multiplier ( = eigenvalue )






ln(c) = l(c, F, n ) = d F(n) / dz

where :
ln(c) it is a multiplier = complex number

It creates polar coordinate system for hyperbolic components of the Mandelbrot set .
Multiplier is used to classify periodic points with stability index = Abs(l)


For periodic points the first derivative of Mandelbrot map  FM(z,c,n) with respect to z is tha same for all the points in the cycle
F'M(Z0,c,n):= F'M(Z1,c,1) * F'M(Z2,c,1) * .. * F'M(Zn,c,1) = 2*Z0 *2 * Z2* .. * 2*Zn-1

F'M(z0,c,1):= 2 * Z0;
F'M(Z0,c,2):= 22*Z0*Z1 = 4*z^3+4*c*z
F'M(Z0,c,3):= 23*Z0*Z1*Z2=8*z^7+24*c*z^5+24*c^2*z^3+8*c*z^3+8*c^3*z+8*c^2*z
F'M(Z0,c,4):=16*z^15+112*c*z^13+336*c^2*z^11+48*c*z^11+560*c^3*z^9+240*c^2*z^9+560*c^4*z^7+480* c^3*z^7+48*c^2*z^7+16*c*z^7+336*c^5*z^5+480*c^4*z^5+144*c^3*z^5+48*c^2*z^5+112*c^6*z^3+240*c^5*z^3+144*c^4*z^3+64*c^3*z^3+16*c^2*z^3+16*c^7*z+48*c^6*z+48*c^5*z+32*c^4*z+16*c^3*z
F'M(Z0,c,5):=32*z^31+480*c*z^29+3360*c^2*z^27+224*c*z^27+14560*c^3* z^25+2912*c^2*z^25+43680*c^4*z^23+17472*c^3*z^23+672*c^2*z^23+96*c*z^23+96096*c^5*z^21+64064*c^4*z^21+7392*c^3*z^21+1056*c^2*z^21+160160*c^6*z^19+160160*c^5*z^19+36960*c^4*z^19+6400*c^3*z^19+480*c^2*z^19+205920*c^7*z^17+288288*c^6*z^17+110880*c^5*z^17+25920*c^4*z^17+4320*c^3*z^17+205920*c^8*z^15+384384*c^7*z^15+221760*c^6*z^15+72000*c^5*z^15+18400*c^4*z^15+960*c^3*z^15+96*c^2*z^15+32*c*z^15+160160*c^9*z^13+384384*c^8*z^13+310464*c^7*z^13+138432*c^6*z^13+48160*c^5*z^13+6720*c^4*z^13+672*c^3*z^13+224*c^2*z^13+96096*c^10*z^11+288288*c^9*z^11+310464*c^8*z^11+185472*c^7*z^11+84000*c^6*z^11+20832*c^5*z^11+2976*c^4*z^11+960*c^3*z^11+96*c^2*z^11+43680*c^11*z^9+160160*c^10*z^9+221760*c^9*z^9+172800*c^8*z^9+99680*c^7*z^9+36960*c^6*z^9+8160*c^5*z^9+2560*c^4*z^9+480*c^3*z^9+14560*c^12*z^7+64064*c^11*z^7+110880*c^10*z^7+109920*c^9*z^7+79520*c^8*z^7+40320*c^7*z^7+13184*c^6*z^7+4480*c^5*z^7+1248*c^4*z^7+128*c^3*z^7+32*c^2*z^7+3360*c^13*z^5+17472*c^12*z^5+36960*c^11*z^5+45600*c^10*z^5+40800*c^9*z^5+26880*c^8*z^5+12288*c^7*z^5+4992*c^6*z^5+1824*c^5*z^5+384*c^4*z^5+96*c^3*z^5+480*c^14*z^3+2912*c^13*z^3+7392*c^12*z^3+11136*c^11*z^3+12160*c^10*z^3+10080*c^9*z^3+6144*c^8*z^3+3136*c^7*z^3+1440*c^6*z^3+480*c^5*z^3+160*c^4*z^3+32*c^3*z^3+32*c^15*z+224*c^14*z+672*c^13*z+1216*c^12*z+1600*c^11*z+1632*c^10*z+1280*c^9*z+832*c^8*z+480*c^7*z+224*c^6*z+96*c^5*z+32*c^4*z
..
F'M(Z0,c,n):= 2n*Z0*Z1*.. *Zn-1;

..........................
How it was computed ?
dFc(2)(Z0)/dZ =
         because it is composition of one function on another
dFc(Fc(Z0))/dZ=
         (using chain rule)
Fc'(Z1)*Fc'(Z0)=
         because F'c(Z)= 2*Z
2*Z0*2*Z1=
22*Z0*Z1

Multiplier map

Multiplier function maps hyperbolic components of the Mandelbrot set conformally onto the unit disk
m(c,F,period) : c --> l(c, F, n )

if Abs(l(c, F, n ))< 1 then c belongs to hyperbolic component of period n
if Abs(l(c, F, n )) = 1 then c belongs to boundary of hyperbolic component of period n

l1 = m(c,F,1) = 1-sqrt(1-4c)     reverse function c = l/2 - l2/4
l2 = m(c,F,2) = 4(1+c)     reverse function c = l/4 - 1



Parametrization of boundary of hyperbolic components of Mandelbrot set by internal angle
gives explicit formula for components of Mandelbrot set

boundary point c : Abs(m(c,F,period)) = 1

for period 1 :
in polar form:
c = c(angle) = ei *angle/2 - (ei *angle)2/4 = ei *angle/2 - e2*i *angle/4
in trigonometric form :
x = cos(angle)/2 - cos(2*angle)/4
y = sin(angle)/2 - sin(2*angle)/4

for period 2 :
c = c(angle) = ei *angle/4 - 1


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Autor: Adam Majewski
adammaj1-at-o2-dot-pl
Feel free to e-mail me. (:-))

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