Forward and backward ( inverse) iteration of complex quadratic polynomial






Function :
  • fc(z) = z*z +c; Gives one value ( image)
  • fc-1(z) =sqrt( z -c); Gives 2 values (preimages). It is multivalued function.

Iteration :
  • forward : zn+1 = fc(zn); is simple
  • backward : zn-1 = fc-1(zn); is not simple. It is impossible to choose good preimage without extre informations !

Orbit :
  • forward : list of points {z0, z1, z2, z3... , zn}
  • backward : binary tree of preimages of zn; One can't choose good path in such tree without extra informations.

Where orbit goes ?
  • forward : to attractor ( for example infinity is always superattracting fixed point for polynomials)
  • backward : to repelling point ( which are in Julia set = IIM/J) or preperiodic points



Example :


external angle : t = 1/3 in turns
radius : r = abs(z) = 2
c = 0
z0 = r*exp(i*t) = 1.732050807568877*%i-1.0

Forward iteration of z0 :

z0 = 1.732050807568877*%i-1.0 = 2*exp(i*1/3)
z1 = -3.464101615137754*%i-2.0 = 4*exp(i*2/3)
z2 = 13.85640646055102*%i-8.0 = 16*exp(i*1/3)

so forward orbit = {1/3, 2/3, 1/3 }


Backward iteration of z2 :

  • z2 = 13.85640646055102*%i-8.0 = 16*exp(i*1/3)
  • 2 preimages :
    • z2m1pv = z2-1 = 3.464101615137754*%i+2.0 = 4*exp(i*1/6) ( principal value of arg )
      • z2m2ppv = z2-2 = %i+1.732050807568877 = 2*exp(i*1/12)
      • z2m2psv = z2-2 = -%i-1.732050807568877 = 2*exp(i*7/12)
    • z2m1sv = z2-1 =-3.464101615137754*%i-2.0 = 4*exp(i*2/3)
      • z2m2spv = z2-2 = 1.0-1.732050807568877*%i = 2*exp(i*10/12)
      • z2m2ssv = z2-2 = 1.732050807568877*%i-1.0 = 2*exp(i*4/12)
So here is 4 preimages of z2 with backward orbits :
  • {1/3, 1/6, 1/12} = {4/12, 2/12, 1/12} = {z2, z2m1pv,z2m2ppv}
  • {1/3, 1/6, 7/12} = {4/12, 2/12, 7/12} = {z2, z2m1pv, z2m2psv}
  • {1/3, 2/3, 5/6} = {4/12, 8/12, 10/12} = {z2, z2m1sv, z2m2spv}
  • {1/3, 2/3, 1/3} = {4/12, 8/12, 4/12} = {z2, z2m1sv, z2m2ssv} = {z2,z1,z0}

so :
z2m1sv is our z1 with angle = 8/12 = 2/3
z2m2ssv is our z0 with angle = 4/12 = 1/3

and here are 4 new points which are preperiodic points falling into period 2 cycle
z2m1pv with angle = 1/6 with forward orbit ={1/6, (2/6, 4/6)}= {1/6, (1/3, 2/3)}
z2m2ppv with angle = 1/12 with forward orbit ={ 1/12 , 2/12 ,( 1/3 , 2/3 )}
z2m2psv with angle = 7/12 with forward orbit ={ 7/12 , 2/12 ,( 1/3 , 2/3 )}
z2m2spv with angle = 5/6 with forward orbit = { 5/6 , ( 2/3 , 1/3 )}




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

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