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Summary
Critical polynomial
Q
n
=
f
c
n
(
z
c
r
)
=
f
c
n
(
0
)
{\displaystyle Q_{n}=f_{c}^{n}(z_{cr})=f_{c}^{n}(0)\,}
so
Q
1
=
f
c
1
(
0
)
=
c
{\displaystyle Q_{1}=f_{c}^{1}(0)=c\,}
Q
2
=
f
c
2
(
0
)
=
c
2
+
c
{\displaystyle Q_{2}=f_{c}^{2}(0)=c^{2}+c\,}
Q
3
=
f
c
3
(
0
)
=
(
c
2
+
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
c
e
n
t
e
r
s
=
{
c
:
f
c
n
(
z
c
r
)
=
0
}
{\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
);
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