# 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