On a dynamical version of a theorem of Rosenlicht

TL;DR

This paper extends Rosenlicht's theorem on orbit separation by rational invariants to a dynamical setting involving semigroups of dominant rational self-maps, which need not form an algebraic variety.

Contribution

It introduces a dynamical analogue of Rosenlicht's theorem for semigroups of rational maps, broadening the scope beyond algebraic group actions.

Findings

Established orbit separation by rational invariants in a dynamical context.

Generalized Rosenlicht's theorem to semigroups of rational maps.

Applicable to semigroups of arbitrary cardinality and without algebraic structure.

Abstract

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.