# Fractals/Iterations in the complex plane/Julia set

   "... a single algorithm for computing all quadratic Julia sets does not exist."

This book shows how to code different algorithms for drawing sets in dynamical plane : Julia, Filled-in Julia or Fatou sets for complex quadratic polynomial. It is divided in 2 parts :

• description of various algorithms
• descriptions of technics for visualisation of various sets in dynamic plane
• Julia set
• Fatou set
• basin of attraction of infinity ( open set)
• basin of attraction of finite attractor

# Algorithms

## Methods based on speed of attraction

Here color is proportional to speed of attraction ( convergence to attractor). These methods are used in Fatou set.

How to find:

• lowest optimal bailout values ( IterationMax) ? 

### Basin of attraction to infinity = exterior of filled-in Julia set and The Divergence Scheme = Escape Time Method ( ETM )

Here one computes forward iterations of a complex point Z0:

$Z_{n+1}=Z_{n}^{2}+C$

Here is function which computes the last iteration, that is the first iteration that lands in the target set ( for example leaves a circle around the origin with a given escape radius ER ) for the iteration of the complex quadratic polynomial above. It is a iteration ( integer) for which (abs(Z)>ER). It can also be improved 

C version ( here ER2=ER*ER) using double floating point numbers ( without complex type numbers) :

 int GiveLastIteration(double Zx, double Zy, double Cx, double Cy, int IterationMax, int ER2)
{
double Zx2, Zy2; /* Zx2=Zx*Zx;  Zy2=Zy*Zy  */
int i=0;
Zx2=Zx*Zx;
Zy2=Zy*Zy;
while (i<IterationMax && (Zx2+Zy2<ER2) ) /* ER2=ER*ER */
{
Zy=2*Zx*Zy + Cy;
Zx=Zx2-Zy2 +Cx;
Zx2=Zx*Zx;
Zy2=Zy*Zy;
i+=1;
}
return i;
}


C with complex type from GSL :

#include <gsl/gsl_complex.h>
#include <gsl/gsl_complex_math.h>
#include <stdio.h>
// gcc -L/usr/lib -lgsl -lgslcblas -lm t.c
// function fc(z) = z*z+c

gsl_complex f(gsl_complex z, gsl_complex c) {
}

int main () {
gsl_complex c = gsl_complex_rect(0.123, 0.125);
gsl_complex z = gsl_complex_rect(0.0, 0.0);
int i;
for (i = 0; i < 10; i++) {
z = f(z, c);
double zx = GSL_REAL(z);
double zy = GSL_IMAG(z);
printf("Real: %f4 Imag: %f4\n", zx, zy);
}
return 0;
}


C++ versions:

 int GiveLastIteration(complex C,complex Z , int imax, int ER)
{
int i; // iteration number
for(i=0;i<=imax-1;i++) // forward iteration
{
if(abs(Z)>ER) break;
}
return i;
}

#include <complex> // C++ complex library

// bailout2 = bailout * bailout
// this function is based on function esctime from mndlbrot.cpp
// from program mandel ver. 5.3 by Wolf Jung
// http://www.mndynamics.com/indexp.html

int escape_time(complex<double> Z, complex<double> C , int iter_max,  double bailout2)
{
// z= x+ y*i   z0=0
long double x =Z.real(), y =Z.imag(),  u ,  v ;
int iter; // iteration
for ( iter = 0; iter <= iter_max-1; iter++)
{ u = x*x;
v = y*y;
if ( u + v <= bailout2 )
{
y = 2 * x * y + C.imag();
x = u - v + C.real();
} // if
else break;
} // for
return iter;
} // escape_time




