# Fractals/Iterations in the complex plane/siegel

For Siegel disc [1] [2]parameter c :

• should be on boundary of hyperbolic component of Mandelbrot set ( internal radius = 1)
• internal angle should be irrational number between 0 and 1 . [3]

Because set of irrationals is uncountable[4] so the number of Julia sets with Siegel disc is infinite.

# DefinitionsEdit

## CenterEdit

Center of Siegel disc is a irrationally indifferent periodic point.

Mane's theorem :

"... appart from its center, a Siegel disk cannot contain any periodic point, critical point, nor any iterated preimage of a critical or periodic point. On the other hand it can contain an iterated image of a critical point." [5]

## Center component of Julia setEdit

In case of c on boundary of main cardioid center component of Julia set is a component containing Siegel disc ( and its center). Critical orbit is a boundary of Siegel disc and center component. All other components are preimages of this component ( see animated image using inverse iterations ).

Inner radius of Siegel Disc =

• radius of inner circle, where inner circle with center at fixed point is the biggest circle inside Siegel Disc.
• minimal distance between center of Siel Disc and critical orbit

Code for computing internal radius :

#### Maxima CAS codeEdit

Here orbit is a list of complex points z.

f(z,c):=z*z+c $GiveCriticalOrbit(c,iMax):= block( ER:2.0, /* Escape Radius */ z:0+0*%i, /* first point = critical point */ orbit:[z], if (abs(z)>ER) then return(orbit), i:0, loop, /* compute forward orbit */ z:rectform(f(z,c)), orbit:endcons(z,orbit), i:i+1, if ((abs(z)<ER) and (i<iMax)) then go(loop), return(orbit) )$

/* find fixed point alfa */
GiveFixed(c):= float(rectform((1-sqrt(1-4*c))/2))$/* distance between point z and fixed point zf */ GiveDistanceFromCenterTo(z):= abs(z-zf)$

/* inner radius of Siegel Disc ; criticla orbit is a boundary of SD */
GiveInnerRadiusOf(orbit):=lmin(map(GiveDistanceFromCenterTo,orbit))$/* outer radius of Siegel Disc ; criticla orbit is a boundary of SD */ GiveOuterRadiusOf(orbit):=lmax(map(GiveDistanceFromCenterTo,orbit))$

/*------------ const ---------------------------------*/
c:0.113891513213121  +0.595978335936124*%i; /* fc(z) = z*z + c */
NrPoints:400000;

/* ----------- main ---------------------------------------------------*/
zf:GiveFixed(c); /* fixed point = center of Siegel disc */
orbit:GiveCriticalOrbit(c,NrPoints)\$


#### C codeEdit

/* fc(z) = z*z + c */
const double Cx, Cy; /* C = Cx + Cy*i */

/*   z fixed ( z=z^2 +c )   it is a  center of Siegel Disc */
const double zfx, zfy; /* zf = zfx + zfy*i */

double GiveDistanceFromCenter(double zx, double zy)
{double dx,dy;

dx=zx-zfx;
dy=zy-zfy;
return sqrt(dx*dx+dy*dy);

}

{ /* compute critical orbit and finds smallest distance from fixed point */
int i; /* iteration */
double Zx=0.0, Zy=0.0; /* Z = Zx + Zy*i */
double Zx2, Zy2; /* Zx2=Zx*Zx;  Zy2=Zy*Zy  */
/* center of Siegel disc */

double Distance;
double MinDistance =2.0;

Zx2=Zx*Zx;
Zy2=Zy*Zy;
for (i=0;i<=40000 ;i++) /* to small number of iMax gives bad result */
{
Zy=2*Zx*Zy + Cy;
Zx=Zx2-Zy2 +Cx;
Zx2=Zx*Zx;
Zy2=Zy*Zy;
/* */

Distance= GiveDistanceFromCenter(Zx,Zy);
if (MinDistance>Distance) MinDistance=Distance; /* smallest distance */
}
return MinDistance;
}


Outer circle with center at fixed point is minimal circle containing Siegel disc.

