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Summary
| Description |
English: dynamical plane with branched periodic external ray 0 for map f(z) = z*z + 0.35. " Julia set for a polynomial ... which belongs to the ray R()(M) in parameter space. The right hand limit ray Ro bounces off infinitely many pre-critical points as it spirals in to the upper fixed point. Both fixed points have rotation number zero." [1] " The countable family of R-analytic curves on which we cannot define the function “external argument”. [2] |
|||
| Date | ||||
| Source | Own work | |||
| Author | Adam majewski | |||
| Other versions |
Papers
Video: |
_%3D_z*z_%2B_0.35.svg)
Licensing
I, the copyright holder of this work, hereby publish it under the following license:
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- Under the following conditions:
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Software
- Maxima CAS
- draw package by Mario Rodríguez Riotorto[3]
- gnuplot
Algorithm
Ray bifurcates at critical point and at it's preimages
Ray consist of segments ( curves between bifurcation points):
- r0: external ray 0 goes from (+) infinity along critical orbit towards critical point z=0 ( horizontal segment). Here bifurcates.
- segment r1 has two subsegments : r1a and r1b. These are two vertical segments from critical point z=0 towards it's two preimages : a(z) = f^-1(z) and b(z) = -a(z). So it is: [a(z), -a]
- each (sub)segment of r1 has 2 preimages
- a(r1a) and b(r1a)
- a(r1b) and b(r1b)
- repeat and infinity ( binary tree)
Ray = r0 + r1 + r2 + ... + rn
Problems
inverse function
_%3D_z*z_%2B_0.35.png){kind=link}
_%3D_z*z_%2B_0.35.png&action=edit&redlink=1){kind=link}
_%3D_z*z_%2B_0.35.png){kind=link}
the standard complex square-root ( csqrt) takes values in the right halfplane, but this is rotated by the squareroot. So some points from one list goes to preimage of anothe lists. Onbe can not use gnuplot option "points joined= true".
The soultion
- use lists with more points ( takes time) and gnuplot option "points joined= false"
- compute only preimages of one r1 subsegment and use symetry. It is 2-4 times faster and solves the problem of csqrt
length of the list
rn is a lis, but it consist of sublist ( list of lists). Each sublist rn[k] is a list of points describint segment (one of n=th preimage's of segment r0);
| n | % | s |
|---|---|---|
| 0 | 1.0 | 493 |
| 1 | 1.0 | 359 |
| 2 | 1.0 | 216 |
| 3 | 1.0 | 132 |
| 4 | 1.0 | 86 |
| 5 | 1.0 | 62 |
| 6 | 1.0 | 48 |
| 7 | 1.0 | 25 |
| 8 | 22 | |
| Example | Example | Example |
| Example | Example | Example |
| 11 | 77/4096 | 16 |
Notation
- n is a level of binary tree of preimages
- % denotes ratio of (sublists with size > PixelSize) / ( all sublists of rn)
- s denotes size ratio = size/PixelSize, where size = cabs( first - last)
- segments = number of segments in the n-th list = 2^n
