  Let $ G $ be a connected, semisimple real algebraic group and $\Gamma < G$ be
a Zariski dense discrete subgroup. Let $N$ denote a maximal horospherical
subgroup of $G$, and $P=MAN$ the minimal parabolic subgroup which is the
normalizer of $N$. Let $\mathcal{E}$ denote the unique $P$-minimal subset of
$\Gamma \backslash G$ and let $\mathcal{E}_0$ be a $P^\circ$-minimal subset. We
consider a notion of a horospherical limit point in the Furstenberg boundary $
G/P $ and show that the following are equivalent for any $[g]\in
\mathcal{E}_0$: (1) $gP\in G/P$ is a horospherical limit point; (2) $[g]NM$ is
dense in $\mathcal{E}$; (3) $[g]N$ is dense in $\mathcal{E}_0$. The equivalence
of (1) and (2) is due to Dal'bo in the rank one case. We also observe that
unlike convex cocompact groups of rank one Lie groups, the $NM$-minimality of
$\mathcal{E}$ does not hold in a general Anosov homogeneous space.
