and their fractals. Some topological properties of GM(Pn) were studied. We prove that GM(Pn) is closed and bounded and hence is compact, and we find a bound of GM(Pn), ∀n≥2. Also, we characterize GM(Pn) as a union of basins of attraction of the attracting periodic point of Pn, ∀n≥2.Finally, we construct (plot) the general Mandelbrot sets ofPn(z), n = 4,5,6,7,8,9, and make a comparison of the properties of Mandelbrot set Mz2+c and our general Mandelbrot sets GM(Pn).


## 2. Definitions

(i)
For any complex mapg, the point zcr, for which the eigenvalues of the Jacobi matrix are equal to zero, is called the general critical point of g [20].(ii)
The general Mandelbrot set of fcz is defined as follows: GMf=c∈⊄:fcnzcris bounded∀n∈N where zcr is critical point of f, and fcnzcr=fcfcn−1zcr [21].B′. The general Mandelbrot set of a function can be defined also as [20](1)GMf=c∈⊄:Jcfis conneced.(iii)
A metric spaceS,ρ is said to be connected iff S is not a union P∪Q of any two disjoint closed sets; it is disconnected otherwise. A set A ⊆ S is called connected iff A,ρ is connected as a subspace of S,ρ, i.e., iff A is not a union of two disjoint sets P,Q≠∅ that are closed in A,ρ, as a subspace of S,ρ [22, 23].(iv)
Letc be any complex number. The smallest closed set in the complex plane that contains all repelling periodic points of a family of complex function fλz is called the Julia set of fλ and is denoted as J(f) [3].(v)
Afixed pointx0 (i.e., fx0=x0) is an attracting fixed point if f′x0<1. The point x0 is a repelling fixed point if f′x0>1. Finally, if f′x0=1, the fixed point is called neutral [3].(vi)
Letf be a function and let x0 be in the domain of f. Then,fx0  = the first iterate of x0 for fffx0  = the second iterate of x0 for fMore generally,fnx0 for the n-th iterate of x0 for fWe call the sequencefnx0n=0∞ of iterates of x0the orbit of x0 [24].(vii)
If a fixed point P off is attracting, then all points near to P are “attracted” toward P, in the sense that their iterates converge to P. The collection of all points whose iterates converge to P is called the basin of attraction of p [1].(viii)
Thebasin of attraction of the period point of period k, P0, is the collection of all points whose iterates converges P0 or f(P0) or … or fk−1P0 [1].


### 2.1. Classification of Fractals

Fractals can be classified according to their self-similarity. There are three types of self-similarity found in fractals.Exact self-similarity is the strongest type of self-similarity, and the fractal appears identical at different scales, for example, the Sierpinski triangle and Koch snowflake exhibit exact self-similarity. The second type is the quasi-self-similarity; here a loose form of self-similarity and the fractal appears approximately (but not exactly) identical at different scales, i.e., the set contains small copies of the entire fractal in distorted and degenerate forms. Finally, statistical self-similarity is the weakest type of self-similarity. The fractal here has numerical or statistical measures which are preserved across scales [16, 25].


### 2.2. The Area

The problem of calculating the area of Mandelbrot set is not simple yet. However, there are several methods which can be used to calculate an approximated area for these sets. The methods that calculate the best approximation to that area are pixel counting method, Monte Carlo method and Grunwall method. We will use the Monte Carlo method by generating random points in the complex plane, and then these points are tested if they are in the general Mandelbrot set or not. The area of a region is then estimated from the ratio of points in the set to those outside it.Remark 1.
The approximate area of original Mandelbrot set using counting method is 1.4880, that using the Monte Carlo method is 1.5 (±0⋅1), and that using Gronwall's method is 1.7274 [6, 11, 26, 27].


