Assume some given holomorphic function T.
The superfunction is holomorphic solution F of equation
T(F(z))=F(z+1)
The Abel function (or abelfunction) is the inverse of superfunction,
G=F-1
The abelfunction is solution of the Abel equation
G(T(z))=G(z)+1
As the superfunction F and the abelfunction G=F-1 are established, the nth iterate of transfer function T can be expressed as follows:
Tn(z)=F(n+G(z))
This expression allows to evaluate the non-integer iterates. The number n of iterate can be real or even complex.
In particular, for integer n, the iterates have the common meaning:
T-1 is inverse function of T,
T0(z)=z
T1(z)=T(z)
T2(z)=T(T(z))
T3(z)=T(T(T(z)))
and so on. For complex m and n, the group property holds:
Tm(Tn(z))=Tm+n(z)
The book is about evaluation of the superfunction F, the abelfunction G and the non-integer iterates of various transfer function T.
The Russian version is also available at my site,
http://mizugadro.mydns.jp/BOOK/202.pdf
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