- points = length(r[n][k]) = kMax + 2
List r11 has 4096 sublists ( segments) but only 77 of them are wider then Pixel size = 0.0024
r0 list: 1 / 1 = 1.0 segments has size > MinSize. MaxSize = 247 pixels r1 list: 2 / 2 = 1.0 segments has size > MinSize. MaxSize = 179 pixels r2 list: 4 / 4 = 1.0 segments has size > MinSize. MaxSize = 108 pixels r3 list: 8 / 8 = 1.0 segments has size > MinSize. MaxSize = 66 pixels r4 list: 16 / 16 = 1.0 segments has size > MinSize. MaxSize = 43 pixels r5 list: 32 / 32 = 1.0 segments has size > MinSize. MaxSize = 31 pixels r6 list: 64 / 64 = 1.0 segments has size > MinSize. MaxSize = 24 pixels r7 list: 128 / 128 = 1.0 segments has size > MinSize. MaxSize = 20 pixels r8 list: 220 / 256 = 0.859375 segments has size > MinSize. MaxSize = 19 pixels r9 list: 148 / 512 = 0.2890625 segments has size > MinSize. MaxSize = 18 pixels r10 list: 224 / 2048 = 0.109375 segments has size > MinSize. MaxSize = 17 pixels r11 list: 200 / 4096 = 0.048828125 segments has size > MinSize. MaxSize = 15 pixels r12 list: 196 / 8192 = 0.02392578125 segments has size > MinSize. MaxSize = 13 pixels
Maxima CAS src code
new code
/*
Batch file for Maxima CAS
save as a s.mac
run maxima :
maxima
and then :
batch("s.mac");
*/
kill(all);
remvalue(all);
/* ---------- functions ---------------------------------------------------- */
/* http://en.wikipedia.org/wiki/Complex_quadratic_polynomial */
/* Forward iteration */
f(z):=float(rectform(z*z+c))$
/*
backward iteration of complex quadratic polynomial
when Im(z-c) changes sign ( from quadrant (-,-) to (-,+)
then preimaga chages quadrant
w = sqrt(z-c)
https://www.geogebra.org/m/YPGT79Zq
*/
b(z):=float(rectform(-sqrt(z-c)))$
a(z):=float(rectform(sqrt(z-c)))$
/* find fixed point alfa of function f(z,c) */
GiveFixed_p(c):= float(rectform((1+sqrt(1-4*c))/2))$
GiveFixed_m(c):= float(rectform((1-sqrt(1-4*c))/2))$
/*
converts complex number z = x*y*%i
to the list in a draw format:
[x,y]
*/
d(z):=[float(realpart(z)), float(imagpart(z))]$
/* give Draw List from one point*/
dl(z):=[d(z)]$
ToPoints(myList):= points(map(d,myList))$
/* gives an orbit of z0 under fc where iMax is the length of the orbit */
GiveForwardOrbit(z0,c, iMax):= block
(
[i,z, orbit],
z:z0,
orbit :[z],
i:1,
while ( i<iMax )
do
(
z:f(z),
orbit : endcons(z, orbit),
i:i+1
),
return(orbit)
)$
/*
Line segment = part of a line that is bounded by two end points
input :
- 2 endpoints : z1, z2
- imax : number of points between endpoints
output:
- list of complex points
*/
GiveLineSegment(z1,z2,iMax):=block
(
[l,z, dz, t , tmin, tmax,dt,i],
dz : z2-z1,
dt : 1/iMax,
tmin: 0,
tmax: 1,
z:z1,
l:[z1],
for t:tmin thru tmax step dt
do
(
z:z1+dz*t,
l :cons(z,l)
),
return(l)
)$
minus(z) := -z;
/*
binary complete tree
to do :
implemented as an array
Pre-order sequence: ABDECFG ( for depth 3)
for(i=0;i<N;i++){ process node in position i; } results in a level-by-level traversal.
