Mandelbrot set for julia sets of arbitrary order.a remark ...

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Mandelbrot set for julia sets of arbitrary order.a remark on the shape of cubic mandelbrot and julia sets Anna V. Tomova [email protected]

Abstract. The Mandelbrot set for Julia sets,associated with fc (z) = z2 + c,c ∈ C is in detail very good studied.This family is of special importance because it provides a model for the onset of chaotic behaviour in physical and biological systems.Moreover it was the first family of dynamical systems for which a useful computergraphical map was constructed by Mandelbrot.In this paper we restrict the attention to the families:fc(z) = z n + c ,fp,q(z) = z3+ pz + q;c,p,q ∈ C.We proof any theorems for the limits of Mandelbrot set for Julia sets of arbitrary order and for cubic Mandelbrot and Julia sets. We consider and make a remark on the shape of Mandelbrot and Julia sets for the dynamical systems in the general case fp,q(z) = zn+ pz + q,n>3 too.

1.

Introduction

In Barnsley (1988) and Barnsley and Liman (1993) we find the following ∧

∧

Definition 1. Let f: C → C denote a polynomial of degree greater than one.Let Ff denote the set of points in C whose orbits do not converge to the Point at Infinity.That is Ff = {z ∈ C:

f

0k

( z) } ∞k= 0 is bounded} .

This set is called the filled Julia set associated with the polynomial f.The boundary of Ff is called the Julia set of the polynomial f,and it is denoted by Jf..In Barnsley(1988) we find an equivalent definition for the Julia set to Definition 1 in the case of polynomials: ∧

∧

Definition 2. The Julia set of a rational function f: C → C ,of degree greater than one,is ∧

the closure of the ser of repulsive periodic points of the dynamical system { C ;f}.

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It is very good known that the family of dynamical systems { C :p(z) = z2 + c,c ∈C} is of special importance because it provides a model for the onset of chaotic behaviour in physical and biological systems :see Peitgen and Richter (1986). Moreover,it was the first family of dynamical systems for which a useful computergraphical map was constructed,by Mandelbrot.See the pages in INTERNET by the way of illustrations:@Yahoo.com/Sciense/Mathematics/Fractals…The Julia set J(c),associated with p(z), is symmetric about the origin,O.We know this because the filled Julia set,of which J(c) isboundary,is the set of points whose orbits remain bounded.The orbit of z ∈C remains bonded if and only if the orbit of -z remains bounded.For some values of c ∈C,O belongs to the filled Julia set,F(c),while for others it is quite far from F(c).This suggests that we try to color the parameter space according to the distance from O to F(c).An approximate method for an estimation of this distance is to look at the “escape time” of the orbit of O.That is,we can color the parameter space according to the number of steps along the orbit of O that are required before it lands in a ball around the Point at Infinity,from where we know that all orbits diverge.The intuitive idea is that the longer an orbit of O takes to reach the ball,the closer O must be to F(c).Of course,if an orbit does not diverge then we know that O ∈F(c).In Barnsley (1988) we find he following Definition 3. The Mandelbrot set for the family of dynamical systems ∧

{ C :z2 - c } is Μ = {c ∈ C: J (c) is connected} . The relationship between escape times of orbits of O and the connectivity of J(c) is provided by the following theorem. ∧

Theorem 1. The Julia set for a member of the family of dynamical systems { C :fc (z) = z2 c},c ∈C is connected if and only if the orbit of the origin does not escape to infinity; that is M = {c ∈ C:

f

0k c

(0)

< ∞ as k → ∞ }.

In Peitgen and Richter (1986) we can found the following Theorem 2. The Mandelbrot set for the family of dynamical systems ∧

{ C :z2 + c} satisfies M ⊂ {c ∈ C: c ≤ 2}. It is known that c = -2 belongs to M.

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Main results.

2.1. Mandelbrot Set for Julia Sets of Arbitrary Order. In Barnsley (1988) we have found the following remark:”…The Random Iteration Algorithm can also be applied to compute Julia sets of cubic and quatric polynomials,and of special polynomials of higher degree such as zn + c when n = 5,6,7,…, and c ∈C.” In this paper we restrict attention to the families: ∧

{ C :p(z) = zn + c}, c ∈ C,n ∈N,n ≥ 3 (1) In this paper we will follow the same Definition 1 for the Julia set Jp and the filled Julia set Fp.We will give the similar definition as Definition 2 for the Mandelbrot set for (1): Definition 4. The Mandelbrot set for the family of dynamical systems (1) is Μ = {c ∈ C: J (c) is connected} .