## 2.1. Classification of Fractals

Fractals can be classified according to their self-similarity. There are three types of self-similarity found in fractals.Exact self-similarity is the strongest type of self-similarity, and the fractal appears identical at different scales, for example, the Sierpinski triangle and Koch snowflake exhibit exact self-similarity. The second type is the quasi-self-similarity; here a loose form of self-similarity and the fractal appears approximately (but not exactly) identical at different scales, i.e., the set contains small copies of the entire fractal in distorted and degenerate forms. Finally, statistical self-similarity is the weakest type of self-similarity. The fractal here has numerical or statistical measures which are preserved across scales [16, 25].


## 2.2. The Area

The problem of calculating the area of Mandelbrot set is not simple yet. However, there are several methods which can be used to calculate an approximated area for these sets. The methods that calculate the best approximation to that area are pixel counting method, Monte Carlo method and Grunwall method. We will use the Monte Carlo method by generating random points in the complex plane, and then these points are tested if they are in the general Mandelbrot set or not. The area of a region is then estimated from the ratio of points in the set to those outside it.Remark 1.
The approximate area of original Mandelbrot set using counting method is 1.4880, that using the Monte Carlo method is 1.5 (±0⋅1), and that using Gronwall's method is 1.7274 [6, 11, 26, 27].


## 3. The General Mandelbrot Set of the PolynomialPnz=zn+z+c

In this section, we construct the general Mandelbrot sets of the polynomials,Pnz=zn+z+c, (n=2,3), denoted by GM(Pn). We determine the area, the fractal, and the boundary of GM(Pn), (n = 2,3). Also, we show that the Julia sets of P2 and P3 are connected for certain values of the parameter c∈ GM(Pn) and disconnected for a value of parameters which does not belong to MG(Pn).Remark 2.
For the general cases (n > 3), the same arguments, used in this section, can be applied with some more complicated details and have similar results.


### 3.1. The General Mandelbrot Set ofP2, GM(P2)

To construct GM(P2), we calculate the general critical points of P2z=z2+z+c . We use the notations z=x+iy,c=cre+cim, and f=u+iv. Then, the equation has the following form:(2)P2z=x+iy2+x+iy+c=x2+2ixy−y2+x+iy+c.Letu =  x2−y2+x+creandv=2xy+y+cim.Its Jacobi matrix isJ=2x+1−2y2y2x+1.Its characteristic polynomial is2x+1−λ−2xy2xy2x+1−λ=2x+1−λ2+4y2. If λ=0, then we get 2x+12+4y2=0.Ify = 0, then 2x+12=0, and this follows x=−1/2. Thus, the point −1/2,0 is a critical point for the polynomial P2z. Also, if x = 0, then the other critical point is (0, i/2). The construction of GM(P2) is shown in Figures 2 and 3.Figure 2
The general Mandelbrot set for the critical point−1/2,0ofP2z.Figure 3
The general Mandelbrot for the critical point0,i/2 of P2z.It is clear that the Mandelbrot set of this polynomial is similar to the original Mandelbrot set without deformations.Letz0 be a fixed point for the polynomial Pnz. Then, z0 is attracting if Pn′z0<1. Thus, the solutions of the equation Pn′z0=1 are the boundary points of the stable (attracting) point. That is, the curve Pn′z0=1 bounds all the attracting fixed points of Pnz. Similarly, the boundary equation Pnm′z0=1 bounds all the periodic points of period m which are attracting. In Sections 3.1.1. and 3.2.1. we use this criterion to draw the boundaries of GM P2 and GM P3, respectively, and it is clear that this criterion is true to draw the boundaries of GM Pn, ∀n.