*/
/*
- read input z
- compute 2 preimages a and b=-a
- output list : [a,b]
Give2Preimages(z):= block(
[p],
p: a(z),
return([p,-p])
)$
*/
Give2Preimages(z):= [a(z), b(z)]$
/*
regular complete binary tree depth
- read input A
- compute full tree of preimages depth d
( number = sum(2^(d-1))
- output: unsorted list zz of complex points
lists
zz is output list
t1 input = previous level of the tree
t2 next level
temp temporary list
length of precritical orbit : 4 094 for depth : 11
length of precritical orbit : 262 142 for depth : 17
if length is to big then save time is too long
*/
GiveBTree(z, depthMax):=block(
[ zz, t1, t2, tmp],
zz :[],
t1 :[z],
t2 :[],
tmp :[],
for depth : 1 thru depthMax step 1 do (
for z in t1 do (
tmp : Give2Preimages(z),
zz : append(tmp, zz),
t2 : append(tmp, t2)
),
tmp:[],
t1:t2,
t2:[]
),
zz:flatten(zz),
return (zz)
)$
/*
input: list zz
output: list of 2 sublists
*/
Give2PreimagesOfList(zz):=block(
[La,Lb, LL],
La:[],
Lb:[],
LL:[],
for z in zz do (
tmp : Give2Preimages(z),
/* trick, sorry, but I do not know how to do it better
the standard square-root takes values in the right halfplane, but this is rotated by the squareroot of -c
if (z = last(zz))
then (
La:cons(tmp[2], La),
Lb:cons(tmp[1], Lb)
)
else(*/
La:cons(tmp[1], La),
Lb:cons(tmp[2], Lb)
/* )*/
/*??? check the values */
),
LL:[La , Lb],
return(LL)
)$
GiveSortedBTree(zz, depthMax):=block(
[G, T, P , L],
/* local input list */
T :[], /* local output list */
G: [], /* global output list */
P: [],
L:[],
L:zz,
for depth : 1 thru depthMax step 1 do (
P : Give2PreimagesOfList(L),
T : cons(P[1], T),
T : cons(P[2], T),
G : cons(P[1], G),
G : cons(P[2], G),
P:[],
L:[],
L:T,
T:[]
),
return (G)
)$
/*
Gives external dynamic ray ( as a list of points in a draw format)
input:
- zcr: critical point z
- c : parameter of the function fc(z) = z^2 + c
- kMax : number of points in the line segment
- parameter n : every n-th element is removed from the list to make it shorter, see ShortList
*/
GiveRay(zcr, depth_Max, k_Max):= block(
[Ray,t,zz],
Ray:[],
R:[],
t:[],
zz:[],
/* level 0 = r0 = parent node is a critical orbit */
/* level 1 : segments between zcr and a or b(zcr) */
zz: GiveLineSegment(zcr, a(zcr) , k_Max),
/*ray:zz, */
Ray : GiveSortedBTree(zz, depth_Max),
/*
zz : GiveLineSegment(zcr, b(zcr), kMax),
for z in zz do (
t: GiveBTree(z, depthMax),
ray:append(t,ray)
),
*/
for L in Ray do(
L:map(d, L),
L: points(L),
R:cons(L,R)
),
return(R)
)$
compile(all)$
/* ===================== const ===================================================== */
half :1.2$
yMax: half$
yMin: - half$
iSide: 1000$
PixelSize: (yMax - yMin)/iSide$
HalfPixelSize2: (PixelSize/2)*(PixelSize/2)$
/*
integer constant values
proportional to the :
* quality of the image ( number of the detailes, but also precision of curves )
* size of the svg file ( but not the png file )
* time of computation
*/
iMax:12; /* number of the orbit points: forward and backward */
kMax: 50; /* number of points in the line segment */
depthMax:15$ /* depth of binary tree = number of preimages */
/* parameter of the function : fc(z) = z^2 + c */
c:0.35;
zcr : 0; /* critical point */
zfp : GiveFixed_p(c) $ /* fixed point */
zfm : GiveFixed_m(c) $ /* fixed point */
/* ============================ computations =============================== */
/* precritical orbit = backward orbit of the critical point */
