Julia sets for polynomial functions

• 
  The elephants zone in z^2+z^3+c fractal
• roots

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

Map${\displaystyle z(1+z{)}^d}$ has d petals. "Although the parabolic point at z = 0 has only one petal, the map f also has a preparabolic critical point b = −1 of local degree d. Thus f has d petals at b. "^[2]

For example "the filled Julia set of f (z) = z(1 + z) 3 has three petals at z = −1." ( Curtis T McMullen )

"The condition

${\displaystyle z^{kn}\in R}$

describes a set of lines through the origin. Note that the map g

${\displaystyle g(z)=z(1+z^{kn})}$

leaves these lines invariant. Then

${\displaystyle z\to \lambda \ast z(1+z^{kn})}$

is the composition of g with the periodic rotation "^[3]

${\displaystyle z\to \lambda z}$

Example

• f(z) = z^3 + z + 0.6*I^[4]
• f(z) = z^6+A*z +c ; degree 6, A=(-34603008+0*i)*2^-25, c=(-262144+0i)*2^-25 from another parameter space slice showing 5 attracting period-2 cycles by marcm200^[5]
• London towers" Period-3 cycle and Period-18 cycle , z^5+A*z+c c=(638976+7536640*i) * 2^-25 A=(37094161+719769*i) * 2^-25 by marcm200^[6]

Z = Zⁿ + c^p

L = (m - 1) * p

• 
  Z = Z² + C²
  L = (2 - 1)* 2 =2
• 
  Z = Z² + C³
  L = (2 - 1)* 3 =3
• 
  Z = Z²+C⁶ - 1
  L = (2 - 1)* 6 =6
• 
  Z = Z³ + C²
  L = (3 - 1)* 2 =4
• 
  Z = Z³ + C³
  L = (3 - 1)* 3 =6
• 
  Z = Z⁴ + C⁴
  L = (4 - 1)* 4 =12

• 
  ${\displaystyle f(z):=z^5-1}$

  "the mulitbrot family, ... they all have a single critical point, namely the origin"   Mark McClure[7]

Multibrot sets^{[8][9][10][11]}

L = m - 1

Compare with it's inverse parameter plane : Z^n + 1/c^[12]

How to compute powers of z :

${\displaystyle Z^2=(x+iy{)}^2}$${\displaystyle \mathrm{R}\mathrm{e}\mathrm{a}\mathrm{l}=x^2-y^2}$${\displaystyle \mathrm{I}\mathrm{m}\mathrm{a}\mathrm{g}=2xy}$

${\displaystyle Z^3=(x+iy{)}^3}$${\displaystyle \mathrm{R}\mathrm{e}\mathrm{a}\mathrm{l}=x^3-3y^{2}x}$${\displaystyle \mathrm{I}\mathrm{m}\mathrm{a}\mathrm{g}=3x^{2}y-y^3}$

${\displaystyle Z^4=(x+iy{)}^4}$${\displaystyle \mathrm{R}\mathrm{e}\mathrm{a}\mathrm{l}=x^4-6x^2{y}^2+y^4}$${\displaystyle \mathrm{I}\mathrm{m}\mathrm{a}\mathrm{g}=4x^{3}y-4x{y}^3}$

${\displaystyle Z^5=(x+iy{)}^5}$${\displaystyle \mathrm{R}\mathrm{e}\mathrm{a}\mathrm{l}=x^5-10x^3{y}^2+5x{y}^4}$${\displaystyle \mathrm{I}\mathrm{m}\mathrm{a}\mathrm{g}=5x^{4}y-10x^2{y}^3+y^5}$

${\displaystyle Z^6=(x+iy{)}^6}$${\displaystyle \mathrm{R}\mathrm{e}\mathrm{a}\mathrm{l}=x^6-15x^4{y}^2+15x^2{y}^4-y^6}$${\displaystyle \mathrm{I}\mathrm{m}\mathrm{a}\mathrm{g}=6x^{5}y-20x^3{y}^3+6x{y}^5}$

${\displaystyle Z^7=(x+iy{)}^7}$${\displaystyle \mathrm{R}\mathrm{e}\mathrm{a}\mathrm{l}=x^7-21x^5{y}^2+35x^3{y}^4-7x{y}^6}$${\displaystyle \mathrm{I}\mathrm{m}\mathrm{a}\mathrm{g}=7x^{6}y-35x^4{y}^3+21x^2{y}^5-y^7}$