#### 3.1.1. The Boundary of the Mandelbrot Set of the Fixed Point ofP2

For the polynomialP2, the fixed points may be found by solving the equation z2+z+c=z which is(3)z2+c=0The solution isz1=ci and z2=−ci, i.e., z1 is the first fixed point of P2 and z2 is the second fixed point of P2.The stability of each period point is determined from the derivative of the map at the point. Now,(4)dP2dz=2z+1,(5)2z+1=reiθ,(6)z=−1+reiθ2,where r≥0and0≤θ≤2π. Substituting (6) in (3), we have −1+reiθ/22+c=0. The solution for c is(7)−1+reiθ22+c=0.One of the fixed points, sayz1, is stable (attracting) as long as dPc/dzz1<1. Therefore, the boundary of the point of period one is given by dP2/dzz1z1=2z1+1=r=1.In the particular case, letc=c1+ic2. Then, from (7), the boundary is given by the following parametric equations: c1+ic2=−1/4+cosθ/2+isinθ/2−cos2θ/4−isin2θ/4. Then,(8)c1=−14+cosθ2−cos2θ4,c2=sinθ2−sin2θ4.We plot the curve in Figure4.Figure 4
The boundary of the fixed point forP2z.


#### 3.1.2. The Fractals of GM(P2)

GM(P2) admits, like the original Mandelbrot set, quasi-self-similarity. Also, it is similar around the x-axis.


#### 3.1.3. The Area of GM(P2)

Using Monte Carlo method, the area of the general Mandelbrot set ofP2z is 1.6456. Note that this area is very close to the area of the original Mandelbrot set, M(f).


#### 3.1.4. The Julia Set ofP2

Applying the definitionB′, if we take c=−0⋅3∈GMP2, then the Julia set JcP2 is connected (Figure 5), and if c=0⋅3∉GMP2, then we obtain that JcP2 is disconnected (Figure 6).Figure 5
The Julia set ofP2z=z2+z+c is connected where (0.3 ∈ GM(P2)).Figure 6
The Julia set ofP2z=z2+z+c is disconnected where (0.3 ∉ GM(P2)).


### 3.2. The General Mandelbrot Set of the PolynomialP3z, GM(P3)

ConsiderP3z=z3+z+c. Using the notations z=x+iy,c=cre+cim and f=u+iv, the equation has the following form:(9)P3z=x+iy3+x+iy+c=x3+3ix2y−3xy2−iy3+x+iy+c.Letu =  x3−3xy2+x+creandv=3x2y−y3+y+cim.Its Jacobi matrix isJ=3x2−3y2+1−6xy6xy3x2−3y2+1.Also, its characteristic polynomial is(10)3x2−3y2+1−λ−6xy6xy3x2−3y2+1−λ=3x2−3y2+1−λ2+36x2y2.Letλ=0; then, we get 3x2+3y22+6x2−6y2+1=0.Ifx = 0, then we have 3y22−6y2+1=0,9y4−6y2+1=0.Letu =  y2; then, 9u2−6u+1=0, and this follows 3u−12=0, u=1/3.y2=1/3⇒y=±1/3. Thus, the points 0,1/3and0,−1/3 are critical points for the polynomial P3z. Figures 7 and 8 show the general Mandelbrot set of the critical points 0,1/3and0,−1/3.Figure 7
The general Mandelbrot setP3 for the critical point 0,1/3.Figure 8
The general Mandelbrot setP3 for the critical point 0,−1/3.Also, ify = 0, the other critical points of P3z are the points i/3,0and−i/3,0. The general Mandelbrot set GM(P3) for these points is shown in Figures 9 and 10.Figure 9
The general Mandelbrot setP3 for the critical point i/3,0.Figure 10
The general Mandelbrot setP3 for the critical point −i/3,0.


#### 3.2.1. The Boundary of the Mandelbrot Set of the Fixed Point ofP3z

The fixed point of this polynomial may found by solving the following equation:z3+z+c=z, which implies z3+c=0 . Thus, c=−z3.Letc=c1+ic2 and z=x+iy; by substituting them in (1), we get(11)c1+ic2=−x3+3xy2−3x2yi+y3i.Now, substitutex=rcosθ and y=rsinθ, and we get(12)c1+ic2=−r3cos3θ+3r3cosθsin2θ−3r3icos2θ+r3isin3θ.Now, the derivative ofP3z is dP3/dz=3z2+1.The boundary consists of allz such that 3z2+1=1.Then,=z=2/3. Substitute r =  2/3 in (9), and we get(13)c1=−233cos3θ+3cosθsin2θ,c2=233−3cos2θ+sin3θ.We plot this for0≤θ≤2π (see Figure 11).Figure 11
The boundary of fixed point of period one forP3z.