precritical : GiveBTree(zcr, depthMax)$
l_precritical : length(precritical)$
zz: GiveLineSegment(zcr, a(zcr) , 10)$
rayA : Give2PreimagesOfList(zz)$ /*GiveSortedBTree(zz, 10)$*/
rt:[]$
r2:[]$
r3:[]$
r4:[]$
r5:[]$
r6:[]$
r7:[]$
r8:[]$
r9:[]$
r10:[]$
r11:[]$
r12:[]$
r13:[]$
r14:[]$
r15:[]$
r16:[]$
r17:[]$
r18:[]$
for L in rayA do(
rt: Give2PreimagesOfList(L),
r2:cons(rt[1],r2),
r2:cons(rt[2],r2)
)$
for L in r2 do rayA:cons(L,rayA)$
for L in r2 do(
rt: Give2PreimagesOfList(L),
r3:cons(rt[1],r3),
r3:cons(rt[2],r3)
)$
for L in r3 do rayA:cons(L,rayA)$
for L in r3 do(
rt: Give2PreimagesOfList(L),
r4:cons(rt[1],r4),
r4:cons(rt[2],r4)
)$
for L in r4 do rayA:cons(L,rayA)$
for L in r4 do(
rt: Give2PreimagesOfList(L),
r5:cons(rt[1],r5),
r5:cons(rt[2],r5)
)$
for L in r5 do rayA:cons(L,rayA)$
for L in r5 do(
rt: Give2PreimagesOfList(L),
r6:cons(rt[1],r6),
r6:cons(rt[2],r6)
)$
for L in r6 do rayA:cons(L,rayA)$
for L in r6 do(
rt: Give2PreimagesOfList(L),
r7:cons(rt[1],r7),
r7:cons(rt[2],r7)
)$
for L in r7 do rayA:cons(L,rayA)$
for L in r7 do(
rt: Give2PreimagesOfList(L),
r8:cons(rt[1],r8),
r8:cons(rt[2],r8)
)$
for L in r8 do rayA:cons(L,rayA)$
for L in r8 do(
rt: Give2PreimagesOfList(L),
r9:cons(rt[1],r9),
r9:cons(rt[2],r9)
)$
for L in r9 do rayA:cons(L,rayA)$
for L in r9 do(
rt: Give2PreimagesOfList(L),
r10:cons(rt[1],r10),
r10:cons(rt[2],r10)
)$
for L in r10 do rayA:cons(L,rayA)$
for L in r9 do(
rt: Give2PreimagesOfList(L),
r10:cons(rt[1],r10),
r10:cons(rt[2],r10)
)$
for L in r10 do rayA:cons(L,rayA)$
for L in r10 do(
rt: Give2PreimagesOfList(L),
r11:cons(rt[1],r11),
r11:cons(rt[2],r11)
)$
for L in r11 do rayA:cons(L,rayA)$
for L in r11 do(
rt: Give2PreimagesOfList(L),
r12:cons(rt[1],r12),
r10:cons(rt[2],r12)
)$
for L in r12 do rayA:cons(L,rayA)$
for L in r12 do(
rt: Give2PreimagesOfList(L),
r13:cons(rt[1],r13),
r13:cons(rt[2],r13)
)$
for L in r13 do rayA:cons(L,rayA)$
for L in r13 do(
rt: Give2PreimagesOfList(L),
r14:cons(rt[1],r14),
r14:cons(rt[2],r14)
)$
for L in r14 do rayA:cons(L,rayA)$
for L in r14 do(
rt: Give2PreimagesOfList(L),
r15:cons(rt[1],r15),
r15:cons(rt[2],r15)
)$
for L in r15 do rayA:cons(L,rayA)$
/*
zz: GiveLineSegment(zcr, b(zcr) , 10)$
rayB : Give2PreimagesOfList(zz)$ /* GiveSortedBTree(zz, 10)$*/
*/
rayB:[]$
Ray:[]$
for L in rayA do(
/*rayB:append(map(minus,L), rayB), */
L:map(d, L),
L: points(L),
Ray:cons(L,Ray)
)$
/* rayB: comjugate(rayA */
for L in rayA do(
L:map(conjugate,L),
L:map(d, L),
L: points(L),
Ray:cons(L,Ray)
)$
/*
ray:
ray0 = along critical orbit
ray1 : from a(zcr) to b(zcr)
*/
critical_orbit : [zcr, half]$ /* GiveForwardOrbit(zcr, c, iMax)$ *//* critical orbit = forward orbit of the critical point */
ray1:[a(zcr), b(zcr)]$
critical_orbit: ToPoints( critical_orbit)$
precritical: ToPoints( precritical)$
ray1:ToPoints(ray1)$
path:"~/Dokumenty/branched_ray/maxima/n0/"$ /* pwd, if empty then file is in a home dir , path should end with "/" */
fileName: sconcat(path, string(depthMax), "_", string(kMax))$
/* draw it using draw package by */
load(draw);
/* if graphic file is empty (= 0 bytes) then run draw2d command again */
draw2d(
user_preamble="set key top right; unset mouse",
terminal = 'png,
file_name = fileName,
title= " dynamical plane : branched periodic external ray 0/1 for map f(z) = z*z + 0.35 ",
/* only upper box */
dimensions = [iSide, iSide],