Delphi version ( using user defined complex type, cabs and f functions )

function GiveLastIteration(z,c:Complex;ER:real;iMax:integer):integer;
var i:integer;
begin
i:=0;
while (cabs(z)<ER) and (i<iMax) do
begin
z:= f(z,c);
inc(i);
end;
result := i;
end;


where :

type complex = record x, y: real; end;

function cabs(z:complex):real;
begin
cabs:=sqrt(z.x*z.x+z.y*z.y)
end;

function f(z,c:complex):complex; // complex quadratic polynomial
var tmp:complex;
begin
tmp.x := (z.x*z.x) - (z.y*z.y) + c.x;
tmp.y := 2*z.x*z.y + c.y ;
result := tmp;

end;


Delphi version without explicit definition of complex numbers :

function GiveLastIteration(zx0,zy0,cx,cy,ER2:extended;iMax:integer):integer;
// iteration of z=zx+zy*i under fc(z)=z*z+c
// where c=cx+cy*i
// until abs(z)<ER  ( ER2=ER*ER )  or i>=iMax
var i:integer;
zx,zy,
zx2,zy2:extended;
begin
zx:=zx0;
zy:=zy0;
zx2:=zx*zx;
zy2:=zy*zy;

i:=0;
while (zx2+zy2<ER2) and (i<iMax) do
begin
zy:=2*zx*zy + cy;
zx:=zx2-zy2 +cx;
zx2:=zx*zx;
zy2:=zy*zy;
//
inc(i);
end;
result := i;
end;


Euler version by R. Grothmann ( with small change : from z^2-c to z^2+c) 

function iter (z,c,n=100) ...

h=z;
loop 1 to n;
h=h^2 + c;
if totalmax(abs(h))>1e20; m=#; break; endif;
end;
return {h,m};
endfunction


Lisp version

This version uses complex numbers. It makes the code short but is also inefficient.

((DEFUN GIVELASTITERATION (Z_0 _C IMAX ESCAPE_RADIUS)
(SETQ Z Z_0)
(SETQ I 0)
(LOOP WHILE (AND (< I IMAX) (< (ABS Z) ESCAPE_RADIUS)) DO
(INCF I)
(SETQ Z (+ (* Z Z) _C)))
I)


Maxima version :

