Jabotinsky's iterative logarithm

[andydude]

The categories (non-mathematical) are different:
\( \text{ilog} \) maps a function to a function (or better a formal powerseries to a formal powerseries) while the Abel function maps values to values. And before writing \( \text{ilog}^{-1} \) you should assure that it is invertible, which stronly seems not to be the case.

I realize why this is the case now. It would be like trying to reconstruct g(t) from g'(0). However, since: \( \mathcal{A}[f](x) = \int \frac{dx}{\mathcal{J}[f](x)} \) and \( f^{\circ t}(x) = \mathcal{A}[f]^{-1}(\mathcal{A}[f](x) + t) \) it could be argued that it should be possible to invert the iterative logarithm, provided the Abel function is invertible.

Also, I've been starting to realize more and more, that this is really amazing! Jabotinsky was a master of iteration.
In "Analytic Iteration" cited above, he gives this formula (3.10) in original and my notations:

\( L(F^{\circ s}(z)) = \frac{\partial}{\partial z}F^{\circ s}(z) \cd L(z) = \frac{\partial}{\partial s}F^{\circ s}(z) \)

\( \mathcal{J}[f](f^{\circ t}(x)) = \frac{\partial}{\partial x}f^{\circ t}(x) \cd \mathcal{J}[f](x) = \frac{\partial}{\partial t}f^{\circ t}(x) \)

where the relationship between L and ilog are \( L(x) = \mathcal{J}[f](x) = \text{ilog}(f) \).
What I find most interesting about this formula is that right after it, Jabotinsky says:

This double equation is fundamental in the theory of iteration. It can be used and extended in many ways.

what did he mean by this? Obviously, he knew how important this was, but from reading it, it seems that he did not realize its connection to Abel functions. Does Ecalle mention the relationship to Julia functions or Abel functions?

Andrew Robbins