root of component of M-set







Root point of hyperbolic component of Mandelbrot set :

Definition :
Parameter point is parabolic if and only if the corresponding quadratic map has a periodic orbit with some root of unity as a multiplier.

parameter point = point of parametr plane
for complex quadratic map Fc(Z)= (Z*Z) + c
   c is parameter
   z is variable

Theorem:
Every parabolic point c < > 1/4 is a landing point for exactly two external rays with angles which are periodic under doubling. [DH2]



Computing root points on boundary of main cardioid in maxima:
(%i30) c(a):=0.5*exp(%i*2*%pi*a) - 0.25* exp(2*%i*2*%pi*a);
(%o30) c(a):=0.5*exp(%i*2*%pi*a)-0.25*exp(2*%i*2*%pi*a)
(%i31) c(0);
(%o31) 0.25
(%i32) c(1);
(%o32) 0.25
(%i33) c(0.5);
(%o33) -0.75
(%i34) c(1/3);
(%o34) 0.5*((sqrt(3)*%i)/2-1/2)-0.25*(-(sqrt(3)*%i)/2-1/2)
(%i35) rectform(%);
(%o35) 0.375*sqrt(3)*%i-0.125
(%i36) float(%), numer;
(%o36) 0.64951905283833*%i-0.125


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Autor: Adam Majewski adammaj1-at-o2-dot--pl

Feel free to e-mail me. (:-))
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