yrange = [-half, half],
xrange = [-half, half],
/*
full image = both
dimensions = [1000, 2000],
yrange = [-0.8, 0.8],
xrange = [-0.1, 0.75],
*/
xlabel = "zx ",
ylabel = "zy",
point_type = filled_circle,
points_joined = true,
point_size = 0.5,
color = black,
critical_orbit,
ray1,
point_size = 0.1,
key="",
color=black,
Ray,
key = "preimages of critical point",
point_size = 0.5,
points_joined = false,
color = red,
precritical,
/* big points */
point_size = 1.1,
key= "fixed points",
color = blue,
points(dl(zfp)),
key = "",
points(dl(zfm)),
key= "critical point",
color = black,
points(dl(zcr))
);
/*
print("depth max =", depthMax, " kMax = ", kMax, " number of points in the ray = ", length(ray));
*/
print("file", fileName, " saved to",path)$
print("file name: depthMax_kMax.png")$
print("Pixel size : ", PixelSize)$
print("length of precritical orbit : ", l_precritical, " for depth : ", depthMax)$
old code
/*
Batch file for Maxima CAS
save as a s.mac
run maxima :
maxima
and then :
batch("s.mac");
*/
kill(all);
remvalue(all);
/* ---------- functions ---------------------------------------------------- */
/* http://en.wikipedia.org/wiki/Complex_quadratic_polynomial */
/* Forward iteration */
f(z):=float(rectform(z*z+c))$
/* backward iteration of complex quadratic polynomial */
b(z):=float(rectform(sqrt(z-c)))$
a(z):=float(rectform(-sqrt(z-c)))$
/* find fixed point alfa of function f(z,c) */
GiveFixed_p(c):= float(rectform((1+sqrt(1-4*c))/2))$
GiveFixed_m(c):= float(rectform((1-sqrt(1-4*c))/2))$
/*
converts complex number z = x*y*%i
to the list in a draw format:
[x,y]
*/
d(z):=[float(realpart(z)), float(imagpart(z))]$
/* give Draw List from one point*/
dl(z):=[d(z)]$
ToPoints(myList):= points(map(d,myList))$
/* gives an orbit of z0 under fc where iMax is the length of the orbit */
GiveForwardOrbit(z0,c, iMax):= block
(
[i,z, orbit],
z:z0,
orbit :[z],
i:1,
while ( i<iMax )
do
(
z:f(z),
orbit : endcons(z, orbit),
i:i+1
),
return(orbit)
)$
/*
Line segment = part of a line that is bounded by two end points
input :
- 2 endpoints : z1, z2
- imax : number of points between endpoints
output:
- list of complex points
*/
GiveLineSegment(z1,z2,iMax):=block
(
[l,z, dz, t , tmin, tmax,dt,i],
dz : z2-z1,
dt : 1/iMax,
tmin: 0,
tmax: 1,
z:z1,
l:[z1],
for t:tmin thru tmax step dt
do
(
z:z1+dz*t,
l :cons(z,l)
),
return(l)
)$
/*
input :
- list a
- parameter n : every n-th element is removed from the list to make it shorter
output : list b
https://stackoverflow.com/questions/49658018/how-to-remove-many-elements-from-the-list-by-checking-its-index-in-maxima-cas
kMax:100
depthMax 12; length(ray) = 58548
depthMax 13; length(ray) = 65076
depthMax 14; length(ray) = 71604
depthMax 15; length(ray) = 78132
depthMax 16; length(ray) = 84660
depthmax 17; ray = 91188
depthmax 18 ray = 97716
depth max = 19 number of points in the ray = 104244
depth max = 20 number of points in the ray = 110772
epth max = 20 kMax = 200 number of points in the ray = 219372
*/
ShortList(a, n):=block(
[l, b],
iMax:length(a),
if ( iMax > 20) then (
b:[],
for i:1 thru iMax step 1 do
if (mod(i,n)=0) then b:cons(a[i],b)
)
else b: a, /* do nothing */
return (b)
)$
/*
binary complete tree
to do :
implemented as an array
Pre-order sequence: ABDECFG ( for depth 3)
for(i=0;i<N;i++){ process node in position i; } results in a level-by-level traversal.
