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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 items (1) and (2) is due to Dal’bo in the rank one case. We also show 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.