/* easy to read but very slow version, uses complex type numbers */
GiveLastIteration(z,c):=
block([i:0],
while abs(z)<ER and i<iMax
do (z:z*z + c,i:i+1),
i)$ /* faster version, without use of complex type numbers, compare with c version, ER2=ER*ER */ GiveLastIter(zx,zy,cx,cy,ER2,iMax):= block( [i:0,zx2,zy2], zx2:zx*zx, zy2:zy*zy, while (zx2+zy2<ER2) and i<iMax do ( zy:2*zx*zy + cy, zx:zx2-zy2 +cx, zx2:zx*zx, zy2:zy*zy, i:i+1 ), return(i) );  #### Boolean Escape time Algorithm: for every point z of dynamical plane (z-plane) compute iteration number ( last iteration) for which magnitude of z is greater than escape radius. If last_iteration=max_iteration then point is in filled-in Julia set, else it is in its complement (attractive basin of infinity ). Here one has 2 options, so it is named boolean algorithm. if (LastIteration==IterationMax) then color=BLACK; /* bounded orbits = Filled-in Julia set */ else color=WHITE; /* unbounded orbits = exterior of Filled-in Julia set */  In theory this method is for drawing Filled-in Julia set and its complement ( exterior), but when c is Misiurewicz point ( Filled-in Julia set has no interior) this method draws nothing. For example for c=i . It means that it is good for drawing interior of Filled-in Julia set. ##### ASCII graphic ; common lisp (loop for y from -2 to 2 by 0.05 do (loop for x from -2 to 2 by 0.025 do (let* ((z (complex x y)) (c (complex -1 0)) (iMax 20) (i 0)) (loop while (< i iMax ) do (setq z (+ (* z z) c)) (incf i) (when (> (abs z) 2) (return i))) (if (= i iMax) (princ (code-char 42)) (princ (code-char 32))))) (format t "~%"))  ##### PPM file with raster graphic Filled-in Julia set for c= =-1+0.1*i. Image and C source code #### Integer escape time = Level Sets of the Basin of Attraction of Infinity = Level Sets Method= LSM/J Escape time measures time of escaping to infinity ( infinity is superattracting point for polynomials). Time is measured in steps ( iterations = i) needed to escape from circle of given radius ( ER= Escape Radius). One can see few things: Level sets here are sets of points with the same escape time. Here is algorithm of choosing color in black & white version.  if (LastIteration==IterationMax) then color=BLACK; /* bounded orbits = Filled-in Julia set */ else /* unbounded orbits = exterior of Filled-in Julia set */ if ((LastIteration%2)==0) /* odd number */ then color=BLACK; else color=WHITE;  Here is the c function which: • uses complex double type numbers • computes 8 bit color ( shades of gray) • checks both escape and attraction test unsigned char ComputeColorOfLSM(complex double z){ int nMax = 255; double cabsz; unsigned char iColor; int n; for (n=0; n < nMax; n++){ //forward iteration cabsz = cabs(z); if (cabsz > ER) break; // esacping if (cabsz< PixelWidth) break; // fails into finite attractor = interior z = z*z +c ; /* forward iteration : complex quadratic polynomial */ } iColor = 255 - 255.0 * ((double) n)/20; // nMax or lower walues in denominator return iColor; }   "if a 2-variable function z = f(x,y) has non-extremal critical points, i.e. it has saddle points, then it's best if the contour z heights are chosen so that the saddle points are on a contour, so that the crossing contours appear visually."Alan Ableson  How to choose parameters for which Level curves cross critical point ( and it's preimages ) ? • choose the parameter c such that it is on an escape line, then the critical value will be on an escape line as well • choose escape radius equal to n=th iteration of critical value // find such ER for LSM/J that level curves croses critical point and it's preimages ( only for disconnected Julia sets) double GiveER(int i_Max){ complex double z= 0.0; // criical point int i; ; // critical point escapes very fast here. Higher valus gives infinity for (i=0; i< i_Max; ++i ){ z=z*z +c; } return cabs(z); }  #### Normalized iteration count (real escape time or fractional iteration or Smooth Iteration Count Algorithm (SICA)) Math formula : $\nu (z)=\lim \limits _{i\to \infty }(i-\log _{2}\log _{2}|z_{i}|)\ .$ Maxima version : GiveNormalizedIteration(z,c,E_R,i_Max):= /* */ block( [i:0,r], while abs(z)<E_R and i<i_Max do (z:z*z + c,i:i+1), r:i-log2(log2(cabs(z))), return(float(r)) )$


In Maxima log(x) is a natural (base e) logarithm of x. To compute log2 use :

log2(x) := log(x) / log(2);


description:

#### Level Curves of escape time Method = eLCM/J

These curves are boundaries of Level Sets of escape time ( eLSM/J ). They can be drawn using these methods:

• edge detection of Level Curves ( =boundaries of Level sets).
• Algorithm based on paper by M. Romera et al.
• Sobel filter
• drawing lemniscates = curves $L_{n}=\{z:abs(z_{n})=ER\}\,$ , see explanation and source code
• drawing circle $L_{0}=\{z:abs(z)=ER\}\,$  and its preimages. See this image, explanation and source code
• method described by Harold V. McIntosh
/* Maxima code : draws lemniscates of Julia set */
c: 1*%i;
ER:2;
z:x+y*%i;
f[n](z) := if n=0 then z else (f[n-1](z)^2 + c);
load(implicit_plot); /* package by Andrej Vodopivec */
ip_grid:[100,100];
ip_grid_in:[15,15];
implicit_plot(makelist(abs(ev(f[n](z)))=ER,n,1,4),[x,-2.5,2.5],[y,-2.5,2.5]);


Density of level curves

  "The spacing between level curves is a good way to estimate gradients: level curves that are close together represent areas of steeper descent/ascent." 

