(05/02/2009, 06:14 PM)Tetratophile Wrote: A function might have no fixed points, and superiterations of a function may not even be continuous. I just wanna know what areas of mathematics do they use when doing real-valued iteraitons?
But before, one should really look at integer functions where your definition is unambigous. I am not especially aware that someone defined/used such a construct. If so I guess it belongs to recursion theory, like the Ackermann function.
But as Andrew mentioned: As soon we get a parabolic fixed point, this fixed point stays forever. That means you can do regular iteration there and have even an analytic hierarchy.