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https://creativecommons.org/licenses/by-sa/3.0CC BY-SA 3.0 Creative Commons Attribution-Share Alike 3.0 truetrue

• 
  distance between points of critical orbit in case of attracting fixed point

/* 
gives distance between 2 complex points on plane
interior points of unit circle under iteration of f(z)=z*z

alfa fixed point  za : -0.368684439039160  -0.337745147130762*%i
 interior zi : 1.331250000000000  +0.375000000000000*%i
 exterior ze : 0.055303418766281  -0.066996098099950 *%i
 siegel zcr : 0 
 inside siegel zs1 : -0.162500000000000  -0.143750000000000 *%i
 inside siege zs2 : -0.256250000000000  -0.187500000000000*% i
*/
kill(all);

GivePlaneDistance(c1,c2):=float(abs(c2-c1))$

f(z):=rectform(z*z+c);

 GiveList(z0, iMax):= block
(
  [zprev,znext,dist,d_list],
   zprev:z0,
   znext:f(zprev),
   dist:GivePlaneDistance(zprev,znext),
   d_list:[[1,dist]],
   zprev : znext,
   for i:2 step 1 thru iMax while (abs(znext)<10 )
    do
    ( 
      znext:f(zprev),
      dist:GivePlaneDistance(zprev,znext),
      d_list:endcons([i,dist],d_list),
      zprev : znext
    ),
   return(d_list)
)$

/* -------------  main -----------------------*/

/*  rotation number or internal angle */
t : (-1+sqrt(5))/2; /* golden Mean Siegel disc */

angle : 2*%pi*t;  /* from turns to radians */
d : exp(angle*%i);   /* point on unit circle */
c  : float(rectform(d/2-d*d/4));  /*  point on boundary of main cardioid */

i_max:100;

zi : 1.331250000000000  +0.375000000000000*%i;
i_list:GiveList(zi,i_max);

zs1 : -0.162500000000000  -0.143750000000000 *%i;
s1_list:GiveList(zs1,i_max);

zs2 : -0.256250000000000  -0.187500000000000*%i;
s2_list:GiveList(zs2,i_max);

ze : 0.055303418766281  -0.066996098099950*%i;
e_list:GiveList(ze,i_max);

za : -0.368684439039160  -0.337745147130762*%i;
a_list:GiveList(za,i_max);

zcr : 0.0;
cr_list:GiveList(zcr,i_max);

/* ----------------- draw ------------ */

load(draw); 

draw2d(
  terminal  = svg,
  file_name = "distance",
  title = "Distance between points of orbit for fc(z)=z*z -0.390540870218399  -0.586787907346969*i",
  yrange = [0,4],
  points_joined =true,
  point_size    =  0.4,
  point_type = filled_circle,

  color         = black,
  key="exterior but near boundary ",
  points(e_list),
 
  color         = red,
  key="critical point ",
  points(cr_list),

  color         = blue,
  key="Siegel 2",
  points(s2_list),
  
  color         = blue,
  key="Siegel 1",
  points(s1_list),

  color         = green,
  key="interior, not Siegel ",
  points(i_list)
);