# 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 superfunction \(F=G^{-1}\), and iterative application of the Transfer equation in order to bring the argument of the superfunction to the range of values where the asymptotic expansion provides the required precision.

## Keywords

Abel equation, Abel function, Iteration of function, Superfunction, Schroeder equation, Schroeder function, Zooming equation, Zooming function