Last modified on 1 June 2014, at 07:00

Fractals/Multibrot sets

Julia sets for polynomial functions

" The dynamics of polynomials is much better understood than the dynamics of general rational maps" due to the Bottcher’s theorem[1]

z(1-z^q)Edit

z^n+ z + cEdit

Example [2]

f (z) = z^3 + z + 0.6*I


z^n + c^pEdit

Z = Zn + cp

L = (m - 1) * p

z^n + cEdit

Multibrot sets [3][4][5]

L = m - 1

Compare with it's inverse parameter plane : Z^n + 1/c [6]

How to compute powers of z :

 Z^2 = (x+iy)^2   \mathrm{Real} = x^2 - y^2    \mathrm{Imag} = 2  x  y  
 Z^3 = (x+iy)^3   \mathrm{Real} = x^3 - 3 y^2  x     \mathrm{Imag} = 3  x^2  y  - y^3 
 Z^4 = (x+iy)^4   \mathrm{Real} = x^4  - 6  x^2  y^2 + y^4     \mathrm{Imag} = 4x^3  y - 4  x  y^3  
 Z^5 = (x+iy)^5   \mathrm{Real} = x^5  - 10  x^3  y^2 + 5  x  y^4     \mathrm{Imag} = 5x^4  y - 10  x^2  y^3 + y^5  
 Z^6  = (x+iy)^6    \mathrm{Real} = x^6  - 15  x^4  y^2 +15  x^2  y^4 - y^6    \mathrm{Imag} = 6x^5  y - 20  x^4  y^2 + 6 x  y^5  
 Z^7   = (x+iy)^7   \mathrm{Real} = x^7  - 21  x^5  y^2 + 35  x^3  y^4 -  7  x  y^6    \mathrm{Imag} = 7x^6  y - 35  x^4  y^3 + 21 x^2  y^5 - y^7  


Fortan source code :

! Fortran program by P.M.J. Trevelyan
! http://philiptrevelyan.co.uk/
     PROGRAM FRACTAL
      IMPLICIT NONE
      INTEGER I,J,ITERATION,N,M
      PARAMETER(N=2000,M=50)
      REAL*8 U,V,X,Y,P,Q
      OPEN(99,FILE='Fractal_quad.dat')
 25   FORMAT(2F9.5,I3)
      DO 10, I=1,N
      DO 20, J=1,N
C Define first point z(n)=U+iV and k=X+iY
      U=0.D0
      V=0.D0
      X=I*3.2D0/(N-1.D0)-2.1D0
      Y=J*2.8D0/(N-1.D0)-1.4D0
      DO 30, ITERATION=1,M
C Calculate z(n+1) = z(n)**2 + k where z(n+1)=P+iQ
      P=U**2-V**2+X 
      Q=2.D0*U*V+Y
      U=P
      V=Q
C If |z|>2 stop iterating
      If (U**2+V**2.GT.4.D0) GOTO 100
 30       CONTINUE
 100      WRITE(99,25) X,Y,ITERATION
 20     CONTINUE
 10   CONTINUE
      STOP
      END

Parameter planes for positive integer powers :

Negative powers

When d is negative the set surrounds but does not include the origin. There is interesting complex behaviour in the contours between the set and the origin, in a star-shaped area with (1 − d)-fold rotational symmetry. The sets appear to have a circular perimeter, however this is just an artifact of the fixed maximum radius allowed by the Escape Time algorithm, and is not a limit of the sets that actually extend in all directions to infinity.

z^2+cEdit

See complex quadratic polynomials

z^3 + cEdit

z_{n+1} = z^3_n + c

It can be computed by :[7]

   X_{n+1} = X^3_n - 3X_n Y^2_n + C_x

   Y_{n+1} = 3 X_n^2 Y_n - Y_n^3 + C_y

z^5 + cEdit

zn + 1/cEdit

It is an inverted parameter plain of z^n + c.

