Let $N=(g_t)_{t\ge 0}$ be a one-parameter subsemigroup of a Lie group $G$ acting on the homogeneous space $G/H$. Let $x\in G/H$ be such that the trajectory $N.x$ is dense in $G/H$.
Let $U\subset G/H$ be any open subset, we know that $U \cap N.x$ by density of $N.x$. But is it true that for any $T\ge 0$, there exists $t > T$ such that $g_t.x\in U$?
(Some relevant background: In the common situation when the action of $N$ on $G/H$ is ergodic, $N.x$ is dense for almost every point in $G/H$ w.r.t. a $G-invariant$ measure)