Summary

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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 ${\displaystyle \phi ,2\phi ,4\phi ...}$

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

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

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

The one that is closer to ${\displaystyle z_{l-1,j}}$ 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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Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled GNU Free Documentation License.http://www.gnu.org/copyleft/fdl.htmlGFDLGNU Free Documentation Licensetruetrue

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