Number of vertices : V = (n - 1)

z^n + m*z^(-d)Edit

McMullen maps :

z^n + \frac{m}{z^d}

where : z and d are >=1

"These maps are known as `McMullen maps', since McMullen [8] first studied these maps and pointed out that when (n;m) = (2; 3) and m is small, the Julia set is a Cantor set of circles." [9]

z^n + m*zEdit

Description [10][11]

z^2+m*zEdit

See complex quadratic polynomial

z^3+m*zEdit

dynamic planesEdit

z^3 + zEdit

It can be found using Maxima CAS :


(%i2) z:zx+zy*%i;
(%o2) %i*zy+zx
(%i6) realpart(z+z^3);
(%o6) -3*zx*zy^2+zx^3+zx
(%i7) imagpart(z+z^3);
(%o7) -zy^3+3*zx^2*zy+zy

Finding roots and its multiplicity :

z^3 + z = z
z^3 = 0

so root z=0 has multiplicity 3.

(%i1) z1:z^3+z;
(%o1) z^3+z
(%i2) solve(z1=z);
(%o2) [z=0]
(%i3) multiplicities;
(%o3) [3]

It means that there is a flower with 2 petals around fixed point z=0. [12]

z^4 + mzEdit

How to compute iteration :

(%i1) z:zx+zy*%i;
(%o1) %i*zy+zx
(%i2) m:mx+my*%i;
(%o2) %i*my+mx
(%i3) z1:z^4+m*z;
(%o3) (%i*zy+zx)^4+(%i*my+mx)*(%i*zy+zx)
(%i4) realpart(z1);
(%o4) zy^4-6*zx^2*zy^2-my*zy+zx^4+mx*zx
(%i5) imagpart(z1);
(%o5) -4*zx*zy^3+4*zx^3*zy+mx*zy+my*zx


See also f(z) = c(z^4-4z). It is a family 4.1 in program Mandel by Wolf Jung[13] ( see main menu / New / 4. Quartic polynomials / 4.1 )

Here c = -m/4 and Mandelbrot set is rotated by 180 degrees.

Parameter planeEdit

Parameter plane and Mandelbrot set for f(z) = z^4 + m*z


Period 1 componentsEdit

Maxima CAS code :

(%i1) f:z^4+m*z;
(%o1) z^4+m*z
(%i2) e1:f=z;
(%o2) z^4+m*z=z
(%i3) d:diff(f,z,1);
(%o3) 4*z^3+m
(%i4) e2:d=w;
(%o4) 4*z^3+m=w
(%i5) s:eliminate ([e1,e2], [z]);
(%o5) [-(m-w)*(w+3*m-4)^3]
(%i6) s:solve([s[1]], [m]);
(%o6) [m=-(w-4)/3,m=w]

It means that there are 2 period 1 components :

  • one with radius = 1 and center =0 ( m=w )
  • second with radius 1/3 and center=4/3 ( m=-(w-4)/3 )

Dynamic planesEdit

z^4+zEdit

Julia set for f(z) = z^4 + z

Finding roots and its multiplicity :

z^4 + z = z
z^4 = 0

so root z=0 has multiplicity 4.

(%i1) z1:z^4+z;
(%o1) z^4+z
(%i2) solve(z1=z);
(%o2) [z=0]
(%i3) multiplicities;
(%o3) [4]

It means that there are 3 petals around fixed point z=0 [14]


How to compute iteration :


(%i17) z:x+y*%i;
(%o17) %i*y+x
(%i18) realpart(z+z^4);
(%o18) y^4-6*x^2*y^2+x^4+x
(%i19) imagpart(z+z^4);
(%o19) -4*x*y^3+4*x^3*y+y

z^4-izEdit

Critical orbits for f(z)=z^4-iz

First compute multiplier for internal angle=3/4 :

(%i1) m:exp(2*%pi*%i*3/4);
(%o1) -%i

Then find how to compute iteration :

(%i1) z:x+y*%i;
(%o1) %i*y+x
(%i2) z1:z^4-%i*z;
(%o2) (%i*y+x)^4-%i*(%i*y+x)
(%i3) realpart(z1);
(%o3) y^4-6*x^2*y^2+y+x^4
(%i4) imagpart(z1);
(%o4) -4*x*y^3+4*x^3*y-x

