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

Keywords

Iteration, Transfer function, Superfunction, Abel function