Jabotinsky's iterative logarithm

[bo198214]

Starting with the Abel functional equation:
\( \mathcal{A}[f](f^{\circ t}(x)) = \mathcal{A}[f](x) + t \)
\( \frac{\partial}{\partial x} \mathcal{A}[f](f^{\circ t}(x)) f'(x) = \frac{\partial}{\partial x} \mathcal{A}[f](x) \)
\( \mathcal{J}[f](f^{\circ t}(x)) = f'(x) \mathcal{J}[f](x) \)
...
\( \mathcal{J}[f](f^{\circ t}(x)) = f'(x) \mathcal{J}[f](x) \)

For me it looks as if there always \( t=1 \) must be set ...

now we know 2 things about 1 function, as opposed to 1 thing about 2 functions...