It is a parabolic Julia set with 12 petal flower [15]

Critical points :

(%i12) s:GiveListOfCriticalPoints(f(z))
(%o12) [0.31498026247372*%i-0.54556181798586,-0.62996052494744*%i,0.31498026247372*%i+0.54556181798586]
(%i13) multiplicities
(%o13) [1,1,1]
(%i14) length(s)
(%o14) 3

with arguments in turns :

[0.41666666666667,0.75,0.083333333333334] = [5/12 , 9/12, 1/12]

Attracting vectors

Multiplier of fixed point −i is a fourth root of unity ( q=4), thus we examine 4-th iteration :

(%i1) z1:z^4-%i*z;
(%o1) z^4-%i*z
(%i2) z2:z1^4-%i*z1;
(%o2) (z^4-%i*z)^4-%i*(z^4-%i*z)
(%i3) z3:z2^4-%i*z2;
(%o3) ((z^4-%i*z)^4-%i*(z^4-%i*z))^4-%i*((z^4-%i*z)^4-%i*(z^4-%i*z))
(%i4) z4:z3^4-%i*z3;
(%o4) (((z^4-%i*z)^4-%i*(z^4-%i*z))^4-%i*((z^4-%i*z)^4-%i*(z^4-%i*z)))^4-%i*(((z^4-%i*z)^4-%i*(z^4-%i*z))^4-%i*((z^4-%i*z)^4-%i*(z^4-%i*z)))
(%i6) taylor(z4,z,0,20);
(%o6)/T/ z+(-76*%i-84)*z^13+(-36*%i+720)*z^16+(1812*%i-2556)*z^19+...

Next term after z is a :

(-76*%i-84)*z^13

so here :

  • k=13 and n=m*q = k-1 = 12
  • a = -76*%i-84

Attracting vectors satisfy :

nav^n = -1

so here :

-12(76i + 84)v^{12} = -1

v^{12} = \frac{1}{912i + 1008}

One can solve it in Maxima CAS :

(%i14) s:map('float,s);
(%o14) [1.007236559448514*%i+1.521106958434882,1.632845927320289*%i+0.81369898815363,
1.820935547602145*%i-0.11173896888541,1.521106958434882*%i-1.007236559448514,
0.81369898815363*%i-1.632845927320289,
-0.11173896888541*%i-1.820935547602145,-1.007236559448514*%i-1.521106958434882,
-1.632845927320289*%i-0.81369898815363,0.11173896888541-1.820935547602145*%i,
1.007236559448514-1.521106958434882*%i,
1.632845927320289-0.81369898815363*%i,0.11173896888541*%i+1.820935547602145]


With arguments in turns :

[0.093087406197659,0.17642073953099,0.25975407286433,0.34308740619766,0.42642073953099,0.50975407286433,
0.59308740619766,0.67642073953099,0.75975407286433,0.84308740619766,0.92642073953099,0.009754072864326]

different then arguments of critical points. Thus critical orbits form distorted 12-arms star


Find the fixed points :

(%i1) f:z^4-%i*z;
(%o1) z^4-%i*z
(%i2) s:solve(f=z);
(%o2) [z=((%i+1)^(1/3)*(sqrt(3)*%i-1))/2,z=-((%i+1)^(1/3)*(sqrt(3)*%i+1))/2,z=(%i+1)^(1/3),z=0]
(%i4) multiplicities;
(%o4) [1,1,1,1]
(%i3) s:map(rhs,s);
(%o3) [((%i+1)^(1/3)*(sqrt(3)*%i-1))/2,-((%i+1)^(1/3)*(sqrt(3)*%i+1))/2,(%i+1)^(1/3),0]
(%i5) s:map('float,s);
(%o5) [0.5*(%i+1.0)^(1/3)*(1.732050807568877*%i-1.0),-0.5*(%i+1.0)^(1/3)*(1.732050807568877*%i+1.0),(%i+1.0)^(1/3),0.0]
(%i6) s:map(rectform,s);
(%o6) [0.7937005259841*%i-0.7937005259841,-1.084215081491351*%i-0.29051455550725,0.29051455550725*%i+1.084215081491351,0.0]


