Fractional iterate
Fractional iterate is concept used to construct non-integer iterates of functions.
For a given function $~T~$, holomorphic in vicinity of its fixed point $~L~$, the function $t_r=T^r$ is called as $~r$th fractional iterate, iff $~r\!=\!m/n~$ for some integer numbers $~m, n~$ and
(1) $~ ~ ~ t_r^m(z)=T^n(z)~$
for all $~z~$ in some vicinity of $~L~$.
If such a function $t_r$ is also regular in vicinity of $~L~$, then such a fractional iterate is called regular iterate of function $~T~$ with number of iteration $~r~$ at fixed point $~L~$.
Specification of fractional iterate
In general, specifying non–integer iterate of some function $~T~$, one should provide some additional information that provides the uniqueness. Such a condition can be indication of the fixed point $~T~$, at which the iterate is regular.
An alternative could be specification of behavior at infinity, or specification of the superfunction $F$ and the Abel function $G=F^{-1}$, used to construct the fractional iterate $T^r$ with
(2) $~ ~ ~ R^r=F\big(r+G(z)\big)~$
If the only one way of the construction of non-integer iterate is indicated, then the simple writing $T^r$ is sufficient to indicate the $r$th iterate of function $T$.
References
http://eretrandre.org/rb/files/Baker1962_53.pdf I.N.Baker. Permutable power series and regular iteration. Imperial College of Science and Technology, London (1960).
http://math.eretrandre.org/hyperops_wiki/index.php?title=Regular_iteration