half composite function

Question

There is half derivative. Is there any definition of half composite function? Or is it possible to define half composite of a function?

1) $f^n=f \circ \cdots \circ f $ ($n$-times, $n \in \mathbb Z^{+}$)

2) if $f$ has an inverse the we can define $f^{-n}$.

3) Can we define $f^{1/2}$ or $f^{\pi}$?

Answer 1

Can be defined, but may or may not exist for a given function $f$.

For example, see half iterate of $x^2+c$

In MO, see https://mathoverflow.net/questions/tagged/fractional-iteration