Compute the multiplier of fixed points :

(%i7) d:diff(f,z,1);
(%o7) 4*z^3-%i


Check the stability of fixed points :

(%i9) for z in s do disp(abs(ev(d)));
4.999999999999998
5.0
4.999999999999999
1
(%o9) done

Point z=0 is a parabolic point.

z^4 -zEdit

Critical points, orbits for f(z) = z^4 - z
Julia set for f(z) = z^4 - z
Julia set for f(z) = z^4 - z with various number of star branches ( animation ) - repelling petals

It is a special case of polynomial from family :

f_{\lambda}(z) = z^4 + \lambda z

Here

\lambda = -1 = e^{\pi i}

so internal angle \theta is :

\theta = \frac{p}{q} = \frac{1}{2}

(%i2) m:exp(2*%pi*%i/2);
(%o2) -1


Because :

|\lambda| = 1

it is a parabolic Julia set. Point \lambda = -1 = e^{\pi i} is between two period one components ( root point ).

Periodic points

Point z=0 is a root of multiplicity seven

k = m*q + 1 = 3*2 +1 = 7

for equation :

f^2_{\lambda}(z) = z


One can check it in Maxima CAS using numerical  :

(%i1) z1:z^4-z;
(%o1) z^4-z
(%i2) z2:z1^4-z1;
(%o2) (z^4-z)^4-z^4+z
(%i3) eq2:z2-z=0;
(%o3) (z^4-z)^4-z^4=0
(%i4) allroots(eq2);
(%o4) [z=0.0,z=0.0,z=0.0,z=0.0,z=0.0,z=0.0,z=0.0,z=1.259921049894873,
z=0.7937005259841*%i-0.7937005259841,z=-0.7937005259841*%i-0.7937005259841,
z=1.084215081491351*%i-0.29051455550725,z=-1.084215081491351*%i-0.29051455550725,
z=0.29051455550725*%i+1.084215081491351,
z=1.084215081491351-0.29051455550725*%i,z=1.091123635971722*%i-0.62996052494744,
z=-1.091123635971722*%i-0.62996052494744]
(%i5) expand(eq2);
(%o5) z^16-4*z^13+6*z^10-4*z^7=0
(%i6) factor(eq2);
(%o6) z^7*(z^3-2)*(z^6-2*z^3+2)=0

and symbolic methods :

(%i1) z1:z^4-z;
(%o1) z^4-z
(%i2) solve(z1=z);
(%o2) [z=(2^(1/3)*sqrt(3)*%i-2^(1/3))/2,z=-(2^(1/3)*sqrt(3)*%i+2^(1/3))/2,z=2^(1/3),z=0]
(%i3) multiplicities;
(%o3) [1,1,1,1]
(%i4) z2:z1^4-z1;
(%o4) (z^4-z)^4-z^4+z
(%i5) solve(z2=z);
(%o5) [z=(2^(1/3)*sqrt(3)*%i-2^(1/3))/2,z=-(2^(1/3)*sqrt(3)*%i+2^(1/3))/2,z=2^(1/3),z=((%i+1)^(1/3)*(sqrt(3)*%i-1))/2,z=-((%i+1)^(1/3)*(sqrt(3)*%i+1))/2,z=(%i+1)^(1/3),z=(sqrt(3)*(1-%i)^(1/3)*%i-(1-%i)^(1/3))/2,z=-(sqrt(3)*(1-%i)^(1/3)*%i+(1-%i)^(1/3))/2,z=(1-%i)^(1/3),z=0]
(%i6) multiplicities;
(%o6) [1,1,1,1,1,1,1,1,1,7]

