Regular iterate

Regular iterate of some function \(T\), referred below as a transfer function, at its fixed point \(L\) is such iterate \(T^n\) that is regular at \(L\) even at non–integer values of \(n\).

In particular, for integen numbers \(m\) and \(n\!\ne\!0\), the regular iterate \(f=T^{m/n}\) is supposed to be fractional iterate of function \(T\), id est, for \(z\) in vicinity of point \(L\),

(1) \( ~ ~ ~ f^n(z)=T^m(z)\)

The regular iterate can be evaluated with regular iteration of the asymptotic expansion of the Abel function \(G\) in vicinity of \(L\) and corresponding expansion of the superfunciton \(F=G^{-1}\), and iterative application of the Transfer equation in order to bring the argument of the superfunciton to the range of values where the asymptotic expansion provides the required precision.

Keywords
Iteration of function, Superfunciton, Abel function, Abel equation, Schroeder equation, Schroeder function, Zooming equation, Zooming function