#### 3.2.2. The Fractal of GM(P3)

The polynomialP3z admits quasi-self-similarity and is similar around y-axis (see Figures 7–10).


#### 3.2.3. The Area of the Polynomial GM(P3)

Applying the Monte Carlo algorithm, we find that the area ofP3z is 2.1952.


#### 3.2.4. The Julia Set of the PolynomialP3

Applying definition B′, if we takec=0⋅2i∈GMP3, then we note that the Julia set JcP3 is connected (Figure 12) and for c=0⋅3+0⋅8i∉GMP3 then JcP3 is disconnected (Figure 13).Figure 12
The Julia set ofP3z=z3+z+c is connected where 0.2i ∈GMP3.Figure 13
The Julia set ofP3z=z3+z+c is connected where 0.3 + 0.8i ∉GMP3.


## 3.1. The General Mandelbrot Set ofP2, GM(P2)

To construct GM(P2), we calculate the general critical points of P2z=z2+z+c . We use the notations z=x+iy,c=cre+cim, and f=u+iv. Then, the equation has the following form:(2)P2z=x+iy2+x+iy+c=x2+2ixy−y2+x+iy+c.Letu =  x2−y2+x+creandv=2xy+y+cim.Its Jacobi matrix isJ=2x+1−2y2y2x+1.Its characteristic polynomial is2x+1−λ−2xy2xy2x+1−λ=2x+1−λ2+4y2. If λ=0, then we get 2x+12+4y2=0.Ify = 0, then 2x+12=0, and this follows x=−1/2. Thus, the point −1/2,0 is a critical point for the polynomial P2z. Also, if x = 0, then the other critical point is (0, i/2). The construction of GM(P2) is shown in Figures 2 and 3.Figure 2
The general Mandelbrot set for the critical point−1/2,0ofP2z.Figure 3
The general Mandelbrot for the critical point0,i/2 of P2z.It is clear that the Mandelbrot set of this polynomial is similar to the original Mandelbrot set without deformations.Letz0 be a fixed point for the polynomial Pnz. Then, z0 is attracting if Pn′z0<1. Thus, the solutions of the equation Pn′z0=1 are the boundary points of the stable (attracting) point. That is, the curve Pn′z0=1 bounds all the attracting fixed points of Pnz. Similarly, the boundary equation Pnm′z0=1 bounds all the periodic points of period m which are attracting. In Sections 3.1.1. and 3.2.1. we use this criterion to draw the boundaries of GM P2 and GM P3, respectively, and it is clear that this criterion is true to draw the boundaries of GM Pn, ∀n.


### 3.1.1. The Boundary of the Mandelbrot Set of the Fixed Point ofP2

For the polynomialP2, the fixed points may be found by solving the equation z2+z+c=z which is(3)z2+c=0The solution isz1=ci and z2=−ci, i.e., z1 is the first fixed point of P2 and z2 is the second fixed point of P2.The stability of each period point is determined from the derivative of the map at the point. Now,(4)dP2dz=2z+1,(5)2z+1=reiθ,(6)z=−1+reiθ2,where r≥0and0≤θ≤2π. Substituting (6) in (3), we have −1+reiθ/22+c=0. The solution for c is(7)−1+reiθ22+c=0.One of the fixed points, sayz1, is stable (attracting) as long as dPc/dzz1<1. Therefore, the boundary of the point of period one is given by dP2/dzz1z1=2z1+1=r=1.In the particular case, letc=c1+ic2. Then, from (7), the boundary is given by the following parametric equations: c1+ic2=−1/4+cosθ/2+isinθ/2−cos2θ/4−isin2θ/4. Then,(8)c1=−14+cosθ2−cos2θ4,c2=sinθ2−sin2θ4.We plot the curve in Figure4.Figure 4
The boundary of the fixed point forP2z.


