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.
(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 \(T\).
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).