 "The density of the contour lines tells how steep is the slope of the terrain/function variation. When very close together it means f is varying rapidly (the elevation increase or decrease rapidly). When the curves are far from each other the variation is slower" 

How to control level curves

• shape of target set
• manually
• drawing equipotential lines
• changing level sets ( level curves are boundaries of level sets)

### Basin of attraction of finite attractor = interior of filled-in Julia set

• How to find periodic attractor ?
• How many iterations is needed to reach attractor ?

#### Components of Interior of Filled Julia set ( Fatou set)

• use limited color ( palette = list of numbered colors)
• find period of attracting cycle
• find one point of attracting cycle
• compute number of iteration after when point reaches the attractor
• color of component=iteration % period
• use edge detection for drawing Julia set

#### Internal Level Sets

See :

• algorithm 0 of program Mandel by Wolf Jung

How to choose size of attracting trap such that level curves cross at critical point ?

It depends on

• dynamic type ( superattracting/attracting, parabolic, repelling)
• period ( child period in the parabolic case)

• petal in the parabolic case :
• for period 1 and 2 : radius of a circle with parabolic point on it's boundary
• for higher periods sector of the circle with center at parabolic point
• for other cases ( except repelling) it is radius of the circle with attractor as a center

int local_setup(double cx){

c = cx;
zp = GiveFixed(c);

switch ( DynamicType){
case repelling: // no  interior = no attracting fixed point = only escaping points

break;
case attracting:
delta = sqrt(1.0 - 4.0* creal(c));  // delta is a distance between alfa and beta fixed points
AR =  delta /20.0;

break;

case superattracting: // cabs(zp - zcr_last ) < PixelWidth
AR = 30.0* PixelWidth * iWidth / 5000 ; //
break;

case parabolic:
// zcr_last < parabolic_trap_center < zp
int i; /* nr of point of critical orbit */
complex double z = zcr;
for (i=1;i<IterMax ; ++i)
{ z = f(z); }
zcr_last = z;
//
AR = (zp - zcr_last)/2.0;
parabolic_trap_center = ( creal(zp) + creal(zcr_last))/ 2.0;
break;
default:
}

AR2 = AR*AR;

return 0;
}

// and print program info
fprintf (stdout, "DynamicType value is setup manually; Once can do it also numerically ( from multiplier of fixed point alfa or from some other properities)\n");
switch ( DynamicType){
case repelling:
fprintf (stdout, "\tThere is only one Fatou basin: basin of infinity \n");
fprintf (stdout, "\tthere is no interior = Julia set is disconnected \n");
fprintf (stdout, "\tcritical point z=0 is repelling = attracted to infinity \n");
break;
case attracting:
fprintf (stdout, "\tbasin type is attracting \n");
fprintf (stdout, "\tzcr_last =  %.16f \talfa fixed point zp = %.16f\n", creal (zcr_last), creal(zp));//
fprintf (stdout, "\tdelta =  %.16f is the distance between fixed points\n", delta);//
fprintf (stdout, "\tAtracting Radius AR is set manually  = %.16f = %f * PixelWidth = %f * ImageWidth \n", AR, AR / PixelWidth, AR /ImageWidth );
break;

case superattracting:
fprintf (stdout, "\tbasin type is superattracting \n");
fprintf (stdout, "\tzcr =  %.16f  = zp = %.16f\n", creal (zcr), creal(zp));//
fprintf (stdout, "\tAtracting Radius AR is set manually  = %.16f = %f *PixelWidth = %f *ImageWidth \n", AR, AR / PixelWidth, AR /ImageWidth);
break;

case parabolic:
fprintf (stdout, "\tbasin type  is parabolic \n");
fprintf (stdout, "\tzcr_last =  %.16f < parabolic_trap_center = %.16f < zp = %.16f\n", creal (zcr_last), creal (parabolic_trap_center), creal(zp));//
fprintf (stdout, "\tzp - zcr_last =  %.16f AR*2 = %.16f \t difference = %.16f\n", creal (zp - zcr_last), AR *2.0, creal (zp - zcr_last) -  AR *2.0);//
fprintf (stdout, "\tAtracting Radius AR is tuned  = (zp - zcr_last)/2 = %.16f = %f *PixelWidth = %f *ImageWidth \n", AR, AR / PixelWidth, AR /ImageWidth);
fprintf (stdout, "\tparabolic_trap_center z =  %.16f %+.16f*i  \n", creal (parabolic_trap_center), cimag (parabolic_trap_center));// parabolic_trap_center
break;
default:

}


Steps for parabolic basin

• choose component with critical point inside
• choose trap

Trap is a disk

• inside component with critical point inside
• trap has parabolic point on it's boundary
• center of the trap is the midpoint between last point of critical orbit and fixed point
• radius of the trap is half of distance between fixed point and last point of critical orbit

### Decomposition of target set

Examples

#### Binary decomposition

Here color of pixel ( exterior of Julia set) is proportional to sign of imaginary part of last iteration ( cimag) = the radial borders of are at dyadic angles (shows external rays at dyadic angles).

Main loop is the same as in escape time.

• Escape radius ( ER) should be bigger: ER = 200
• for superattracting case small: AR = PixelWidth

In other words target set is decompositioned in 2 parts ( binary decomposition) :

$T_{b}+=\{z:abs(z_{n})>ER~~{\mbox{and}}~~im(z_{n})>0\}\,$

$T_{b}-=\{z:abs(z_{n})>ER~~{\mbox{and}}~~im(z_{n})<=0\}\,$

Algorithm in pseudocode ( Im(Zn) = Zy ) :

if (LastIteration==IterationMax)
then color=BLACK;   /* bounded orbits = Filled-in Julia set */
else   /* unbounded orbits = exterior of Filled-in Julia set  */
if (Zy>0) /* Zy=Im(Z) */
then color=BLACK;
else color=WHITE;


Description

unsigned char ComputeColorOfBD (complex double z)
{

double cabsz;

int i;			// number of iteration
for (i = 0; i < IterMax_LSM; ++i)
{

cabsz = cabs(z); // numerical speed up : cabs(zp-z) = cabs(z) because zp = zcr = 0

//
if ( cabsz > ER  ||  cabsz < AR ) // if z is inside target set ( orbit trap)
{
if (cimag(z) > 0) // binary decomposition of target set
{  return 0;}
else {return 255; }

}

z = f(z);

}

return iColorOfUnknown;

}


Attracting case is used for "field lines" coloring method by Gertbuschmann

These curves :

• are boundaries of binare decomposition boxes
• are not field lines of potential = external rays

Notes

• if the escape radius is too low, then binary (or ternary, etc) decomposition rays will have visible discontinuities at iteration bands. increasing escape radius makes discontinuities smaller, but changes the aspect ratio
• mrob says exp(pi) is the best escape radius for binary decomposition as it makes boxes have square aspect ratio (possibly more visible when using exponential map transformation?)
##### rotation of target set by internal angle
 // for MBD
double t0 = 1.0 / 3.0; // period = 3

// Modified BD
unsigned char ComputeColorOfMBD (complex double z)
{

double cabsz;
double turn;

int i;			// number of iteration
for (i = 0; i < IterMax_LSM; ++i)
{

cabsz = cabs(z); // numerical speed up : cabs2(zp-z) = cabs2(z) because zp = zcr = 0

//  if z is inside target set ( orbit trap) = exterior of circle with radius ER
if ( cabsz > ER  ) // exterior
{
if (creal(z) > 0) // binary decomposition of target set
{  return 0;}
else {return 255; }

}

if ( cabsz  < AR ) // if z is inside target set ( orbit trap) = interior of circle with radius AR
{
turn = c_turn(z);
if (turn < t0 || turn > t0+0.5) // modified binary decomposition of target set
{  return 0;}
else {return 255; }

}

z = f(z);

}

return iColorOfUnknown;

}

##### Constant number of iterations

Here exterior of Julia set is decompositioned into radial level sets.