### 3.1.2. The Fractals of GM(P2)

GM(P2) admits, like the original Mandelbrot set, quasi-self-similarity. Also, it is similar around the x-axis.


### 3.1.3. The Area of GM(P2)

Using Monte Carlo method, the area of the general Mandelbrot set ofP2z is 1.6456. Note that this area is very close to the area of the original Mandelbrot set, M(f).


### 3.1.4. The Julia Set ofP2

Applying the definitionB′, if we take c=−0⋅3∈GMP2, then the Julia set JcP2 is connected (Figure 5), and if c=0⋅3∉GMP2, then we obtain that JcP2 is disconnected (Figure 6).Figure 5
The Julia set ofP2z=z2+z+c is connected where (0.3 ∈ GM(P2)).Figure 6
The Julia set ofP2z=z2+z+c is disconnected where (0.3 ∉ GM(P2)).


## 3.1.1. The Boundary of the Mandelbrot Set of the Fixed Point ofP2

For the polynomialP2, the fixed points may be found by solving the equation z2+z+c=z which is(3)z2+c=0The solution isz1=ci and z2=−ci, i.e., z1 is the first fixed point of P2 and z2 is the second fixed point of P2.The stability of each period point is determined from the derivative of the map at the point. Now,(4)dP2dz=2z+1,(5)2z+1=reiθ,(6)z=−1+reiθ2,where r≥0and0≤θ≤2π. Substituting (6) in (3), we have −1+reiθ/22+c=0. The solution for c is(7)−1+reiθ22+c=0.One of the fixed points, sayz1, is stable (attracting) as long as dPc/dzz1<1. Therefore, the boundary of the point of period one is given by dP2/dzz1z1=2z1+1=r=1.In the particular case, letc=c1+ic2. Then, from (7), the boundary is given by the following parametric equations: c1+ic2=−1/4+cosθ/2+isinθ/2−cos2θ/4−isin2θ/4. Then,(8)c1=−14+cosθ2−cos2θ4,c2=sinθ2−sin2θ4.We plot the curve in Figure4.Figure 4
The boundary of the fixed point forP2z.


## 3.1.2. The Fractals of GM(P2)

GM(P2) admits, like the original Mandelbrot set, quasi-self-similarity. Also, it is similar around the x-axis.


## 3.1.3. The Area of GM(P2)

Using Monte Carlo method, the area of the general Mandelbrot set ofP2z is 1.6456. Note that this area is very close to the area of the original Mandelbrot set, M(f).


## 3.1.4. The Julia Set ofP2

Applying the definitionB′, if we take c=−0⋅3∈GMP2, then the Julia set JcP2 is connected (Figure 5), and if c=0⋅3∉GMP2, then we obtain that JcP2 is disconnected (Figure 6).Figure 5
The Julia set ofP2z=z2+z+c is connected where (0.3 ∈ GM(P2)).Figure 6
The Julia set ofP2z=z2+z+c is disconnected where (0.3 ∉ GM(P2)).


## 3.2. The General Mandelbrot Set of the PolynomialP3z, GM(P3)

ConsiderP3z=z3+z+c. Using the notations z=x+iy,c=cre+cim and f=u+iv, the equation has the following form:(9)P3z=x+iy3+x+iy+c=x3+3ix2y−3xy2−iy3+x+iy+c.Letu =  x3−3xy2+x+creandv=3x2y−y3+y+cim.Its Jacobi matrix isJ=3x2−3y2+1−6xy6xy3x2−3y2+1.Also, its characteristic polynomial is(10)3x2−3y2+1−λ−6xy6xy3x2−3y2+1−λ=3x2−3y2+1−λ2+36x2y2.Letλ=0; then, we get 3x2+3y22+6x2−6y2+1=0.Ifx = 0, then we have 3y22−6y2+1=0,9y4−6y2+1=0.Letu =  y2; then, 9u2−6u+1=0, and this follows 3u−12=0, u=1/3.y2=1/3⇒y=±1/3. Thus, the points 0,1/3and0,−1/3 are critical points for the polynomial P3z. Figures 7 and 8 show the general Mandelbrot set of the critical points 0,1/3and0,−1/3.Figure 7
The general Mandelbrot setP3 for the critical point 0,1/3.Figure 8
The general Mandelbrot setP3 for the critical point 0,−1/3.Also, ify = 0, the other critical points of P3z are the points i/3,0and−i/3,0. The general Mandelbrot set GM(P3) for these points is shown in Figures 9 and 10.Figure 9
The general Mandelbrot setP3 for the critical point i/3,0.Figure 10
The general Mandelbrot setP3 for the critical point −i/3,0.


