Summary

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Summary

Critical polynomial

${\displaystyle Q_{n}=f_{c}^{n}(z_{cr})=f_{c}^{n}(0)\,}$

so

${\displaystyle Q_{1}=f_{c}^{1}(0)=c\,}$

${\displaystyle Q_{2}=f_{c}^{2}(0)=c^{2}+c\,}$

${\displaystyle Q_{3}=f_{c}^{3}(0)=(c^{2}+c)^{2}+c\,}$

These polynomials are used for finding :

• centers of period n Mandelbrot set components. Centers are roots of n-th critical polynomials ${\displaystyle centers=\{c:f_{c}^{n}(z_{cr})=0\}\,}$ ( points where critical curve Qn croses x axis )
• Misiurewicz points

Maxima CAS src code

kill(all);
remvalue(all);

/* ---------- functions ---------------------- */

/* http://en.wikipedia.org/wiki/Complex_quadratic_polynomial  */
f(z,c):=z*z+c;

fn(p, z, c) :=
  if p=0 then z
  elseif p=1 then f(z,c)
  else f(fn(p-1, z, c),c);

GiveDrawList(Period,cMin, cMax, cStep):=
 block
 (
  [MyList:[]],
  for c:cMin while c <= cMax step cStep do
    MyList:cons([c,fn(Period,0,c)],MyList),
 
   return(MyList)
)$

compile(all);

/* ---------- constant ---------------------------*/

nMax:5$
cMin:-2;
cMax:0.25;
cStep:0.003;

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

ColorList:[red,blue,green, black, purple , brown, cyan, violet,gray ]$

DrawLists:[]$ /* empty list */

for n:0 thru nMax do 
(
 Period : 2^n,
 DrawList:GiveDrawList( Period,cMin, cMax, cStep),
 DrawLists:cons(points(DrawList),DrawLists),
 DrawLists:cons(key =concat("period =", string(Period)),DrawLists),
 DrawLists:cons(color=ColorList[n+1],DrawLists)
 
 
);

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

path:"~/maxima/batch/skeleton/even/test/"$ /* result of pwd; to save image in the same directory as mac file  */
 FileName:string(nMax)$ /* without extension which is the terminal name */

load(draw); 

draw2d(
  terminal  = svg,
  file_name = concat(path,FileName),
  title = "Critical curves diagram for real quadratic map fc(z) = z^2 + c",
  dimensions=[2000,2000],
  user_preamble = "",
  xlabel     = "c",
  ylabel     = "z",
  xaxis        = true,
  points_joined =true,
  point_size    =  0.2,
  point_type = filled_circle,
  DrawLists
);