Summary

Compare with

• Figure 27 on page 36 in Peitgen, Richter : The beauty of fractals. ISBN 0-387-15851-0
• Figure 3 on page 17 in Rational iteration: complex analytic dynamical systems by Norbert Steinmetz, Walter de Gruyter, 1993 , ISBN 3110137658,

Maxima src code

c:-1;   /*       define c value  */
iMax:500000; /* maximal number of reversed iterations */
probability:0.5; /* one preimage is more contractive then other so it should be chosen .....*/
f(z,c):=expand(z*z+c);
finverseplus(z,c):=expand(sqrt(z-c));
finverseminus(z,c):=expand(-sqrt(z-c)); 
/* define z-plane ( dynamical ) */
zxMin:-2.0;
zxMax:2.0;
zyMin:-2.0;
zyMax:2.0; 
/* define image size : width:iXmax-0+1;      height:iYmax-0+1 ;    */
iXmax:1000;
iYmax:1000; 
/* 
zx:zxMin+PixelWidth*iX ;
zy:zyMin+PixelHeight*iY; 
so
iX:fix((zx-zxMin)/PixelWidth);
iY:fix((zy-zyMin)/PixelHeight);
*/
PixelWidth:(zxMax-zxMin)/iXmax;
PixelHeight:(zyMax-zyMin)/iYmax;
GiveJuliaPoints(c,iMax,probability):=
block( 
 xyv:[],
 /* */
 array(Hits,fixnum,iXmax,iYmax), /* 2D array of hits pixels . Hit > 0 means that point was in orbit */
 fillarray (Hits,[0]), /* no hits for beginning */
 /* compute fixed points of f(z,c) :   z=f(z,c)   */
 fixed:float(rectform(solve([z*z+c=z],[z]))),
 /* Find which is repelling  */
 if (abs(2*rhs(fixed[1]))<1) 
  then (
   beta:rhs(fixed[1]),
   alfa:rhs(fixed[2])
   ) 
  else (
   alfa:rhs(fixed[1]),
   beta:rhs(fixed[2])
  ),
 /* choose repelling fixed point as a starting point of inversed iteration */
 z:beta,
 /* reversed iteration of beta */
 for i:1 thru iMax  step 1 do 
 (
 if random(1.0)>probability
  then z:finverseplus(z,c)
  else z:finverseminus(z,c),
  iX:fix((realpart(z)-zxMin)/PixelWidth),
  iY:fix((imagpart(z)-zyMin)/PixelHeight),
 /* save hits values to draw it later */
 Hits[iX,iY]:Hits[iX,iY]+1
 ),
/* */
for iX:0 thru iXmax  step 1 do
for iY:0 thru iYmax  step 1 do
if (Hits[iX,iY]>0) then 
(zx:zxMin+PixelWidth*iX ,
zy:zyMin+PixelHeight*iY,
xyv:cons([zx,zy,Hits[iX,iY]],xyv)),
return(xyv)
)$
JuliaPoints:GiveJuliaPoints(c,iMax,probability);	
/* draw reversed orbit of beta  using draw package */
load(draw);
draw3d(
file_name = "iimj_5",
terminal  = 'screen,
pic_width  = iXmax,
pic_height = iYmax,
yrange = [zyMin,zyMax],
xrange = [zxMin,zxMax],
title= concat("Julia set for c=",string(c),"  using IIM (",string(iMax)," points, prob= 0.5)"),
xlabel     = "Z.re ",
ylabel     = "Z.im",
point_type    = dot,
points_joined = impulses,
line_width    = 2,
color         = red,
point_size    = 5,
/*color         = black,*/
points(JuliaPoints)
);

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