### 3.2.1. The Boundary of the Mandelbrot Set of the Fixed Point ofP3z

The fixed point of this polynomial may found by solving the following equation:z3+z+c=z, which implies z3+c=0 . Thus, c=−z3.Letc=c1+ic2 and z=x+iy; by substituting them in (1), we get(11)c1+ic2=−x3+3xy2−3x2yi+y3i.Now, substitutex=rcosθ and y=rsinθ, and we get(12)c1+ic2=−r3cos3θ+3r3cosθsin2θ−3r3icos2θ+r3isin3θ.Now, the derivative ofP3z is dP3/dz=3z2+1.The boundary consists of allz such that 3z2+1=1.Then,=z=2/3. Substitute r =  2/3 in (9), and we get(13)c1=−233cos3θ+3cosθsin2θ,c2=233−3cos2θ+sin3θ.We plot this for0≤θ≤2π (see Figure 11).Figure 11
The boundary of fixed point of period one forP3z.


### 3.2.2. The Fractal of GM(P3)

The polynomialP3z admits quasi-self-similarity and is similar around y-axis (see Figures 7–10).


### 3.2.3. The Area of the Polynomial GM(P3)

Applying the Monte Carlo algorithm, we find that the area ofP3z is 2.1952.


### 3.2.4. The Julia Set of the PolynomialP3

Applying definition B′, if we takec=0⋅2i∈GMP3, then we note that the Julia set JcP3 is connected (Figure 12) and for c=0⋅3+0⋅8i∉GMP3 then JcP3 is disconnected (Figure 13).Figure 12
The Julia set ofP3z=z3+z+c is connected where 0.2i ∈GMP3.Figure 13
The Julia set ofP3z=z3+z+c is connected where 0.3 + 0.8i ∉GMP3.


## 3.2.1. The Boundary of the Mandelbrot Set of the Fixed Point ofP3z

The fixed point of this polynomial may found by solving the following equation:z3+z+c=z, which implies z3+c=0 . Thus, c=−z3.Letc=c1+ic2 and z=x+iy; by substituting them in (1), we get(11)c1+ic2=−x3+3xy2−3x2yi+y3i.Now, substitutex=rcosθ and y=rsinθ, and we get(12)c1+ic2=−r3cos3θ+3r3cosθsin2θ−3r3icos2θ+r3isin3θ.Now, the derivative ofP3z is dP3/dz=3z2+1.The boundary consists of allz such that 3z2+1=1.Then,=z=2/3. Substitute r =  2/3 in (9), and we get(13)c1=−233cos3θ+3cosθsin2θ,c2=233−3cos2θ+sin3θ.We plot this for0≤θ≤2π (see Figure 11).Figure 11
The boundary of fixed point of period one forP3z.


## 3.2.2. The Fractal of GM(P3)

The polynomialP3z admits quasi-self-similarity and is similar around y-axis (see Figures 7–10).


## 3.2.3. The Area of the Polynomial GM(P3)

Applying the Monte Carlo algorithm, we find that the area ofP3z is 2.1952.


