Summary

DescriptionDomains for Fatou coordinate for f(z) = z+a 2z ^2+O(z^3).png	English: Domains for Fatou coordinate for ${\displaystyle f(z)=z+a_{2}z^{2}+O(z^{3})}$
Date	25 May 2014, 09:35:57
Source	Own work
Author	Adam majewski

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compare with

M 597 LECTURE NOTES TOPICS IN MATHEMATICS COMPLEX DYNAMICS by LUKAS GEYER, page 39

Maxima CAS src code

/*

 b batch file for maxima



There are 2 complex planes :
* u-plane 
* z-plane 

http://arxiv.org/abs/1404.4735
Near parabolic renormalization for unisingular holomorphic maps by Arnaud Chéritat

1 step transformation from u plane go to z plane using z=h(u)


 
*/

kill(all);
remvalue(all);


/*

*/



iMax:1000; /* number of points to draw */
uxMin: -80;
 /* */


/* =================== functions ============ */


/* 

uy:f(ux)

"Let θ ∈ [π/2, π] and W_θ(r) denote the following domain: 
it contains a right half plane and 
is bounded by the arc of circle of center 0, radius r 
and argument ranging from −(θ − π/2) to θ − π/2, 
and by the two half lines continuing this arc tangentially to the circle (see Figure 22).
circle is centered in origin and have radius=r 

For θ = π/2 and r = R_0[g] = 1/|c g |r 0 ,
this domain is exactly the half plane, image of D attr in the u-coordinates of g.""


*/

f(ux):= if (ux<uxt) 
             then (a*ux + b )      /* line segment */
             else  sqrt(r2-ux*ux); /*  circle segment */



/*

1 step transformation from u plane go to z plane using z=h(u)

h(0)

expt: undefined: 0 to a negative exponent.
 -- an error. To debug this try: debugmode(true);

h(u):= if (u=0.0) then infinity else -1/u$
*/
h(u):= if (u=0.0) then infinity else -1/u$

m(u):= -realpart(u)+imagpart(u)*%i$ /* minus */


/* 

 inverse function of f
 ux = fi(uy)

 */
fi(uy):=block
(
 [s,ux],
 if (uy>uyt) 
    then s: (uy - b)/a
    else s:sqrt(r2-uy*uy),
 s:float(s),
 return(s)
)$



/* converts complex number into list for draw package  */
 draw_format(z):= if (z=infinity) then [1000.0,1000.0] else  [float(realpart(z)),float(imagpart(z))];


 /*  line im(u) = const in a draw format : point(list ) 
  uu is a list of u values
  zz is a list of z values
  u = ux + uy*%i

*/
GiveHorizontalLines(uy):= 
block(
  [uu,zz],
  uu:makelist ( uy*%i + (1.0 +k/10), k, -1000, 1000 ),
  zz:map(h,uu),
  uu:map(draw_format,uu),
  zz:map(draw_format,zz),
  [points(uu),points(zz)] /* list of 2 sublists : first is a uuh list , second is a zzh list */
  
)$ 





compile(all);



/* ============== compute ===============  */
r : 2; /* radius of the circle */
r2: r*r;

t:1/8; /* angle in turns at which line is tangent to circle  */
/* point in which line is tangent to circle  wt : wxt+wyt*i */
uxt:r*cos(2*%pi*t);
uyt:r*sin(2*%pi*t);

a:-1;
b:2*uyt;


uxMax: r;
uxStep: (uxMax-uxMin)/iMax;



/* point to point method of drawing 
 compute first point of curve, create list and save point to this list */

 
 uListA1:[]; 
 

 for ux:uxMin step uxStep  while (ux<= uxMax) do
  (  
     uy:f(ux),
     uListA1:cons(ux+uy*%i,uListA1),  
     uListA1:endcons(ux-uy*%i,uListA1)
 )$


 uListR1: map(m, uListA1)$
 zListA1:map(h,uListA1)$
 zListR1: map(m, zListA1)$

 uListA0 : makelist (r+%i*k/10, k, -400, 700 )$ /* line re(w)=0 */
 zListA0: map(h,uListA0)$
 
 uListA01 : makelist (r+1+ %i*k/10, k, -400, 700 )$ /* line re(w)=0 */
 zListA01 : map(h,uListA01)$

 /* horizontal lines */
 uh:[0,1,2,3,4,5,6,-1,-2,-3,-4,-5,-6]$ /* list of  values for lines im(u) = const */
 uzh:map(GiveHorizontalLines,uh)$
 uuh:map(first,uzh)$
 zzh:map(second,uzh)$
 
 
 


/* single important points */
 z0 : 0; /* origin z=0 */
 

/* convert list to draw format */
 uListA01:map(draw_format,uListA01)$ 
 uListA0:map(draw_format,uListA0)$ 
 uListA1:map(draw_format,uListA1)$ 
 uListR1:map(draw_format,uListR1)$ 

 zListA01:map(draw_format,zListA01)$ 
 zListA0:map(draw_format,zListA0)$ 
 zListA1:map(draw_format,zListA1)$ 
 zListR1:map(draw_format,zListR1)$ 


 
 z0:draw_format(z0);

/* =================  draw ======================================*/

 path:""$ /* pwd  ; if you put here working directory name then graphic file will be saved in that dir */
 FileName:concat(string(r),"r")$ /* without extension which is the terminal name */

 load(draw); /* Mario Rodríguez Riotorto   http://riotorto.users.sourceforge.net/gnuplot/points/index.html */
 draw(
  terminal  = 'svg,
  file_name = concat(path,FileName),
  columns  = 2,
  dimensions=[1000,500], /*  x = y*columns  */
  
  gr2d(title = " u plane ",
   /* xrange = [0,3],  xtics={-10, -2, 0,2}, */
   yrange = [-20.0,20.0],
   xrange = [-20.0,20.0],
   grid = false,
   xaxis       = false,
   points_joined =true,
   
   point_size    = 0.2,
   point_type = filled_circle,
   color=green,
   points(uListA0),
   color         = yellow,
   points(uListA01), 

   color         = red,
   points(uListA1), 
   
   color         = blue,
   points(uListR1),

  /* key = "orbits = invariant cirves",*/
   color = gray,
   uuh
   
   ),

  gr2d(title = " z plane : z = -1/u with petals ",
   yrange = [-0.8,0.8],
   xrange = [-0.8,0.8],
   points_joined =true,
   grid = false,
   point_size    = 0.2,
   point_type = filled_circle,
   key = "small attracting petal",
   color         = yellow,
   points(zListA01), 
   key = "small attracting petal",
   color=green,
   points(zListA0),
   key = "big attracting petal",
   color         = red,
   points(zListA1), 
   key = "big repelling petal",
   color         = blue,
   points(zListR1),
   key = "",
   /*key = "orbits = invariant cirves",*/
   color = gray,
   zzh,

   points_joined =false,
   color = black,
   point_size    = 0.8,
   key="fixed point",
   points([z0])
  )

 );

Image Magic src code

My gnuplot does not have png terminal, svg file is very big so :

convert 2r.svg a.png