It is because main loop is without bailout test and number of iterations ( iteration max) is constant.

  for (Iteration=0;Iteration<8;Iteration++)
/* modified loop without checking of abs(zn) and low iteration max */
{
Zy=2*Zx*Zy + Cy;
Zx=Zx2-Zy2 +Cx;
Zx2=Zx*Zx;
Zy2=Zy*Zy;
};
iTemp=((iYmax-iY-1)*iXmax+iX)*3;
/* --------------- compute  pixel color (24 bit = 3 bajts) */
/* exterior of Filled-in Julia set  */
/* binary decomposition  */
if (Zy>0 )
{
array[iTemp]=255; /* Red*/
array[iTemp+1]=255;  /* Green */
array[iTemp+2]=255;/* Blue */
}
if (Zy<0 )
{
array[iTemp]=0; /* Red*/
array[iTemp+1]=0;  /* Green */
array[iTemp+2]=0;/* Blue */
};


It is also related with automorphic function for the group of Mobius transformations 

##### BDM in the attracting domain

BDM in the attracting domain ( usually interior of Julia sets ) gives (pseudo) Field lines

Explanation by Gert Buschmann The equipotential lines for iteration towards infinity Field lines for an iteration of the form ( 1 − z 3 / 6 ) ( z − z 2 / 2 ) 2 + c {\displaystyle {\frac {(1-z^{3}/6)}{(z-z^{2}/2)^{2}}}+c}

In each Fatou domain (that is not neutral) there are two systems of lines orthogonal to each other: the equipotential lines (for the potential function or the real iteration number) and the field lines.

If we colour the Fatou domain according to the iteration number (and not the real iteration number $\nu (z)$ , as defined in the previous section), the bands of iteration show the course of the equipotential lines. If the iteration is towards ∞ (as is the case with the outer Fatou domain for the usual iteration $z^{2}+c$ ), we can easily show the course of the field lines, namely by altering the colour according as the last point in the sequence of iteration is above or below the x-axis (first picture), but in this case (more precisely: when the Fatou domain is super-attracting) we cannot draw the field lines coherently - at least not by the method we describe here. In this case a field line is also called an external ray.

Let z be a point in the attracting Fatou domain. If we iterate z a large number of times, the terminus of the sequence of iteration is a finite cycle C, and the Fatou domain is (by definition) the set of points whose sequence of iteration converges towards C. The field lines issue from the points of C and from the (infinite number of) points that iterate into a point of C. And they end on the Julia set in points that are non-chaotic (that is, generating a finite cycle). Let r be the order of the cycle C (its number of points) and let z* be a point in C. We have $f(f(\dots f(z^{*})))=z^{*}$  (the r-fold composition), and we define the complex number α by

$\alpha =(d(f(f(\dots f(z))))/dz)_{z=z^{*}}.$

If the points of C are $z_{i},i=1,\dots ,r(z_{1}=z^{*})$ , α is the product of the r numbers $f'(z_{i})$ . The real number 1/|α| is the attraction of the cycle, and our assumption that the cycle is neither neutral nor super-attracting, means that 1 < 1/|α| < ∞. The point z* is a fixed point for $f(f(\dots f(z)))$ , and near this point the map $f(f(\dots f(z)))$  has (in connection with field lines) character of a rotation with the argument β of α (that is, $\alpha =|\alpha |e^{\beta i}$ ).