## 3.2.4. The Julia Set of the PolynomialP3

Applying definition B′, if we takec=0⋅2i∈GMP3, then we note that the Julia set JcP3 is connected (Figure 12) and for c=0⋅3+0⋅8i∉GMP3 then JcP3 is disconnected (Figure 13).Figure 12
The Julia set ofP3z=z3+z+c is connected where 0.2i ∈GMP3.Figure 13
The Julia set ofP3z=z3+z+c is connected where 0.3 + 0.8i ∉GMP3.


## 4. Topological Properties of General Mandelbrot Set ofPnz

In this section, we study some topological properties of GMPn, n = 2,3, ….Proposition 1.
The general Mandelbrot set ofPn ,GM(Pn) is closed.Proof.
Consider the complement of GM(Pn), i.e., ⊄-GM(Pn), Pnz=zn+z+c.
Letz0∈⊄-GM(Pn); then, by definition of GM(Pn), Pnkz0⟶∞ask⟶∞. Since Pnz is continuous, then there exists a neighborhood for z0, say Br0z0, such that ∀z∈Br0z0, Pnkz0⟶∞ as k⟶∞, i.e., Brz0⊆⊄−GMPn.
This implies that⊄-GM(Pn) is an open set in ⊄. Therefore, GM(Pn) is closed.Proposition 2.
LetPn=zn+z+c, then there exists an integer k, such that Pnmz⟶∞ for z>k.Proof.
LetPnz=zn+z+c; then, Pnz≥zn−z−c (remember that a+b≥a−b,∀a,b∈⊄).
Consider ǥn(z) = zn−2z−c.
Sincez is positive, then whenever z is large enough, we obtain that g(z) is positive, i.e., zn−2z−c≥0. Choose k, an integer large enough, such that ǥn(z) > 0, ∀z>k.
Thus,zn−z−c≥z. That is, Pnz≥z, ∀z>k. Therefore, the function Pnz is monotonically increasing.
Now, sincePn is continuous (polynomial), then ∀z0∈⊄ with z0>k, there exists a neighborhood Brz0 such that Pnz>z, ∀z∈Brz0.
Therefore,Pnz0 is unbounded for z0>k.
Thus, the sequencePnz0⟶∞, ∀z>k (every unbounded monotonically increasing sequence is divergent).Proposition 3.
GMPn is bounded.Proof.
To proveGMPn is bounded, we show that GMPn=c:c<k,for somek∈R+}. Let z be a fixed point of Pn; then, zn+z+c=z⇒zn+c=0⇒zn=c⇒z=c1/n. Thus, z=c1/n is the equation boundary of all fixed points with Pn′z<1.
The first derivativePn′z=nzn−1+1.
Ifnzn−1+1=0,thenz0=−1/nn−1 is the critical point of Pn.
Now ifz0∉GMPn, then Pnz0≥c1/n. Thus,(14)−1nn−1n+−1nn−1+c≥c1/n,iff−1nn−1n+−1nn−1+c≥c1/n,iffc−c1/n+−1nn−1n+−1nn−1≥0.
Letu =  c1/n; then, un=c. Hence, the inequality becomes un−u+A≥0, where A =  −1/nn−1n+−1/nn−1. Now let u1,u2,u3,…,un be the solutions of un−u+A≥0, and let k=maxu1,u2,…,un. Thus, c=kn. Therefore, ∀c∈⊄, with c>k, then c∉GMPn.
Similarly, ifz0 is a periodic point of period m, then by the same argument and using Proposition 2, we obtain the same result. Thus, GMPn=c:c<k for some k∈R+}. Thus, GMPn is bounded.Example 1.
Consider the casen = 2, i.e., P2z  =  z2+z+c.
The fixed point ofP2 satisfies z2+z+c=z.(15)⇔z2=−c,⇔z=c.
P2′z  = 2z + 1, and hence z0=−1/2 is the critical point of f.
Now, if(16)P2z0≥c⇔−122+−12+c≥c⇔−14+c≥c⇔−14+c≥c⇔c−c+−14≥0.
Letu=c; then, u2−u+1/4≥0.(17)⇔u−122≥0⇔u≥12.
Sincec=u2, we get c=1/4. Thus, c≥1/4. Now if z0 is a periodic point of period m, then P2mz0>P2z0 Proposition 2 and this implies that z0∉GMP2 iff z0>1/4. Therefore, GM P2=c:c<1/4, and GM P2 is bounded. From the above results, we conclude the following theorem.Theorem 1.
LetPnz=zn+z+c and let GMPn be the general Mandelbrot set of Pn, n = 2,3, …, and then GMPn is compact.Proof.
