On a dynamical version of a theorem of Rosenlicht

Consider the action of an algebraic group $G$ on an irreducible algebraic variety $X$ all defined over a field $k$. M. Rosenlicht showed that orbits in general position in $X$ can be separated by rational invariants. We prove a dynamical analogue of this theorem, where $G$ is replaced by a semigroup of dominant rational self-maps of $X$. Our semigroup $G$ is not required to have the structure of an algebraic variety and can be of arbitrary cardinality.