The relationship between escape times of orbits of O and the connectivity of J(c) is provided by the following theorem,which is similar to Theorem 1 ∧

Theorem 3. The Julia set for a member of the family of dynamical systems { C :pc (z) = zn + c},c ∈C n ∈N,n ≥ 3 is connected if and only if the orbit of the origin does not escape to infinity; that is M = {c ∈ C:

0k

p

c

(0)

< ∞ as k → ∞ }.

Proof. This is essentially the same as the sketch of the proof of Theorem 1,p.316 in Barnsley (1988).This theorems follows from Brolin,Theorem 11.2,which says that the Julia set of a polynomial,of degree greater than one,is connected if and only if none of the finite critical points lie in the basin of attraction of the Point at Infinity.We find pc’(z) = nzn - 1,pc(z) possesses one finit critical point,O.Hence J(c) is connected if and only if k → ∞ . The proof is completed.

0k

p

c

(0) < ∞ as

Let us now consider the equation: zn - z - c = 0. (2)

We will proof the following Theorem 4. The equation (2) has an unique positive root Rc>1.If n = 2m it has an unique negative root too.If n = 2m + 1 the number of the negative roots is two or 0.

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Proof. We can use the graphics method (Kurosh ,1968) for proof the fact that (2) has an unique positive root Rc for every n.Then we have: pc(z) = zn - z - c = (z - Rc)(zn-1 + Rczn-2 + Rc2 zn-3 +…+ Rcn-2z + Rcn-1-1) Rc(Rcn-1 - 1) = c , where Rc > 1 is the unique positive root of (2).We can use the graphics method for proof the fact that if n=2m (2) has an unique negative root too and if n = 2m + 1 the number of the negative roots of (2) is two or 0. Remark.See also Decart’sTheorem for the number of the posirtive roots of the polynomials equations on the page 245 in (Kurosh ,1968).We can use the formulas of Viet for confirm the affirmation of theorem 4 too. On Fig.1 we show the graphics for the functions x4 and x + 5.On Fig.2 we show the graphics for the functions x5 and x + 1.The figures confirm the affirmation of theorem 4.

30 25 20 15 10 5

-4

-2

2

4

Figure 1 15 10 5

-4

-2

2

4

-5 -10

Figure 2 Now we proof the following Theorem 5. The Julia set for a member of the dynamical system(1) lies in the circle z ≤ Rc where Rc>1 is the unique positive root of the equation (2).

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Proof. The method of the proof is borrowed from Beardon and Rippon (1994).We have: Pc ( z) ≥ z − c n

zn n = n Rc − c Rc

zn = n (Rc + c ) - c Rc

=

 zn  zn zn zn  z    > n R c > Rc if z > Rc R + ( − 1 ) = + c − 1 R c    Rn R cn c R cn  Rc  c  c  Rc n

Then if z > Rc we have the estimate: nk

 z  p 0c k ( z) >   R c → ∞ as k → ∞ (3)  Rc 

The proof is completed. Collorary.All others roots of (2) lie in the circle z ≤ R c . Let us consider the case (1) in detail:pc(0)=c,pc02(0)=cn+ c .Let c is the solution of the i ( π + 2 kπ )

equation cn + c = 0;that is c1=0,c2,…,n = e n −1 ,k = 0,…,n-2.Then cj,j=1,2,…,n belong to M.Let n = 2m and we solve the equation:pc02 (0) = - c; which solution are c1 = 0, c2,…,n = i ( π + 2 kπ )

2 e n −1 ,k = 0,1,…,2m-2.Then cj,j = 1,2,…,2m belong to M. This way we can proof own main result in the following n−1

Now we will proof the following Theorem 5. The Julia set for a member of the dynamical system(1) lies in the circle z ≤ Rc where Rc>1 is the unique positive root of the equation (2). Proof.We have proved that ck and 0 belong to M if n=2m.Now we must proof that if c 1

> 2 n−1 then

0k pc (0) → ∞ as k → ∞ .We consider the dynamical system:

∧

{ C: f ( p) = p n + c, c ∈ C }.