## FoldingEdit

Folding in scholaredia[8]

## Siegel disk implosionEdit

" ... an arbitrary small change of the multiplier of the Siegel point may lead to an implosion of the Siegel disk - its inner radius collapses to zero " [9]

Examples :

• from Golden Mean Siegel Disc with rotation number = [0;1,1,1,1,1, ...]= 0.618033988957902 to parabolic Julia set with rotation number (internal angle) = 5/8 = [0;1,1,1,1,1]
• SEMI-CONTINUITY OF SIEGEL DISKS UNDER PARABOLIC IMPLOSION  : P(n) = [0, 2, 2, n + r] with r = (√5−1)/ 2 , the first three images show the Siegel disk of P(n) for n = 10, 500, 10000, and the last is the virtual Siegel disk of p/q = 2/5 they tend to. Here P(n) > p/q.[10]

# TypesEdit

for c :

• on boundary of main cardioid are siegel discs around fixed point alfa
• on boundary of period n component are periodic Siegel discs around n-periodic points

These Siegel Discs are :

• bounded (because the Julia set is bounded)
• not smooth (differentiable) in general

## Around fixed pointEdit

• Julia set consist of infinitely many curves bounding open regions
• $f_c\,$ maps each region into "larger" one, until the region containing the fixed point is reached
• inside component containing Siegel disc $f_c\,$ rotates points on invariant loops around the fixed point[11]

so in other words :

"$K_c$ has infinitely many components. One of these contains the alpha fixed point. It is mapped to itself by $f_c$. This is the Siegel disk."

"The Siegel disk is one component and it has infinitely many preimages. If you zoom in to z = 0 with large nmax, you shall see that there are two components touching at z=0". ( Wolf Jung )

# VisualisationEdit

Visualisation of dynamical plane :

• critical orbit by forward iteration of critical point
• Julia set by
• boundary of center component by drawing of critical orbit
• whole Julia set by inverse iteration of critical orbit
• interior
• whole interior by average velocity by Chris King
• Siegel disc orbits by forward iteration of point inside or on boundary of Siegel Disc

## OptimisationEdit

### Trick 1Edit

If point ( z0 or its forward image zn) is inside inner circle then it is interior point.

# ExamplesEdit

CFE Rotation number c internal adres center z period inner R outer R
[0; 1, 1, 1,...] 0.618033988957902 -0.390540870218399-0.586787907346969i -0.368684439039160-0.337745147130762i 1 0.25 0.4999
[0;3,2,1000,1...] .2857346725405882 0.113891513213121+0.595978335936124i -0.111322907374331+0.487449700270424i 1 .1414016645283217 .5285729063154562

## Boundary of main cardioidEdit

• examples by Jay Hill[12]

### ParametrizationEdit

The boundary of main cardioid is described by 2 simultaneous equations :

• describing fixed point ( period 1 ) under f
• describing multiplier[13] ( stability ) of fixed point ( it should be indifferent)

$\begin{cases} z_1^2 + c = z_1 \\ abs(\lambda (z_1)) = 1 \end{cases}$

where :

$z_1 = \left\{ z : f_c(z) = z \right\} \,$

$\lambda (z_1) = 2*z_1 \,$

First equation :

$z_1^2 + c = z_1$

$c = - z_1^2 + z_1$

Second equation :

$\lambda (z_1) = 1 \,$

$Abs(2z_1) = 1\,$

$2z_1 = e^{2*\Pi*t*i} \,$

$z_1 = e^{\Pi*t*i} = cos(t) + i*sin(t)\,$

As a result one gets function describing relation between parameter c and internal angle t :

$c = c(t) = \frac\lambda2\left(1-\frac\lambda2\right)= \frac{e^{\Pi*t*i}}{2} - \frac{e^{2*\Pi*t*i}}{4}$

It is used for computing :

• c point of boundary of main cardioid
• when t is irrational number then filled Julia set has Siegel disc

One can compute boundary point c

$c = c_x + c_y*i$

of period 1 hyperbolic component ( main cardioid) for given internal angle ( rotation number) t using this code by Wolf Jung[14]

t *= (2*PI); // from turns to radians
cx = 0.5*cos(t) - 0.25*cos(2*t);
cy = 0.5*sin(t) - 0.25*sin(2*t);



### examplesEdit

#### the Golden MeanEdit

approximated of golden mean by finite continued fractions

rotation number t is the Golden Mean :[15]

$t = \frac{\sqrt{5} - 1}{2} \approx 0.618033988749895$

Continued_fraction expansion :[16]

$t = [0; 1, 1, 1, \dots] = [ 1, 1, 1, \dots]= \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{1 + \ddots}}}$

In Maxima CAS :

