Summary

Long description

conformal mapping from : the circle with center=0 and radius=1

given by equation : ${\displaystyle abs(w)=1\,}$

where : ${\displaystyle w=x+y*i=r*cos(t)+i*r*sin(t)=8*P\,}$

onto boundary of 3 period 3 hyperbolic components of Mandelbrot set^[1] is given by 3 equations :

${\displaystyle c=\gamma _{3}(P)=g_{3}({\frac {w}{8}})\,}$

code of functions gamma by Robert P. Munafo^[2].

These functions are computed in Maxima from boundary equation ^[3]

(%i3) b3:c^3+2*c^2+(1-P)*c+(P-1)^2=0$
(%i4) solve(b3,c);
(%o4) [c=(-(sqrt(3)*%i)/2-1/2)*(((P-1)*sqrt(27*P^2-22*P+23))/(6*sqrt(3))-(27*P^2-36*P+25)/54)^(1/3)+
(((sqrt(3)*%i)/2-1/2)*(3*P+1))/(9*(((P-  1)*sqrt(27*P^2-22*P+23))/(6*sqrt(3))-(27*P^2-36*P+25)/54)^(1/3))-2/3,c=((sqrt(3)*%i)/2-1/2)*
(((P-1)*sqrt(27*P^2-22*P+23))/(6*sqrt(3))-(27*P^2-36*P+25)/54)^(1/3)+
((-(sqrt(3)*%i)/2-1/2)*(3*P+1))/(9*(((P-1)*sqrt(27*P^2-22*P+23)) /(6*sqrt(3))-
(27*P^2-36*P+25)/54)^(1/3))-2/3,c=(((P-1)*sqrt(27*P^2-22*P+23))/(6*sqrt(3))-(27*P^2-36*P+25)/54)^(1/3)+
(3*P+1)/(9*(((P-1)*sqrt(27*P^2-22*P+23))/(6*sqrt(3))-(27*P^2-36*P+25)/54)^(1/3))

Maxima source code

NmbrOfPnts:400;
/* ------------------------ definitions -------------------------*/
/* code of functions gamma by Robert P. Munafo */
/* conformal maps from circle to period 3 components  */
gamma3a(P):=(-(sqrt(3)*%i)/2-1/2)*(((P-1)*sqrt(27*P^2-22*P+23))/(6*sqrt(3))-(27*P^2-36*P+25)/54)^(1/3)+
(((sqrt(3)*%i)/2-1/2)*(3*P+1))/(9*(((P-1)*sqrt(27*P^2-22*P+23))/(6*sqrt(3))-(27*P^2-36*P+25)/54)^(1/3))-2/3;
g3a(w):=gamma3a(w/8);
gamma3b(P):=((sqrt(3)*%i)/2-1/2)*(((P-1)*sqrt(27*P^2-22*P+23))/(6*sqrt(3))-(27*P^2-36*P+25)/54)^(1/3)+
((-(sqrt(3)*%i)/2-1/2)*(3*P+1))/(9*(((P-  1)*sqrt(27*P^2-22*P+23))/(6*sqrt(3))-(27*P^2-36*P+25)/54)^(1/3))-2/3;
g3b(w):=gamma3b(w/8);
gamma3c(P):=(((P-1)*sqrt(27*P^2-22*P+23))/(6*sqrt(3))-(27*P^2-36*P+25)/54)^(1/3)+(3*P+1)/(9*(((P-1)*sqrt(27*P^2-22*P+23))/
(6*sqrt(3))-    (27*P^2-36*P+25) /54)^(1/3))-2/3; 
g3c(w):=gamma3c(w/8);
/* exponential for of complex number with angle in turns */
l(t,r):=r*%e^(%i*t*2*%pi);
/* ------------------------- */
GiveIntRayOfCircleL(angle,NmbrOfPnts):=
makelist(rectform(ev(l(angle,j/NmbrOfPnts), numer)),j, 0,NmbrOfPnts);
/* ---------------------- */
GiveUnitCircleL(NmbrOfPnts):=
makelist(rectform(ev(l(j/NmbrOfPnts,1), numer)),j, 0,NmbrOfPnts);
/* ------------------  computes points of circle and ray ------------------*/
circleList:GiveUnitCircleL(100); 
/* ------------------- conformal mapping from  circle to 3 period 3 components ----------------*/
c3aList:map(g3a, circleList);
c3bList:map(g3b, circleList);
c3cList:map(g3c, circleList);
/* ---------- ------------ drawing -------------------------*/
load(draw); /* Mario Rodríguez Riotorto   http://www.telefonica.net/web2/biomates/maxima/gpdraw/index.html archive copy at the Wayback Machine */
draw(file_name = "hypComp3",
pic_width=1000, 
pic_height= 500,
terminal  = 'png,
columns  = 2,
gr2d(title = " unit circle D={w:abs(w)=1 } in standard plane ",
 points_joined =true,
 color         = red,
 point_type = 0,
 points(map(realpart,circleList),map(imagpart,circleList))
 ),
gr2d(title      = "3 images of circle under 3 maps gamma3(w) in parameter plane",
 points_joined =false,
 color         = blue,
 point_type = 0,
 points(map(realpart,c3aList),map(imagpart,c3aList)),
 color         = green,
 points(map(realpart,c3bList),map(imagpart,c3bList)),
 color         = magenta,
 points(map(realpart,c3cList),map(imagpart,c3cList)))  );

References

1. ↑ Boundaries of 53 hyperbolic components of Mandelbrot set for periods 1-6 made with polynomial maps from the unit circle
2. ↑ MuEncy by Robert P. Munafo
3. ↑ Computing boundary of hyperbolic components from boundary equations
4. ↑ Walter Hannah thesis, see page 11
5. ↑ A Parameterization of the Period 3 Hyperbolic Components of the Mandelbrot Set Dante Giarrusso; Yuval Fisher Proceedings of the American Mathematical Society, Vol. 123, No. 12. (Dec., 1995), pp. 3731-3737

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