Then we can prove the estimate (see the proof of the theorem 5): nk

 p  f 0 k ( p) ≥   R c → ∞ as k → ∞ and p > Rc  Rc 

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where Rc is the positive root of the equation pn - p - c = 0

We denote : f(pc(0))=[pc(0)]n + c =cn + c = pc02(0); f02(pc(0)) = [f(pc(0)]n + c = [pc02(0)]n + c = (cn + c)n + c = pc03(0)… f0k(pc(0)) = pc0(k+1)(0). The first proof’s method Let now we substitute p for pc(0) in (3): we have the estimate: nk

 pc (0)  0k  Rc → ∞ f (pc (0)) >  as k → ∞ and   Rc  where Rc is the unique positive root of the equation [pc(0)] 1 n−1

n

c.This way we can substitute for c 2 and verify that is R = 2 of the equation pn - p - c = 0.Then we can write (3) in the form

p c (0) > Rc (4)

- pc(0) - c = 0. But pc(0) =

1 n−1

is the unique positive root

nk

0(k+1)

p

c

 c    (0) > 1   n−1  2 

2

1 n−1

1

→ ∞ as k → ∞ and c > 2 n−1 (5)

The proof is complete.

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The second proof’s method:

f ( p c (0)) ≥

p c (0)

(

2    p c (0)  1    2 n −1 

)

1  n1−1   2 + 2 n −1 - c =  

(

)2

p c (0)

2

n

1 n −1

1 n −1

+

p c (0)

n

2

- c >

n

2

1 n −1

f 02 ( p c (0) )    p c (0)  1    2 n −1 

1 n −1

n

≥

n2

2

1

if p c (0) = c > 2 n−1 ;

1 n −1

(

f ( p c (0))

2

1 n −1

   p c (0)  - c > 1   2 n −1 

f 0 k ( p c (0)) >

  p c ( 0) 1  n  2 −1

   

)  2 n

1 n −1

1  + 2 n −1 

-

c =

(

f ( p c (0)

2

1 n −1

)2 n

1 n −1

+

n2

2

1 n −1

1

if p c (0) = c > 2 n −1 …

nk

2

1 n −1

1

if p c (0) = c > 2 n−1

Then we can write (3) in the form (4). The proof is completed. Let z0 is a fixed point for (1).Then pc’(z0) = nz0n-1 = λ and z0= n−1

 λ    n

1 n −1

 λ    n

1 n −1

λ  λ ,  n  n

n n −1

+ c =

.If λ < 1 then z0 is an attractive fixed point for pc(z) and M ⊃ {c ∈C:c =

 λ  1 −  , λ < 1}.}.  n

We solve the equation pc02(0)=0 and find pc02’(z)=n2zn-1(zn+c)n-1.Then the circles n 1 c + ( −1) n −1 < belong to M. 2 n−1

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2.2. The application of Mandelbrot and Julia sets theory for dynamical systems (1). a) On figure 3,4,7 we use ESCAPE TIME ALGORITHM and show the Mandelbrot sets of order n = 5,6,50 for Julia sets for the dynamical systems (1). b) In figure 5,7,8 we use the ESCAPE TIME ALGORITHM to show the application of Julia set theory for an illustration of the trajectories of the dynamical systems (1) in the cases n 2 = 5 :c = (1+i) ; n = 6:c = -1.148657;n = 50:c = - 49 2 . The Julia sets are connected in all 2 cases.

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2.3. The Mandelbrot Set for Cubic Julia Sets Let us now restrict the attention to the families: {C:fp,q(z) = z3+ pz + q;p,q∈ C} (6) and consider the shape of cubic Julia sets. We will follow the same Definition 1 for the Julia set J and the filled Julia set F f and the same definition as Definition 2 for the f p ,q p ,q Mandelbrot set for (6) .Let us now consider the equation:

z 3 − ( p + 1) z − q = 0 (7)

We will proof the following Theorem 7. If q = 0 ,the roots of (8) are 0 and ± 1 + p . Let q ≠ 0.Then the equation (7) has an unique positive root Rp.q > p + 1.It has or two negative roots r1,r2 : r1 ≤ r2 < Rp,q too or two complex roots z1, z2 = z1 , z1 = z 2 < Rp,q. Proof. The first proof’s method. The case q = 0 is clear:the roots of (7) are 0 and ± 1 + p . Let us now consider the case q ≠ 0.If q ≠ 0,we use the graphical method for proof the facts that (7) has an unique positive root Rp.q and that the number of the negative roots is two or 0 ( Kurosh ,1968) . We can see of the graphics for the first function y1(x) = xn and for the second function y2(x) = (1 + p ) x + q that : r1 ≤ r2 < Rp,q too,if (8) has 3 reals roots.We have: tomova02

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z 3 − ( p + 1) z − q = ( z − R p ,q )( z 2 + R p ,q z + R p2,q − 1 − p ) .This formula shows that if (7) has 1 real and 2 complex roots z1, z2 = z1 , z1 = z 2 < Rp,q.