*/
/* binary tree
- read input A
- compute 2 preimages b and c
- output list : [b,c]
*/
Give2Preimages(z):= [a(z), b(z)]$
/*
regular complete binary tree depth
- read input A
- compute full tree of preimages depth d
( number = sum(2^(d-1))
- output list zz of complex points
lists
zz is output list
t1 input = previous level of the tree
t2 next level
temp temporary list
*/
GiveBTree(z, depthMax, n):=block(
[ zz, t1, t2, tmp],
zz :[],
t1 :[z],
t2 :[],
tmp :[],
for depth : 1 thru depthMax step 1 do (
for z in t1 do (
tmp : Give2Preimages(z),
zz : append(tmp, zz),
t2 : append(tmp, t2)
),
tmp:[],
t1:t2,
t2:[],
if depth > 2 then t1:ShortList(t1,n)
),
zz:flatten(zz),
return (zz)
)$
/*
Gives external dynamic ray ( as a list of points in a draw format)
input:
- zcr: critical point z
- c : parameter of the function fc(z) = z^2 + c
- kMax : number of points in the line segment
- parameter n : every n-th element is removed from the list to make it shorter, see ShortList
*/
R2(zcr, c, depthMath, kMax, n):= block(
ray:[],
/* level 0 = r0 = parent node is a critical orbit */
/* level 1 : segments between zcr and a or b(zcr) */
zz: GiveLineSegment(zcr, a(zcr) , kMax),
ray:zz,
for z in zz do (
t: GiveBTree(z, depthMax, n),
ray:append(t,ray)
),
zz : GiveLineSegment(zcr, b(zcr), kMax),
ray:append(zz,ray),
for z in zz do (
t: GiveBTree(z, depthMax, n),
ray:append(t,ray)
),
ray:flatten(ray),
ray:map(d,ray),
return(ray)
)$
/*
compile(all)$
*/
/* ===================== const ===================================================== */
half :1.2$
yMax: half$
yMin: - half$
iSide: 1000$
PixelSize: (yMax - yMin)/iSide$
/*
integer constant values
proportional to the :
* quality of the image ( number of the detailes, but also precision of curves )
* size of the svg file ( but not the png file )
* time of computation
*/
iMax:15; /* number of the orbit points: forward and backward */
kMax: 200; /* number of points in the line segment */
depthMax:20$ /* depth of binary tree = number of preimages */
n:2$ /* every n-th element is removed from the list to make it shorter , see ShortList */
/* parameter of the function : fc(z) = z^2 + c */
c:0.35;
zcr : 0; /* critical point */
zfp : GiveFixed_p(c) $ /* fixed point */
zfm : GiveFixed_m(c) $ /* fixed point */
/* ============================ computations =============================== */
/* critical orbit = forward orbit of the critical point */
critical_orbit : GiveForwardOrbit(zcr, c, iMax)$
/* precritical orbit = backward orbit of the critical point */
precritical : GiveBTree(zcr, depthMax, n)$
/* ray: GiveRay(zcr,c)$ */
ray:R2(zcr, c, depthMax, kMax, n)$
critical_orbit: ToPoints( critical_orbit)$
precritical: ToPoints( precritical)$
path:"~/Dokumenty/branched_ray/maxima/sl2/"$ /* pwd, if empty then file is in a home dir , path should end with "/" */
fileName: sconcat(path, string(depthMax), "_", string(kMax),"_", string(n))$