Number of petals = 6 [16]

m*q = 3*2 = 6

Atracting vectors Denominator of internal angle \theta = \frac{p}{q} = \frac{1}{2} is q = 2 so one have to check second iteration of function :

(%i5) z1:z^4-z;
(%o5) z^4-z
(%i6) z2:z1^4-z1;
(%o6) (z^4-z)^4-z^4+z
(%i8) expand(z2);
(%o8) z^16-4*z^13+6*z^10-4*z^7+z

Next term after z is a -4z^7. Then :

  • k = 7 and n=m*q = k-1 = 6
  • a = -4

Attracting vectors satisfy :

nav^n = -1

so here :

-24v^6 = -1


v^6 = \frac{1}{24}

One can solve it using Maxima CAS :

(%i10) s:solve(z^6=1/24);
(%o10) [z=(sqrt(3)*%i+1)/(2^(3/2)*3^(1/6)),z=(sqrt(3)*%i-1)/(2^(3/2)*3^(1/6)),z=-1/(sqrt(2)*3^(1/6)),z=-(sqrt(3)*%i+1)/(2^(3/2)*3^(1/6)),z=-(sqrt(3)*%i-1)/(2^(3/2)*3^(1/6)),z=1/(sqrt(2)*3^(1/6))]
(%i11) s:map(rhs,s);
(%o11) [(sqrt(3)*%i+1)/(2^(3/2)*3^(1/6)),(sqrt(3)*%i-1)/(2^(3/2)*3^(1/6)),-1/(sqrt(2)*3^(1/6)),-(sqrt(3)*%i+1)/(2^(3/2)*3^(1/6)),-(sqrt(3)*%i-1)/(2^(3/2)*3^(1/6)),1/(sqrt(2)*3^(1/6))]
(%i12) s:map('float,s);
(%o12) [0.29439796075012*(1.732050807568877*%i+1.0),0.29439796075012*(1.732050807568877*%i-1.0),-0.58879592150024,-0.29439796075012*(1.732050807568877*%i+1.0),-0.29439796075012*(1.732050807568877*%i-1.0),0.58879592150024]
(%i13) s:map(rectform,s);
(%o13) [0.50991222566388*%i+0.29439796075012,0.50991222566388*%i-0.29439796075012,-0.58879592150024,-0.50991222566388*%i-0.29439796075012,0.29439796075012-0.50991222566388*%i,0.58879592150024]
(%i14) s:map(carg_t,s);
(%o14) [0.5235987755983/%pi,1.047197551196598/%pi,1/2,1-1.047197551196598/%pi,1-0.5235987755983/%pi,0]
(%i15) s:map('float,s);
(%o15) [0.16666666666667,0.33333333333333,0.5,0.66666666666667,0.83333333333333,0.0]

So critical points lie on attracting vectors. Thus critical orbits tend straight to the origin under the iteration[17]


How to compute f_{\lambda}(z) :

(%i2) z:x+y*%i;
(%o2) %i*y+x
(%i3) realpart(z^4-z);
(%o3) y^4-6*x^2*y^2+x^4-x
(%i4) imagpart(z^4-z);
(%o4) -4*x*y^3+4*x^3*y-y

Critical points :

s:GiveListOfCriticalPoints(f(z))
(%o8) [0.54556181798586*%i-0.31498026247372,-0.54556181798586*%i-0.31498026247372,0.62996052494744]

These points has arguments in turns : 1/3, 2/3, 0

z^5 + m*zEdit

dynamic planesEdit

z^5 + zEdit

Julia set z+z^5. Image and src code

Finding roots and its multiplicity :

z^5 + z = z
z^5 = 0

so root z=0 has multiplicity 5. It means that there is a flower with 4 petals [18]

around fixed point z=0.

