# 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).