The second proof’s method: We use the graphical method for proof the facts that (7) has one unique positive root Rp.q ( Kurosh ,1968) .Then we have:

R 3p , q = ( p + 1)R p , q + q and R p, q ( R 2p , q − p − 1) = q ,also: R p2,q > p + 1 ≥ 1. Now

p +1 > 0 .If the equation (7) has two 3 negative roots r1,r2 , the graphical method shows that : r1 ≤ r2 < Rp,q .Now we consider the we denote: R p , q = α + β, α > 0, β > 0 ,because αβ =

case when (7) hasn’t negative roots and we denote their two complex conjugate roots as:

z1, 2

2 (α + β) 2 (α − β) 2 α+β α −β = ±i 3 .We find: z1, 2 = +3 = α 2 + β 2 − αβ . 2 2 4 4

It is evident that R p ,q = α + β + 2αβ .Also Rp,q > z1, 2 ,because αβ > 0 .The proof is completed. 2

2

2

15 10 5

-4

-2

2

4

-5 -10 -15

Figure 9

1000 500

-20

-10

10

20

-500 -1000

Figure 10

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-20

-10

10

20

-100 -200

Figure 11 Remark. On Fig.9 we show the graphics for the functions x3 and 2x + 1. On Fig.10 we show the graphics for the functions x3 and 50x + 100.On Fig.11 we show the graphics x3 and 2x + 100. The figures confirm the affirmation of theorem 7. See also Decart’sTheorem for the number of the positive roots of the polynomials equations on the page 245 in (Kurosh ,1968). Now we will proof the following Theorem 8. The Julia set for a member of the dynamical systems (6) lies in the circle z ≤ Rp,q where Rp,q>1 is the unique positive root of the equation (7). Proof.The method of the proof is borrowed from Beardon and Rippon (1994) and it is similar to the proof of the Theorem 5 .We have for z > R p , q :

f p , q ( z ) = z 3 + pz + q ≥ z 3 − pz + q ≥ z 3 − p z − q = z

3

R 3p , q >

z

(( p + 1) R p , q + q ) − p z − q =

3

R 3p , q

R p, q

R 3p , q

R 3p , q − p z − q =

  z3   z2   + z p 2 − 1 + q  3 − 1  R p, q   R p, q     

3

R 3p , q

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We can continue if z > R p , q :

3

f p02,q ( z ) = f p3,q ( z ) + pf p ,q ( z ) + q ≥ f p ,q ( z ) − pf p ,q ( z ) + q ≥ f p3,q ( z ) − p f p , q ( z ) − q = f p,q ( z )

3

R

R 3p ,q f p ,q ( z )

3

R 3p ,q >

3 p,q

f p ,q ( z ) R 3p ,q

R p ,q

− p f p,q ( z ) − q =

f p,q ( z )

3

(( p + 1) R p ,q + q ) − p f p ,q ( z ) − q =

R 3p ,q

 f ( z) 2   f ( z) 3   p ,q   p ,q  + f p ,q ( z ) p  − 1 + − 1 q 2 3     R p ,q   R p,q    

3

R p ,q >

z

32

R p ,q

2

R 3p ,q

This way we can receive the estimate for z > R p , q :

 z   f p0,kq ( z ) >   R p, q   

3k

R p , q → ∞ as k → ∞

The proof is completed.

2.4. Examples We use the system for computer algebra MATHEMATICA 4.0 for computing the following examples: Solve@z^3 - z - 1 ç 0, zD

‚

1ƒ3

1ƒ3 I 1 I9 + 69 MM ‚ 1 i 27 3 69 y ::z “ + 2 3 k 2 2 { 32ƒ3

>,

i 27 3 ‚ 69 y1ƒ3 I1 - É 1 :z “ - I1 + É ‚ 3 M 6 2 { k 2 i 27 3 ‚ 69 y1ƒ3 I1 + É 1 :z “ - I1 - É ‚ 3 M 6 2 { k 2

‚ 3 M I 1 I9 + ‚ 69 MM1ƒ3 2

2 32ƒ3

‚ 3 M I 1 I9 + ‚ 69 MM1ƒ3 2

2 32ƒ3

>, >>

Solve@z^3 - z - 1 ç 0, zD 88z “ -0.662359 - 0.56228 É