Jabotinsky's iterative logarithm

[bo198214]

This must mean that:
\( \text{ilog}(f^{\circ t}) = t \text{ilog}(f) \)
so
\( f^{\circ t} = \text{ilog}^{-1}(t \text{ilog}(f)) \)

At first I thought this was either an Abel or Schroeder function, but it seems that it is neither.

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.

To be shorter I adopt Ecalle's notation and write \( f_\ast \) for \( \text{ilog}(f) \) and write \( f^{\ast} \) for the regular Abel function of \( f \). Then the relation between them both is:
\( \frac{\partial f^\ast(x)}{\partial x}=1/f_\ast(x) \)

This can easily deduced by:
\( \begin{align*}
f^\ast(f^{\circ t}(x))&=t+f^\ast(x)\\
\frac{\partial f^\ast(f^{\circ t})}{\partial t}&=1\\
\frac{\partial f^\ast(x)}{\partial x} \frac{\partial f^{\circ t}}{\partial t}&=1\\
\end{align*}
\)

In particular \( 1/f_\ast \) is a meromorphic function i.e. can be expressed as
\( c_{-n}x^{-n}+\dots+c_{-1}x^{-1}+c_0+c_1x+c_2x^2+\dots \)

If you now integrate this expression \( x^{-1} \) becomes \( \log(x) \). As this is so important, Ecalle calls \( c_{-1} \)
the "residu iteratif" (which he introduces in his first theorem in "Theorie des Invariants Holomorphes") so that we have
\( f^\ast(z) = c_{-1} \log(z) + F(z) \) where \( F(z) \) is a meromorphic function. I think similar considerations can be found in Szerkes' paper too.

Because I've also heard Abel functions referred to as "iterational logarithm" before (I think Peter Walker calls them this).

Ecalle calls the Abel function of \( f \) "iterateur de \( f \)" and he calls \( f_\ast \) "logarithme iteratif de \( f \)".