Categorical iteration theory

[andydude]

Lastly, it is also known that for analytic iteration to exist uniquely, that the function being iterated must have a hyperbolic fixed point

Out of curiosity, why did you exclude the possibility of a parabolic fixed point?

Well, I suppose I'm drawing experience from \( e^x - 1 \), and how although this is \( C^{\infty} \) (infinitely differentiable) it is not \( C^{\omega} \) (analytic). Since \( e^x - 1 \) is infinitely differentiable, and the general iterate is also infinitely differentiable, because \( \eta = e^{1/e} \) is a parabolic fixed point, this allows us to say that the general iterate "perserves the structure" of \( e^x - 1 \) with respect to infinite differentiability (it doesn't preserve analycity). This is really what I'm interested in, is for what functions can the general iterate be found with similar properties.

I suppose I didn't want to exclude parabolic fixed points from the conversation, I just wanted to make sure that I was finding the correct conditions for having an analytic iterate.

Andrew Robbins