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Summary

DescriptionBackward Iteration.svg	English: Backward iteration of complex quadratic polynomial with proper chose of the preimage
Date	26 September 2012
Source	Own work
Author	Adam majewski

iThe source code of this SVG is valid.

This plot was created with Gnuplot by v.

Category:Valid SVG created with Gnuplot#Backward%20Iteration.svg

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Preimages of the circle under map f(z) = z*z+0.25

Summary

"In the dynamic plane, external rays can be drawn by backwards iteration. It is most effective for a periodic or preperiodic angle.

You must keep track of points on the finite collection of rays with angles $\phi ,2\phi ,4\phi ...$

Say $z_{l,j}$ corresponds to a radius $R^{\frac {1}{2l}}$ and the angle $2j\phi$ .

Then $f_{c}(z)$ maps $z_{l,j}$ to $z_{l-1,j+1}$ .

This point, which was constructed before, has two preimages under $f_{c}(z)$ .

The one that is closer to $z_{l-1,j}$ is the correct one. This criterion was proved by Thierry Bousch. The ray will look better when you introduce intermediate points." Wolf Jung

Maxima CAS src code

/* 

batch file for Maxima CAS 
It shows how to choose one from 2 preimages under complex quadratic polynomial

comments  are from Wolf Jung program Mandel 
http://www.mndynamics.com/indexp.html

*/

kill(all);
remvalue(all);

/* 
Say z_{l, j } corresponds to:
- a radius = R^{(1/2) ^l } 
-  the angle = t*(2^j) 
Then fc(z) maps z_{l,j}  to  z_{(l-1),(j+1)}.
Inverse function maps z_{l,j} to 
z_{(l+1),(j-1)}

 */
p(radius, angle, l,j):= radius^((1/2)^l) * %e^(%i*angle*(2^j)*2*%pi);

/* 
circle D={w:abs(w)=1 } where w=l(t,r) 
t is angle in turns ; 1 turn = 360 degree = 2*Pi radians 
*/
tMax:100;
/* exponential for of complex number with angle in turns */
GiveCirclePoint(t):=R*%e^(%i*t*2*%pi)$ /* gives point of unit circle for angle t in turns */
/*-------------- unit circle ------------*/
R:1;
circle_angles:makelist(t/tMax,t,0,tMax-1)$
CirclePoints:map(GiveCirclePoint,circle_angles)$

/* initial points on the rays t and 2*t */

R:20;

t:1/3; /* initial angle */
/* 
initial points on periodic rays 
" 
You must keep track of points on the finite collection of rays with angles t, 2t, 4t... 
Say z_{l, j } corresponds to:
- a radius = R^{(1/2) ^l } 
-  the angle = t*(2^j) 
"
*/
z00:p(R,t,0,0); /* on ray 1/3 */
z01:p(R,t,0,1); /* on ray 2*1/3=2/3 */

z1m1:p(R,t,1,-1); /* preimages of z00  */

pre1:[z1m1,-z1m1];

if (z1m1.z01>0) then z11:z1m1 else z11:-z1m1;

ray1:[z01,z11];

load(draw); /* Mario Rodriguez Riotorto   http://www.telefonica.net/web2/biomates/maxima/gpdraw/index.html */
draw2d(file_name = "iteration6",
	pic_width=1000, 
	pic_height= 1000,
     terminal  = 'svg,
	 
	
	
	title = "Backward iteration with proper chose of preimage ",
		user_preamble = "set angles degrees; set grid polar 30; set xtics 5; set mxtics 5; 
							set size square;set key out;set key top right",
		yrange = [-20,20],
		xrange = [-20,20],
		points_joined =true,
		color         = black,
	     point_type = filled_circle,
		 point_size    = 0.1,
	     points(map(realpart, CirclePoints),map(imagpart, CirclePoints)),
		 

                 points_joined =true,
		 point_size    = 0.9,
                 line_width = 3,
                 color = black,
                 key = "ray 2/3",
                 points(map(realpart,ray1),map(imagpart,ray1)),
		 color         = red,
                 point_size    = 1.0,
                  points_joined =false,
		 key = "z0 on ray 1/3",
		 label(["z00",realpart(z00)+2,imagpart(z00)+2]),
		 points([realpart(z00)],[imagpart(z00)]),
		 color         = green,
		 key = "z0 on ray 2/3",
		 label(["z01",realpart(z01)+2,imagpart(z01)-2]),
		 points([realpart(z01)],[imagpart(z01)]),
                 color         = blue,
		 key = "good preimage of z0 from ray 1/3",
		 points([realpart(z11)],[imagpart(z11)]),
		color         = black,
		 key = "bad preimage of z0 from ray 1/3",
		 points([realpart(-z11)],[imagpart(-z11)])	
		 
		);

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