Ratner's orbit closure for a unipotent semigroup For Ratner's orbit closure theorem, one may refer to the following Wikipedia page.
Let $\{u_t\mathrel: t\in \mathbb{R} \}\subset G$ be a unipotent one-parameter subgroup of a connected Lie group. Let $\Gamma$ be a lattice of $G$. From the theorem, we know that the closure of every full orbit $\{x\cdot u_t\mathrel: t\in \mathbb{R}\}$ is dense in the orbit $xS\subset \Gamma\backslash G$ of a certain subgroup $S$ of $G$.
Do we also have the density of the semigroup orbit $\{x\cdot u_t\mathrel: t\geq 0\}$ in certain orbits of subgroups? Is $\{x\cdot u_t\mathrel: t\geq 0\}$ just dense in $xS$ as well?
 A: $\DeclareMathOperator\supp{supp}$The theorem holds for semigroups as well (well, in the finite volume setting! in the infinite volume setting there are subtleties between two-sided and one-sided averages).
Assume $\mu$ is an $S$-invariant and ergodic probability measure, where $S$ is a semigroup inside a one-parameter unipotent subgroup $U$.
We will show that $\supp(\mu)$ contains a full $U=\langle S\rangle$ orbit, that's enough.
Pick some $x\in \supp(\mu)$.
Consider $\overline{S\cdot x}=P$. For a generic point $x$, $P=\supp(\mu)$ by the ergodic theorem.
Notice that $S\cdot P\subset P$, moreover $S^{2}\cdot P=S\cdot P$, so this is an $S$-invariant subset. By ergodicity $\mu(S\cdot P)=1$, but we also have that $\mu(P)=1$ as well, so $P=S\cdot P$ up to a measure zero set, or in other words $S^{-1}\cdot P=P$ as well, hence $P$ is invariant under $S\cup S^{-1}$, now you may apply the orbit closure theorem.
P.S. you might want to consider the paper by Nimish Shah - "Invariant measures and orbit closures on homogeneous spaces for actions of subgroups generated by unipotent elements" (MSN) in Lie groups and ergodic theory, where he discusses the move from Ratner's theorem to discrete subgroup actions!
