Another aproach to this problem is :
Complex Dynamics by Lennart Carleson,Theodore Williams Gamelin page 40
( how to compute coefficient a ? )
I have made a Maxima program, which is here :
http://commons.wikimedia.org/wiki/File:Parabolic_Julia_set_for_internal_angle_1_over_5_-_new_method.png
It does not work.
TIA
Adam
PS. Below is a program which shows what I expect for different periods :
========== code using vectors ==============
kill(all);
remvalue(all);
/* compute vectors at the given points vector ([x,y], [dx,dy])
vector([x,y], [dx,dy]) plots vector [dx,dy] with origin in [x,y].
*/
GiveCirclePoint(t):=rectform(%e^(%i*t*2*%pi)); /* gives point of unit
circle for angle t in turns */
GiveVector(origin, target):=block(
[x,y,dx,dy],
x:realpart(origin),
y:imagpart(origin),
dx:realpart(target)-x,
dy:imagpart(target)-y,
vector([x,y],[dx,dy])
)$
GiveAttractingVector(origin):=GiveVector(origin, 0)$ /* from origin to 0 */
GiveRepellingVector(target):=GiveVector(0, target)$ /* from 0 to target */
GiveAttractingVectors(Denominator):=(
[AttractingInternalAngles, AttractingCirclePoints,nMax],
nMax:Denominator,
AttractingInternalAngles:makelist(n/nMax ,n,1,nMax),
AttractingCirclePoints:listify(map(GiveCirclePoint,AttractingInternalAngles
)),
listify(map(GiveAttractingVector,AttractingCirclePoints))
)$
GiveRepellingVectors(Denominator):=(
[RepellingInternalAngles, AttractingCirclePoints,nMax],
nMax:Denominator,
RepellingInternalAngles:makelist(n/nMax ,n,1,nMax),
RepellingCirclePoints:listify(map(GiveCirclePoint,RepellingInternalAngles )),
listify(map(GiveRepellingVector,RepellingCirclePoints))
)$
GiveScene(sTitle, vector):=
gr2d(title= sTitle,
user_preamble = "set border 0;set nokey;set size square;set
noxtics ;set noytics;",
yrange = [-1,1],
xrange = [-1,1],
[head_length=0.1, head_angle = 10, color=blue],
vector
)$
compile(all);
/* ----------------*/
scenes:[];
for Denominator:1 thru 10 step 1 do
(
scene:GiveScene(string(1/Denominator),GiveRepellingVectors(Denominator)),
scenes:endcons(scene,scenes))$
/* ----------draw the vector field ----------------------------*/
draw(
terminal = 'svg,
columns =5,
file_name = "/home/a/maxima/parabolic/a_directions/repelling",
pic_width = 1000, /* Since Maxima 5.23, pic_width and
pic_height are deprecated. */
pic_height = 500, /* See option dimensions. To get the same
effect, write dimensions=[800,600] */
scenes
)$