/* draw it using draw package by */
load(draw);
/* if graphic file is empty (= 0 bytes) then run draw2d command again */
draw2d(
user_preamble="set key top right; unset mouse",
terminal = 'png,
file_name = fileName,
title= " dynamical plane : branched periodic external ray 0/1 for map f(z) = z*z + 0.35 ",
/* only upper box */
dimensions = [iSide, iSide],
yrange = [-half, half],
xrange = [-half, half],
/*
full image = both
dimensions = [1000, 2000],
yrange = [-0.8, 0.8],
xrange = [-0.1, 0.75],
*/
xlabel = "zx ",
ylabel = "zy",
point_type = filled_circle,
points_joined = false,
point_size = 0.1,
color = black,
key = "",
points(ray),
key="",
point_size = 0.1,
points_joined = true,
key = "",
color = black,
critical_orbit,
key = "preimages of critical point",
point_size = 0.5,
points_joined = false,
color = red,
precritical,
/* big points */
point_size = 1.1,
key= "fixed points",
color = blue,
points(dl(zfp)),
key = "",
points(dl(zfm)),
key= "critical point",
color = black,
points(dl(zcr))
);
draw2d(
user_preamble="set key top right; unset mouse",
terminal = 'svg,
file_name = fileName,
title= " dynamical plane : branched periodic external ray 0/1 for map f(z) = z*z + 0.35 ",
/* only upper box */
dimensions = [iSide, iSide],
yrange = [-half, half],
xrange = [-half, half],
/*
full image = both
dimensions = [1000, 2000],
yrange = [-0.8, 0.8],
xrange = [-0.1, 0.75],
*/
xlabel = "zx ",
ylabel = "zy",
point_type = filled_circle,
points_joined = false,
point_size = 0.1,
color = black,
key = "",
points(ray),
key="",
point_size = 0.1,
points_joined = true,
key = "",
color = black,
critical_orbit,
key = "preimages of critical point",
point_size = 0.5,
points_joined = false,
color = red,
precritical,
/* big points */
point_size = 1.1,
key= "fixed points",
color = blue,
points(dl(zfp)),
key = "",
points(dl(zfm)),
key= "critical point",
color = black,
points(dl(zcr))
);
print("depth max =", depthMax, " kMax = ", kMax, " number of points in the ray = ", length(ray));
print("file", fileName, " saved to",path)$
print("file name: depthMax_kMax_n.png")$
- ↑ Fixed points of polynomial maps. Part II. Fixed point portraits Lisa R. Goldberg; John Milnor Annales scientifiques de l'École Normale Supérieure (1993) Volume: 26, Issue: 1, page 51-98 ISSN: 0012-9593, see Fig 14 page 91
- ↑ Exploring the Mandelbrot set. The Orsay Notes. October 2009 Adrien DouadyJohn H. HubbardJohn H. Hubbard. PAGE 64
- ↑ draw package ( Maxima-Gnuplot interface) by Mario Rodríguez Riotorto
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| Date/Time | Thumbnail | Dimensions | User | Comment | |
|---|---|---|---|---|---|
| current | 14:02, 9 November 2020 | | 1,000 × 1,000 (47 KB) | Soul windsurfer | better quality |
| 21:48, 7 November 2020 |  | 1,000 × 1,000 (32 KB) | Soul windsurfer | Uploaded own work with UploadWizard |
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_%3D_z*z_%2B_0.35.png){kind=link}