Precondition

Be sure to know the basic math: Math Home
Or: straight to the pictures: Second part with examples

Introduction: The Disturbance Constant c

A usual fixed point iteration looks like this:
        z0 = s
        z1 = f(z0)
        z2 = f(z1)
        zn+1 = f(zn)
A constant c added to the iteration can be viewed as a disturbance disturbing the usual fixed point iteration:
        z0 = s
        z1 = f(z0) + c
        z2 = f(z1) + c
        zn+1 = f(zn) + c
A more general approach is just not to add the constant c:
        z0 = s
        z1 = f(z0,c)
        z2 = f(z1,c)
        zn+1 = f(zn,c)

Disturbing the disturbance

What if the disturbance c is a function(disturbance) of c itself ?
        z0 = s
        c0 = c
        z1 = f(z0,c0)
        c1 = g(c0)
        z2 = f(z1,c1)
        c2 = g(c1)
        zn+1 = f(zn,cn)
        cn+1 = g(cn)

Keeping the perfectness perfect

Unfortunately I'm not a mathematician and proving anything fractal is out of my scope.
As far as I have found out by trial and error is: perfectness (inifinite apple manikins)
is preserved under the condition that the disturbance c stays constant with the start value c periodically.
E.g. g(c) = 1/c, results in c,1/c,c,1/c,c,1/c
E.g. g(c) = -c, results in c,-c,c,-c,c,-c
By the way 1/c and -c are Babbage Functions (Involutions): Wiki Functional Equations

Second part with examples