Böttcher iterate

A way to compute the regular iterates of a function $f$ with super-attracting fixpoint $a$.
For $f(z)=c_m (z-a)^m + c_{m+1} (z-a)^{m+1} + \dots$ the Böttcher iterate is: $$f^t(z)=\lim_{n\to\infty} f^{-n}(f^n(z)^{m^t})$$
Or if $\beta$ is the Böttcher coordinate of $f$ at $a$, then $$f^t(z)=\beta^{-1}\left(\beta(z)^{m^t}\right)$$