In order to colour the Fatou domain, we have chosen a small number ε and set the sequences of iteration $z_{k}(k=0,1,2,\dots ,z_{0}=z)$  to stop when $|z_{k}-z^{*}|<\epsilon$ , and we colour the point z according to the number k (or the real iteration number, if we prefer a smooth colouring). If we choose a direction from z* given by an angle θ, the field line issuing from z* in this direction consists of the points z such that the argument ψ of the number $z_{k}-z^{*}$  satisfies the condition that

$\psi -k\beta =\theta \mod \pi .\,$

For if we pass an iteration band in the direction of the field lines (and away from the cycle), the iteration number k is increased by 1 and the number ψ is increased by β, therefore the number $\psi -k\beta \mod \pi$  is constant along the field line. Pictures in the field lines for an iteration of the form z 2 + c {\displaystyle z^{2}+c}

A colouring of the field lines of the Fatou domain means that we colour the spaces between pairs of field lines: we choose a number of regularly situated directions issuing from z*, and in each of these directions we choose two directions around this direction. As it can happen that the two field lines of a pair do not end in the same point of the Julia set, our coloured field lines can ramify (endlessly) in their way towards the Julia set. We can colour on the basis of the distance to the center line of the field line, and we can mix this colouring with the usual colouring. Such pictures can be very decorative (second picture).

A coloured field line (the domain between two field lines) is divided up by the iteration bands, and such a part can be put into a one-to-one correspondence with the unit square: the one coordinate is (calculated from) the distance from one of the bounding field lines, the other is (calculated from) the distance from the inner of the bounding iteration bands (this number is the non-integral part of the real iteration number). Therefore, we can put pictures into the field lines (third picture).

## DEM/J

This algorithm has 2 versions:

Compare it with version for parameter plane and Mandelbrot set : DEM/M It's the same as M-set exterior distance estimation but with derivative w.r.t. Z instead of w.r.t. C.

## Convergence

In this algorithm distances between 2 points of the same orbit are checked

### average discrete velocity of orbit

It is used in case of :

### Cauchy Convergence Algorithm (CCA)

This algorithm is described by User:Georg-Johann. Here is also Matemathics code by Paul Nylander

## Normality

Normality In this algorithm distances between points of 2 orbits are checked

### Checking equicontinuity by Michael Becker

"Iteration is equicontinuous on the Fatou set and not on the Julia set". (Wolf Jung) 

Michael Becker compares the distance of two close points under iteration on Riemann sphere.

This method can be used to draw not only Julia sets for polynomials ( where infinity is always superattracting fixed point) but it can be also applied to other functions ( maps), for which infinity is not an attracting fixed point.

### using Marty's criterion by Wolf Jung

Wolf Jung is using "an alternative method of checking normality, which is based on Marty's criterion: |f'| / (1 + |f|^2) must be bounded for all iterates." It is implemented in mndlbrot::marty function ( see src code of program Mandel ver 5.3 ). It uses one point of dynamic plane.

## Koenigs coordinate

Koenigs coordinate are used in the basin of attraction of finite attracting (not superattracting) point (cycle).

# Optimisation

## Distance

You don't need a square root to compare distances.

## Symmetry

Julia sets can have many symmetries 

Quadratic Julia set has allways rotational symmetry ( 180 degrees) :

colour(x,y) = colour(-x,-y)


when c is on real axis ( cy = 0) Julia set is also reflection symmetric :

colour(x,y) = colour(x,-y)


Algorithm :

• compute half image
• rotate and add the other half
• write image to file 

# Sets

## Target set

• trap for forward orbit
• it is a set which captures any orbit tending to fixed / periodic point.

## Julia sets

"Most programs for computing Julia sets work well when the underlying dynamics is hyperbolic but experience an exponential slowdown in the parabolic case." ( Mark Braverman )

• when Julia set is a set of points that do not escape to infinity under iteration of the quadratic map ( = filled Julia set has no interior = dendrt)
• IIM/J
• DEM/J
• checking normality
• when Julia set is a boundary between 2 basin of attraction ( = filled Julia set has no empty interior) :
• boundary scanning 
• edge detection

## Fatou set

Interior of filled Julia set can be coloured :

More is here

# Video

One can make videos using :

• zoom into dynamic plane
• changing parametr c along path inside parameter plane
• changing coloring scheme ( for example color cycling )

Examples :