By Proposition1 and Proposition 3, we get that GMPn is closed and bounded. Using Heine–Borel theorem, GMPn is compact.Proposition 4.
LetPnz=zn+z+c; then, Pnmz has (at most) nm−1 attracting periodic orbits, ∀ (m = 1,2,3, …).Proof.
Pnz=zn+z+c is a polynomial of degree n. Then, Pnmz is a polynomial of degree nm. Now, Pnm is a polynomial of degree nm−1. Since z is a complex variable, then Pnm′z=0 has nm−1 distinct roots. That is, Pnm has nm−1 distinct critical points.
But the orbit of every critical point must lie in an attracting periodic orbit. Then, there are (at most)nm−1 periodic orbits.
The following theorem characterizes the general Mandelbrot set of our family as a union of basins of attractions.Theorem 2.
The general Mandelbrot setGMPn consists of the union of the basins of attraction of the attracting periodic orbits, and each basin of a periodic point contains all values of c whose orbits converge to that periodic point.Proof.
By definition,GMPn  =  c:Pnmzcris bounded. That is, c∈GMPn iff the orbit of the critical points of Pnzcr is bounded. But the orbit of any critical point must lie in the orbit of attracting period orbit and hence in a basin of an attracting periodic point. Thus, GMPn is composed of basins of the attracting periodic orbits, and all the values of c∈GMPn converge to these periodic orbits.
Now, we construct the general Mandelbrot set of the polynomialsPnz=zn+z+c for n = 4,5, …, 9 (see Figure 14).(18)Pz=z9+z+c,Pz=z8+z+c,Pz=z7+z+c,Pz=z6+z+c,Pz=z5+z+c,Pz=z4+z+c.
Note that forn≥4 in previous forms, the number of arms extended in the sets equals n−2 arm. Moreover, if we plot the GM of Pnz=zn+z+c, then we find the same behavior. In a future work, we hope to analyze this behavior.
Finally, we give a comparison between the Mandelbrot set generated withPz=z2+c and the general Mandelbrot sets of GMP2 and GMP3 (Table 1).Figure 14
The general Mandelbrot set forPnz=zn+z+c, n = 4,5, …, 9.Table 1
Comparison between the Mandelbrot set generated withP(z) = z^2 + c and the general Mandelbrot sets of GM(P_2) and GM(P_3).
FractalSymmetryAreaTopological propertiesM (z2 + c)Quasi-self-similarityAround thex-axis1.72CompactGMP2Quasi-self-similarityAround thex-axis1.6456CompactGMP3Quasi-self-similarityAround they-axis2.1952G(MPn)Quasi-self-similarityAround some diagonalCompact


## 5. Conclusions

(1)
In this paper, we construct the general Mandelbrot sets of the family of polynomialsPnz=zn+z+c, (n≥2), using the escaping algorithm.(2)
The areas, boundaries, and fractals forPnz are studied and calculated for n = 2,3.(3)
We study some topological properties of GM(Pn). It is proved that GM(Pn) is closed and bounded and hence is compact; also, we find a bound of GM(Pn), ∀n≥2.(4)
GMPn is characterized as a union of basins of attraction of the attracting periodic point of Pn, ∀n≥2.(5)
Finally, we plot the general Mandelbrot sets ofPn(z), n = 4,5,6,7,8,9, and make a comparison of the properties of Mandelbrot set Mz2+c and our general Mandelbrot sets GM(Pn).



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