Abel equation

Summary

The Abel equation is the iterative functional equation $\alpha(f(x)) = \alpha(x) + 1$, where $f$ is a given map and $\alpha$ is unknown. A solution conjugates the map $f$ to the translation $t \mapsto t + 1$. An injective solution gives a natural way to define non-integer iterates of $f$, by $f^t(x) = \alpha^{-1}(\alpha(x) + t)$. It is closely related to Schröder's equation: if $\psi$ solves $\psi(f(x)) = s \psi(x)$ with $s > 0$, $s \ne 1$ and $\psi > 0$, then $\alpha = \log_s \psi$ solves the Abel equation.