It How to compute :

(%i23)  z:x+y*%i;
(%o23) %i*y+x
(%i24) realpart(z+z^5);
(%o24) 5*x*y^4-10*x^3*y^2+x^5+x
(%i25) imagpart(z+z^5);
(%o25) y^5-10*x^2*y^3+5*x^4*y+y

In c programs one must use temporary variable so it can be :

tempx = 5*x*y*y*y*y-10*x*x*x*y*y + x*x*x*x*x + x ; // temporary variable
y = y*y*y*y*y -10*x*x*y*y*y + 5*x*x*x*x*y + y ;
x=tempx;

It can be optimized


"...an escape time algorithm would take forever to generate that type of image, since the dynamics are so slow there. If you want resolution of 1/100, it would take roughly 2*10^8 iterates to move the point z0=0.01 to z=2 by iterating f(z)=z+z^5." ( Mark McClure [19]

z^6+m*zEdit

dynamical planeEdit

z6+z on plane [-1.2;1.2]x[-1.2;1.2]. It has 5 petals [20]


z^14 + m*zEdit

dynamic planeEdit

z^14 - zEdit

Julia set for f(z) = z^14 - z . Image and c src code
Critical orbit for f(z) = z^14 - z . Image and Maxima CAS src code

How to compute iteration :

/* Maxima CAS session */
(%i1) z:x+y*%i;
(%o1) %i*y+x
(%i2) z1:z^14-z;
(%o2) (%i*y+x)^14-%i*y-x
(%i3) realpart(z1);
(%o3) -y^14+91*x^2*y^12-1001*x^4*y^10+3003*x^6*y^8-3003*x^8*y^6+1001*x^10*y^4-91*x^12*y^2+x^14-x
(%i4) imagpart(z1);
(%o4) 14*x*y^13-364*x^3*y^11+2002*x^5*y^9-3432*x^7*y^7+2002*x^9*y^5-364*x^11*y^3+14*x^13*y-y


f(z)=z^14-z, on [-1,2;1,2]x[-1,2;1,2] has 26 petals. Compare with image by Michael Becker.[21]

How to find fixed points :

(%i1) z1:z^14-z;
(%o1) z^14-z
(%i2) solve(z1=z);
(%o2) [z=2^(1/13)*%e^((2*%i*%pi)/13),z=2^(1/13)*%e^((4*%i*%pi)/13),
     z=2^(1/13)*%e^((6*%i*%pi)/13),z=2^(1/13)*%e^((8*%i*%pi)/13),
     z=2^(1/13)*%e^((10*%i*%pi)/13),z=2^(1/13)*%e^((12*%i*%pi)/13),
     z=2^(1/13)*%e^(-(12*%i*%pi)/13),z=2^(1/13)*%e^(-(10*%i*%pi)/13),
     z=2^(1/13)*%e^(-(8*%i*%pi)/13),z=2^(1/13)*%e^(-(6*%i*%pi)/13),
     z=2^(1/13)*%e^(-(4*%i*%pi)/13),z=2^(1/13)*%e^(-(2*%i*%pi)/13),
     z=2^(1/13),z=0]
(%i3) multiplicities;
(%o3) [1,1,1,1,1,1,1,1,1,1,1,1,1,1]
(%i4) z2:z1^14-z1;
(%o4) (z^14-z)^14-z^14+z
(%i5) solve(z2=z);
(%o5) [z=2^(1/13)*%e^((2*%i*%pi)/13),z=2^(1/13)*%e^((4*%i*%pi)/13),
   z=2^(1/13)*%e^((6*%i*%pi)/13),z=2^(1/13)*%e^((8*%i*%pi)/13),
   z=2^(1/13)*%e^((10*%i*%pi)/13),z=2^(1/13)*%e^((12*%i*%pi)/13),
   z=2^(1/13)*%e^(-(12*%i*%pi)/13),z=2^(1/13)*%e^(-(10*%i*%pi)/13),
   z=2^(1/13)*%e^(-(8*%i*%pi)/13),z=2^(1/13)*%e^(-(6*%i*%pi)/13),
   z=2^(1/13)*%e^(-(4*%i*%pi)/13),z=2^(1/13)*%e^(-(2*%i*%pi)/13),
   z=2^(1/13),z=0,0=z^78-7*z^65+21*z^52-35*z^39+35*z^26-21*z^13+7,
   0=z^78-5*z^65+11*z^52-13*z^39+9*z^26-3*z^13+1]
(%i6) multiplicities;
(%o6) [1,1,1,1,1,1,1,1,1,1,1,1,1,27,1,1]

z^15 +m*zEdit

dynamic planeEdit

z^15-zEdit

Julia set for f(z)= z^15-z
Critical points and orbits for f(z)= z^15-z

How to compute iterations :

