Abel function

Summary

An Abel function of a map $f$ is a solution $\alpha$ of the Abel equation $\alpha(f(x)) = \alpha(x) + 1$. It is a coordinate in which one step of $f$ becomes a shift by $1$. When $\alpha$ is injective, it defines a whole family of iterates $f^t = \alpha^{-1} \circ (\alpha + t)$, one for each real $t$ in a suitable range. Abel functions are the standard tool for fractional iteration, for example in the construction of iterates of the exponential function.