Is this THE equation for parabolic fix ?

[tommy1729]

What you have is not a Taylor series, so how did you find this expansion.

This series uses Jean Ecalle's FPS solution; there are many other posts on mathoverflow by Will Jagy and Henryk Trapman and Gottried Helms, about Jean Ecalle's parabolic solution.

Also this does not answer the op.

I know ways to get the Abel , but i forgot why getting to z=0 helps.
Probably perturbation theory.

It sounds like you're looking for the inverse of the parabolic solution for the Abel function, also developed around the fixed point, no? I haven't seen such an "inverse Abel function" formal power series; so it might be novel. The form might be similar to what was posted by Mick, on mathstack, but it would require a 1/x term to be the inverse of Ecalle's solution. Also, such an FPS would also likely have a zero radius of convergence for the same reasons that Ecalle's solution is an asymptotic series; which I briefly explained.

Btw for x + x^N the situation is different for Every N.

True; Will Jagy explains the general case in some of his posts. I just figured I would give the FPS series for iterating the parabolic case; \( f(z)=\exp(z)-1 \), to help you out. Scaling this FPS solution gives the solution for iterating \( \eta=\exp(1/e)\;\;\;f(z)=\eta^z\;\;\; \) which you mentioned earlier.

\( \alpha_\eta(z) = \alpha(\frac{z}{e}-1)\;\;\;\alpha_\eta(\eta^z)=\alpha_\eta(z)+1\;\;\; \) Abel function for \( f(z)=\eta^z \) in terms of the Abel function for \( f(z)=\exp(z)-1\;\;\; \) derived using \( \ln(\ln(\eta^{\eta^z})) \)

What Will Jagy posted or what you call the ecalle FPS method ( what is FPS ? ) , I call the Julia method.

It is now clear again to me why getting closer to z=0 works ; the limit form and Taylor are equivalent (isomorphic).
I feel a bit dumb for forgetting it.

I wanted to post something about matrices but in your link it seems Gottfried already found that.

I wonder if these methods for parabolic generalize easy into methods/expansions that work for both parabolic and hyperbolic.

Thanks for the refreshing link.

I will start a new thread for a question that I think is related.
Also Im working on an idea that could be informally described as " fake parabolic function theory " , but that is for much later.

Although the amount of questions unanswered remains about the same , Im starting to feel a generalization... a deep connection between concepts.

But I have to start a new thread because otherwise Im going a bit offtopic with respect to the OP here.

Maybe we need a Julia-like equation for the inverse Abel to answer most questions ?


regards

tommy1729

" Proof implies both truth and why "
tommy1729