/* Maxima CAS session */
(%i1) z:x+y*%i;
(%o1) %i*y+x
(%i2) z1:z^15-z;
(%o2) (%i*y+x)^15-%i*y-x
(%i3) realpart(z1);
(%o3) -15*x*y^14+455*x^3*y^12-3003*x^5*y^10+6435*x^7*y^8-5005*x^9*y^6+1365*x^11*y^4-105*x^13*y^2+x^15-x
(%i4) imagpart(z1);
(%o4) -y^15+105*x^2*y^13-1365*x^4*y^11+5005*x^6*y^9-6435*x^8*y^7+3003*x^10*y^5-455*x^12*y^3+15*x^14*y-y

Critical points :

(%i1) m:-1;
f:z^15+ m*z;
d:diff(f,z,1);
s:solve(d=0,z)$
s:map(rhs,s)$
s:map(rectform,s)$
s:map('float,s);
multiplicities;
(%o1) -1
(%o2) z^15-z
(%o3) 15*z^14-1
(%o7) 
[0.35757475986465*%i+0.74251163973317,
0.64432745317147*%i+0.51383399763062,
0.80346319222004*%i+0.1833852305369,
0.80346319222004*%i-0.1833852305369,
0.64432745317147*%i-0.51383399763062,
0.35757475986465*%i-0.74251163973317,
-0.8241257452789,
-0.35757475986465*%i-0.74251163973317,
-0.64432745317147*%i-0.51383399763062,
-0.80346319222004*%i-0.1833852305369,
0.1833852305369-0.80346319222004*%i,
0.51383399763062-0.64432745317147*%i,
0.74251163973317-0.35757475986465*%i,
0.8241257452789]
(%o8) [1,1,1,1,1,1,1,1,1,1,1,1,1,1]

It means that here are 14 critical points and 14 critical orbits.

Fixed points :

kill(all);
remvalue(all);


/*------------- functions definitions ---------*/

/* function */
f(z):=z^15 -z;

/* find fixed points  returns a list */
GiveFixedPoints():= block
(
  [s],
  s:solve(f(z)=z),
  /* remove "z="  from list s */
  s:map('rhs,s),
  s:map('rectform,s),
  s:map('float,s),
  return(s)
)$

compile(all);


ff:GiveFixedPoints();
multiplicities;
length(s);


for i:1 thru length(ff) step 1 do
  (z:ff[i],
   disp("z= ",z, " abs(d(z))= ",abs(15*z^14-1)));


Result is :

