Translation equation

Remarks

Example: $F(x, t) = x e^t$ on $X = \R$ is a solution, since $F(F(x, s), t) = x e^s e^t = x e^{s + t}$. Non-example: $F(x, t) = x + t^2$ is not a solution, since $F(F(0, 1), 1) = 2$ while $F(0, 2) = 4$. For a flow, each map $f^t = F(\cdot, t)$ is a bijection of $X$ with inverse $f^{-t}$, and $f^s \circ f^t = f^{s + t}$. If $\alpha \colon X \to \R$ is a bijection, then $F(x, t) = \alpha^{-1}(\alpha(x) + t)$ is a flow, and this is how Abel functions produce fractional iterates. The same equation is studied with $\R$ replaced by $[0, \infty)$, $\Z$, $\C$ or any group or semigroup.