(05/26/2011, 10:09 PM)tommy1729 Wrote: another question is : how many superfunctions can a function have ?
There are different answers.
If you just ask about the number of superfunctions, then there are infinitely many. We discussed that already, when ever you have a superfunction F, F(x+1)=f(F(x)) then also the function \( G(x)=G(x+\theta(x)) \) is a superfunction, for \( \theta \) 1-periodic, this should not be new for you.
If you however ask, how many *regular* super-functions you have at a given fixpoint, i.e. superfunction from regular fractional iterations, i.e. which have an asymptotic powerseries development at the fixpoint, which is equal to the formal fractional iteration powerseries, then there is a clear answer:
You look at the powerseries development of the corresponding function, for simplicity we assume fixpoint at 0.
\( f(x)=f_1 x + f_2 x^2 + \dots \), assume \( f_1\neq 0 \)
Hyperbolic: \( |f_1|\ne 1 \): there is exactly one regular superfunction
Parabolic: \( f(x)=x + f_m x^m + f_{m+1}x^{m+1} + \dots \): There are exactly 2(m-1) regular superfunctions.
For example \( e^x-1=x+x^2/2+\dots \), that's why we have 2*(2-1)=2 regular superfunctions. One from left and one from right.
Generally there are \( 2(m-1) \) petals around the fixpoint, which are alternatingly attractive and repellant (in our example coming from left is attractive and coming from right repellant), on each petal there is defined a different regular Abel function (which is the inverse of a superfunction).
The whole thing is called the Leau-Fatou-flower and is kinda standard in holomorphic dynamics (see for example the book of Milnor mentioned on the forum).