(%i12) ff:GiveFixedPoints()
(%o12) [0.45590621928146*%i+0.94669901916834,0.82151462051137*%i+0.65513604843564,1.024411975933374*%i+0.23381534859391,
1.024411975933374*%i-0.23381534859391,0.82151462051137*%i-0.65513604843564,0.45590621928146*%i-0.94669901916834,-1.050756638653219,-
0.45590621928146*%i-0.94669901916834,-0.82151462051137*%i-0.65513604843564,-1.024411975933374*%i-0.23381534859391,0.23381534859391-
1.024411975933374*%i,0.65513604843564-0.82151462051137*%i,0.94669901916834-0.45590621928146*%i,1.050756638653219,0.0]
(%i13) multiplicities
(%o13) [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
(%i14) length(s)
(%o14) 14
(%i15) for i thru length(ff) do (z:ff[i],disp("z= ",z," abs(d(z))= ",abs(15*z^14-1)))
z= 0.45590621928146*%i+0.94669901916834 ; abs(d(z))= 28.99999999999996
z= 0.82151462051137*%i+0.65513604843564 ; abs(d(z))= 28.99999999999998
z= 1.024411975933374*%i+0.23381534859391 ; abs(d(z))= 28.99999999999999
z= 1.024411975933374*%i-0.23381534859391 ; abs(d(z))=  28.99999999999997
z= 0.82151462051137*%i-0.65513604843564 ; abs(d(z))= 29.00000000000001
z= 0.45590621928146*%i-0.94669901916834 ; abs(d(z))=  28.99999999999995
z= -1.050756638653219 ;                   abs(d(z))= 29.00000000000003
z=  -0.45590621928146*%i-0.94669901916834; abs(d(z))= 28.99999999999995
z= -0.82151462051137*%i-0.65513604843564; abs(d(z))=  29.00000000000001
z= -1.024411975933374*%i-0.23381534859391 ; abs(d(z))= 28.99999999999997
z= 0.23381534859391-1.024411975933374*%i abs(d(z))= 28.99999999999999
z= 0.65513604843564-0.82151462051137*%i ; abs(d(z))= 28.99999999999998
z= 0.94669901916834-0.45590621928146*%i ; abs(d(z))=  28.99999999999996
z= 1.050756638653219 ;                    abs(d(z))= 29.00000000000003
z= 0.0 ;                                   abs(d(z))= 1.0

So only z=0 is parabolic fixed points, the rest of them are repelling

Inside dynamics under generalized power settingEdit

The orbit dynamics of the set can become more complex when the power is something other than 2.0. The iterated function can become multivalued and the structure of the set is then affected by the 'arbitrary' choice of which value is chosen.

ReferencesEdit

  1. ON THE NOTIONS OF MATING by CARSTEN LUNDE PETERSEN AND DANIEL MEYER
  2. Inverse Iteration Algorithms for Julia Sets by Mark McClure
  3. wikipedia : Multibrot_set
  4. High-Order Mandelbrot and Julia Sets Written by Christopher Thomas. Images and Perl program
  5. Stephen Haas : The Hausdorff Dimension of the Julia Set of Polynomials of the Form z^d+c
  6. Desenvolupament_de_fractals_mitjan
  7. De la génération des fractales Théorie et Applications John Bonobo
  8. C. McMullen, Automorphisms of rational maps. In `Holomorphic Functions and Moduli I', 31-60, Springer-Verlag, 1988.
  9. Hyperbolic components of McMullen maps Weiyuan Qiu, Pascale Roesch, Xiaoguang Wang, Yongcheng Yin
  10. "Visualizing the complex dynamics of families of polynomials with symmetric critical points" by Ning Chena, Jing Suna, Yan-ling Suna, Ming Tangb. Chaos, Solitons & Fractals. Volume 42, Issue 3, 15 November 2009, Pages 1611–1622
  11. A topological characterisation of holomorphic parabolic germs in the plane Fr ́ed ́eric Le Roux
  12. One approach to the digital visualization of hedgehogs in holomorphic dynamics Alessandro Rosa
  13. Mandel: software for real and complex dynamics by Wolf Jung
  14. Complex dynamics, Lennart Carleson, Theodore W. Gamelin, Springer, 1993, ISBN 978-0-387-97942-7. Page 40, Figure 2.
  15. F. Bracci, Local holomorphic dynamics of diffeomorphisms in dimension one. Contemporary Mathematics 525, (2010), 1-42. Notes of the PhD course given in A.A 2007/08.
  16. Complex dynamics, Lennart Carleson, Theodore W. Gamelin, Springer, 1993, ISBN 978-0-387-97942-7. Page 41
  17. Mark McClure in stackexchange questions : what-is-the-shape-of-parabolic-critical-orbit
  18. One approach to the digital visualization of hedgehogs in holomorphic dynamics Alessandro Rosa
  19. stackexchange questions : what-is-the-shape-of-parabolic-critical-orbit
  20. Some Julia sets 2 by Michael Becker
  21. Fixpunkte und Periodische Punkte by Michael Becker