Fractional iterate

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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 I.N.Baker. Permutable power series and regular iteration. Imperial College of Science and Technology, London (1960).


Iteration, Transfer function, Superfunction, Abel function