Fortan source code :

! Fortran program by P.M.J. Trevelyan! http://philiptrevelyan.co.uk/PROGRAMFRACTALIMPLICIT NONEINTEGERI,J,ITERATION,N,MPARAMETER(N=2000,M=50)REAL*8U,V,X,Y,P,QOPEN(99,FILE='Fractal_quad.dat')25FORMAT(2F9.5,I3)DO10,I=1,NDO20,J=1,NCDefinefirstpointz(n)=U+iVandk=X+iYU=0.D0V=0.D0X=I*3.2D0/(N-1.D0)-2.1D0Y=J*2.8D0/(N-1.D0)-1.4D0DO30,ITERATION=1,MCCalculatez(n+1)=z(n)**2+kwherez(n+1)=P+iQP=U**2-V**2+XQ=2.D0*U*V+YU=PV=QCIf|z|>2stopiteratingIf(U**2+V**2.GT.4.D0)GOTO10030CONTINUE100WRITE(99,25)X,Y,ITERATION20CONTINUE10CONTINUE      STOP      END

Parameter planes for positive integer powers :

• 
  z² +c
• 
  z³ +c
• 
  z⁴ +c
• 
  z⁵ +c
• 
  z⁶ +c

Negative powers

Whend 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 +c
• 
  z⁻³ +c
• 
  z⁻⁴ +c
• 
  z⁻⁵ +c
• 
  z⁻⁶ +c

Seecomplex quadratic polynomials

${\displaystyle z_{n+1}=z_n^3+c}$

It can be computed by :^[13]

${\displaystyle X_{n+1}=X_n^3-3X_n{Y}_n^2+C_x}$

${\displaystyle Y_{n+1}=3X_n^2{Y}_n-Y_n^3+C_y}$

Examples:

• c= -0.040000000000000036 + I * -0.78^[14]

• 
  Parameter plane and Mandelbrot set
• 
  Dynamic plane and Julia set for${\displaystyle f(z)=z^3+c}$ where c = ?
• 
  zeros of the function${\displaystyle z^3-1}$ in the complex plane
• 
  complex color plot of the function${\displaystyle z^3-1}$ in the complex plane
• 
• 
  c = -0.4730679 -0.5625i

• 
  Parameter plane and Mandelbrot set
• 
  zeros of the function${\displaystyle z^5-1}$ in the complex plane
• 
  complex color plot of the function${\displaystyle z^5-1}$ in the complex plane
• 

• 
  Julia Set fractal using z = z^10 + c where c = -0.925 + 0.19i

It is aninverted parameter plain of z^n + c.

Number of vertices : V = (n - 1)

• 
  Z = Z² + 1/C
• 
  Z = Z³ + 1/C
• 
  Z = Z⁴ + 1/C
• 
  Z = Z⁵ + 1/C
• 
  Z = Z⁶ + 1/C
• 
  Z = Z⁷ + 1/C

McMullen maps :

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

where : n and d are >=1

"These maps are known as `McMullen maps', since McMullen^[15] first studied these maps and pointed out that when (n: d) = (2: 3) and m is small, the Julia set is a Cantor set of circles."^[16]

Description^[17][18]

Seecomplex quadratic polynomial

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 :

${\displaystyle z^3+z=z}$

${\displaystyle z^3=0}$

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 aflower with 2 petals around fixed point z=0.

Compare figures :

• by Alessandro Rosa^[19]
• by Xavier Buff and Adam L. Epstein^[20]

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^[21] ( see main menu / New / 4. Quartic polynomials / 4.1 )

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

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 )

Finding roots and its multiplicity :

${\displaystyle z^4+z=z}$

${\displaystyle z^4=0}$

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^[22]

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

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^[23]

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 :

${\displaystyle na{v}^n=-1}$

so here :

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

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

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

Findthe 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]

Computethe multiplier of fixed points :

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

Checkthe stability of fixed points :

(%i9) for z in s do disp(abs(ev(d)));4.9999999999999985.04.9999999999999991(%o9) done

Point z=0 is a parabolic point.

