(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 is a superfunction, for 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.

, assume

Hyperbolic: : there is exactly one regular superfunction

Parabolic: : There are exactly 2(m-1) regular superfunctions.

For example , that's why we have 2*(2-1)=2 regular superfunctions. One from left and one from right.

Generally there are 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).