(%i2) a:[0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
(%o2) [0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
(%i3) t:cfdisrep(a)
(%o3) 1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/1))))))))))))))))))))))
(%i4) float(t)
(%o4) 0.618033988957902
(%i5) l:%e^(2*%pi*%i*t)
(%o5) %e^(-(17711*%i*%pi)/23184)
(%i6) c:(l*(1-l/2))/2
(%o6) ((1-%e^(-(17711*%i*%pi)/23184)/2)*%e^(-(17711*%i*%pi)/23184))/2
(%i7) float(rectform(c))
(%o7) -.5867879078859505*%i-.3905408691260131


so

$c \approx -0.390540870218399 -0.586787907346969 *i$

Compare with :

• Quadratic Julia sets depicted by combined methods[17]
• Golden mean Siegel disk by Curtis T McMullen [18]
• Siegel Disk Fractal by Jim Muth [19]
• Siegel disk by Davoud Cheraghi[20]
• a bettter Siegel disk program in Mathematica From Roger Bagula[21]
• Xander's image [22]
• Images by Arnaud Chéritat[23]

#### QUADRATIC SIEGEL DISKS WITH SMOOTH BOUNDARIESEdit

Example by XAVIER BUFF AND ARNAUD CHERITAT[24]

α = ( 5 + 1)/2 = [1, 1, 1, 1, . . .],

α(1) = [1, 1, 1, 1, 1, 1, 25, 1, 1, 1, . . .]

α(2) = [1, 1, 1, 1, 1, 1, 25, 1010 , 1, 1, 1, . . .]


#### [3,2,1000,1...]Edit

finite continued fractions aproximation to [0;1,1,1,....]

rotation number t is : [25]

$t = [3,2,1000,1,...] = [0; 3,2,1000,1 \dots] = \cfrac{1}{3 + \cfrac{1}{2 + \cfrac{1}{1000 + \cfrac{1}{1 + \ddots}}}}$

In Maxima CAS one can compute it :

(%i2) kill(all)
(%o0) done
(%i1)  a:[0,3,2,1000,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
(%o1)  [0,3,2,1000,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
(%i2) t:cfdisrep(a)
(%o2) (1)/(3+(1)/(2+(1)/(1000+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+ (1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/1)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))
(%i3) float(t)
(%o3) .2857346725405882
(%i4) l:%e^(2*%pi*%i*t)
(%o4) %e^((2*%i*%pi)/(3+(1)/(2+(1)/(1000+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/1))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))
(%i5) c:(l*(1-l/2))/2
(%o5) ((1-(%e^((2*%i*%pi)/(3+(1)/(2+(1)/(1000+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/1)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))/(2))*%e^((2*%i*%pi)/(3+(1)/(2+(1)/(1000+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+(1)/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/1)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))/(2)
(%i6) float(rectform(c))
(%o6) .5959783359361234*%i+.1138915132131216


So

t = .2857346725405882
c = 0.113891513213121  +0.595978335936124 i


How one can find limit of [3,2,1000,1,...] ?

Here is explanation of Bill Wood

"I don't know if Maxima knows much about the algebra of continued fractions, but it can be of some help hacking out the manipulation details of a derivation. A most useful fact is that

   [a1, a2, a3, ...] = a1 + 1/[a2, a3, ...]


provided the continued fraction converges. If we apply that three times by hand to [3, 2, 1000, 1, 1, ...] we obtain

   3 + 1/(2 + 1/(1000 + 1/[1, 1, ...]))


Now it is known that [1, 1, ...] converges to the Golden Ratio = (1+sqrt(5))/2. So now we can use Maxima as follows:

 (%i20) 3+(1/(2+(1/(1000+(1/((1+sqrt(5))/2))))));
1
(%o20)                    ---------------------- + 3
1
------------------ + 2
2
----------- + 1000
sqrt(5) + 1
(%i21) factor(%o13);
7003 sqrt(5) + 7017
(%o21)                        -------------------
2001 sqrt(5) + 2005
(%i22) %o21,numer;
(%o22)                         3.499750279196346


You set a to [0, 3, 2, 1000, 1, 1, ...], which by our useful fact must be the reciprocal of [3, 2, 1000, 1, 1, ...], and indeed the reciprocal of 3.499750279196346 is 0.2857346725405882, which is what your float(t) evaluates to, so we seem to get consistent results.

If all of the continued fractions for the rotation numbers exhibited on the link you provided do end up repeating 1 forever then the method I used above can be used to determine their limits as ratios of linear expressions in sqrt(5)." Bill Wood