Translation equation

Summary

The translation equation is the functional equation $F(F(x, s), t) = F(x, s + t)$ for an unknown map $F \colon X \times \R \to X$. Its solutions with $F(x, 0) = x$ are the flows on the set $X$, that is, actions of the additive group $\R$ on $X$. So it is the basic equation of continuous-time dynamical systems and of iteration with non-integer exponents. With the real line replaced by a group, it becomes the equation of a group action.