$\DeclareMathOperator\supp{supp}$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 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!

$\DeclareMathOperator\supp{supp}$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!

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 a $S$-invariant and ergodic probability measure, where $S$ is a semigroup inside a unipotent subgroup $U$. We will show that $supp(\mu)$ supports contains a full $U=\langle S\rangle$ orbit, that's enough. Pick some $x\in supp(\mu)$. Consider $\overline{S.x}=P$. For a generic point, $P=supp(\mu)$ by the ergodic theorem. Notice that $S.P\subset P$, moreover $S^{2}.P=S.P$, so this is an $S$-invariant subset. By ergodicity $\mu(S.P)=1$, but we also have that $\mu(P)=1$ as well, so $P=S.P$ up to a measure zero set, or in other words $S^{-1}.P=P$ 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" where he discuss the move from Ratner's theorem to discrete subgroup actions!

$\DeclareMathOperator\supp{supp}$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 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!