Functional super-iteration and hierarchy of functional hyper-iterations

[tommy1729]

Interesting, if \( f(x) = f_1x + f_2x^2 + \cdots \) and \( g(x) = g_1x + g_2x^2 + \cdots \) both have a (non-parabolic) fixed point at 0, then their supercomposition would have a parabolic fixed point.
\( f^{\circ g(x)}(x) = x + x^2 g_1 \ln(f_1) + x^3\ln(f_1)\left(\frac{f_2g_1}{f_1(f_1-1)} + \frac{g_1^2}{2}\ln(f_1) + g_2\right) + \cdots \)
also, one could find the infinite superiterate by looking at the first n coefficients of \( {}^{\circ n}f(x) \):
\( {}^{\circ(\infty)}f(x) = x + \ln(f_1)x^2 + \left(\frac{3}{2}\ln(f_1)^2 + \frac{f_2\ln(f_1)}{f_1(f_1-1)}\right)x^3 + \cdots \)

to be honest , i dont have a clue what you are talking about , how the **** did you arrive at those coefficients ?

im quite skeptical about " super-iteration " ...