It is a special case of polynomial from family :

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

Here

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

so internal angle${\displaystyle \theta }$ is :

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

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

Because :

${\displaystyle |\lambda |=1}$

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

Periodic points

Point z=0 is a root of multiplicity seven

${\displaystyle k=m\ast q+1=3\ast 2+1=7}$

for equation :

${\displaystyle f_{\lambda }^2(z)=z}$

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^[24]

${\displaystyle m\ast q=3\ast 2=6}$

Atracting vectorsDenominator of internal angle${\displaystyle \theta =\frac{p}{q}=\frac{1}{2}}$ is${\displaystyle q=2}$ 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 :

${\displaystyle na{v}^n=-1}$

so here :

${\displaystyle -24v^6=-1}$

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

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^[25]

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

(%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

Finding roots and its multiplicity :

${\displaystyle z^5+z=z}$

${\displaystyle z^5=0}$

so root z=0 has multiplicity 5. It means that there is aflower with 4 petals^[26]

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 variabley=y*y*y*y*y-10*x*x*y*y*y+5*x*x*x*x*y+y;x=tempx;

It can beoptimized

"...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^[27]

"This picture shows the Julia Set of f(z) = z + z^5 which has an indifferent fixed point at z = 0. ( f(0) = 0 and f ' (0) = 1 .)

The 4 lines :

Re z = 0 and Im z = 0 and Re z = Im z and Re z = -Im z

are invariant under iteration of f.

On Im z = 0: f(x) = x + x^5

On Re z = 0: f(ix) = ix + (ix)^5 = ix + i^5 x^5 = i(x+x^5)

On Re z = Im z , f(z) = r e^(i pi/4) + r^5 e^(i 5 pi/4) = e^ (i pi/4)(r - r^5)

On Re z = -Im z, f(z) = = r e^(i 3pi/4) + r^5 e^(i 15 pi/4) = e^ (i 3pi/4)(r - r^5)

Using one dimensional analysis it is easily shown that f(x) = x + x^5 has a repelling fixed point at x = 0 and f(x) = x - x^5 has an attracting fixed point at x = 0. Thus along the four invariant lines 0 is attracting on the first two and repelling on the second two. The points repelled from 0 are shown in shades of blue, while those attracted to 0 are shown in shades of brown. 0 has four attracting petals, which are in shades of brown. (A simply connected region C is a petal for an indifferent fixed point p if p is contained in the boundary of C and for each z in C,

F^n(z) -> p (see Devaney - 1987)"^[28]

c=(-6145144-20171676*i) * 2^-25 = (-6145144 - 20171676 i)/2^25 = -0.1831395626068115234375 - 0.60116279125213623046875 i

Critical points:

• using WolframAlfa

• 
  Julia set for fc(z)= z^6+A*z+c where c = 4.6875e-1 - 5.703125e-1 *I and A = 6.96854889392852783203125e-2 - 1.07958018779754638671875e-1*I.png]]
• 

List:

z = -0.454407 + 0.0918858 I z = -0.227808 - 0.403772 I z = -0.0530308 + 0.460561 I z = 0.313614 - 0.341431 I z = 0.421632 + 0.192756 I

Higher precision

+0.4216319827875524+0.1927564710317439*%i-0.4544068504035357+0.09188580053693407*%i-0.2278080284907348 -0.4037723222177803*%i+0.3136137458861571-0.3414308193839966*%i-0.05303084977943909+0.4605608700330989*%i

dynamical plane

z6+z on plane [-1.2;1.2]x[-1.2;1.2]. It has 5 petals^[29]

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.^[30]

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]

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))= 28.99999999999996z= 0.82151462051137*%i+0.65513604843564 ; abs(d(z))= 28.99999999999998z= 1.024411975933374*%i+0.23381534859391 ; abs(d(z))= 28.99999999999999z= 1.024411975933374*%i-0.23381534859391 ; abs(d(z))=  28.99999999999997z= 0.82151462051137*%i-0.65513604843564 ; abs(d(z))= 29.00000000000001z= 0.45590621928146*%i-0.94669901916834 ; abs(d(z))=  28.99999999999995z= -1.050756638653219 ;                   abs(d(z))= 29.00000000000003z=  -0.45590621928146*%i-0.94669901916834; abs(d(z))= 28.99999999999995z= -0.82151462051137*%i-0.65513604843564; abs(d(z))=  29.00000000000001z= -1.024411975933374*%i-0.23381534859391 ; abs(d(z))= 28.99999999999997z= 0.23381534859391-1.024411975933374*%i abs(d(z))= 28.99999999999999z= 0.65513604843564-0.82151462051137*%i ; abs(d(z))= 28.99999999999998z= 0.94669901916834-0.45590621928146*%i ; abs(d(z))=  28.99999999999996z= 1.050756638653219 ;                    abs(d(z))= 29.00000000000003z= 0.0 ;                                   abs(d(z))= 1.0

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

• multibrot^[31]
• z^2 +cz^5^[32]

Description^[33]

• map :${\displaystyle f(z)=z^5+(0.8+0.8i)z^4+z}$
• coefficients in ascending order ( from ao to an): 0,1,0,0, 0.8+0.8i, 1
• ListOfCriticalPoints [(- 0.7558074500261052 %i) - 0.7558074500261052, 0.2793534499540583 %i - 0.5310341598343944, 0.2793534499540583 - 0.5310341598343943 %i, 0.3674881599064412 %i + 0.3674881599064413]
• fixed points : [(- 0.8 %i) - 0.8, 0.0] with stability : [0.6384000000000008, 1.0]

• 
  Parabolic Julia set for f(z)=z^5+m*z^4+z where m = 0.8+0.4*i
• 
  critical orbits

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

• commons:Category:Complex polynomial maps
• Derivative of iterated fuction
• Mark McClure : visualization of polynomial_julia_sets, coefficients a0, a1, a2, a3, .... for f(z) = an*z^n+ ... + a2*z^2 + a1*z + a0, example  : 1.42+0.37i,-1.5,0,1
• Exploring the 4-Dimensional Mandelbrot Set
• Elliptic curve julia sets by Clifford A. Reiter
• FF: Julia sets: True shape and escape time

1. ↑ON THE NOTIONS OF MATING by CARSTEN LUNDE PETERSEN AND DANIEL MEYER
2. ↑dimension and conformal dynamics II: Geometrically finite rational maps by Curtis T McMullen
3. ↑COMPLEX ANALYTIC DYNAMICS ON THE RIEMANN SPHERE BY PAUL BLANCHARD
4. ↑Inverse Iteration Algorithms for Julia Sets by Mark McClure
5. ↑fractalforums.org: julia-sets-true-shape-and-escape-time
6. ↑fractalforums.org: julia-sets-true-shape-and-escape-time
7. ↑Mark McClure - comment
8. ↑wikipedia : Multibrot_set
9. ↑High-Order Mandelbrot and Julia Sets Written by Christopher Thomas. Images and Perl program
10. ↑Stephen Haas : The Hausdorff Dimension of the Julia Set of Polynomials of the Form z^d+c
11. ↑video on youtube by rrwick
12. ↑Desenvolupament_de_fractals_mitjan
13. ↑De la génération des fractales Théorie et Applications John Bonobo
14. ↑wolfram : julia-set-explorations
15. ↑C. McMullen, Automorphisms of rational maps. In `Holomorphic Functions andModuli I', 31-60, Springer-Verlag, 1988.
16. ↑Hyperbolic components of McMullen maps Weiyuan Qiu, Pascale Roesch, Xiaoguang Wang, Yongcheng Yin
17. ↑"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
18. ↑A topological characterisation of holomorphic parabolic germs in the plane Fr ́ed ́eric Le Roux
19. ↑One approach to the digital visualization of hedgehogs in holomorphic dynamics Alessandro Rosa, fig 5.13 page 26
20. ↑A parabolic Pommerenke-Levin-Yoccoz inequality by Xavier Buff and Adam L. Epstein, fig 1 page 4
21. ↑Mandel: software for real and complex dynamics by Wolf Jung
22. ↑Complex dynamics, Lennart Carleson, Theodore W. Gamelin, Springer, 1993,ISBN 978-0-387-97942-7. Page 40, Figure 2.
23. ↑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.
24. ↑Complex dynamics, Lennart Carleson, Theodore W. Gamelin, Springer, 1993,ISBN 978-0-387-97942-7. Page 41
25. ↑Mark McClure in stackexchange questions : what-is-the-shape-of-parabolic-critical-orbit
26. ↑One approach to the digital visualization of hedgehogs in holomorphic dynamics Alessandro Rosa
27. ↑stackexchange questions : what-is-the-shape-of-parabolic-critical-orbit
28. ↑Fractals by Anne Burns, Department of Mathematics, C.W. Post Campus, Long Island University
29. ↑Some Julia sets 2 by Michael Becker
30. ↑Fixpunkte und Periodische Punkte by Michael Becker
31. ↑multibrot by Claude
32. ↑z^2 +cz^5
33. ↑math.stackexchange question: whats-the-difference-between-a-total-basin-of-attraction-and-an-immediate-basin

• Book:Fractals