Dense subsets of boundaries of CAT(0) groups

In this paper, we study dense subsets of boundaries of CAT(0) groups. Suppose that a group $G$ acts geometrically on a CAT(0) space $X$ and suppose that there exists an element $g_0\in G$ such that (1) $Z_{g_0}$ is finite, (2) $X\setminus F_{g_0}$ is not connnected, and (3) each component of $X\setminus F_{g_0}$isconvexandnot$g_0$-invariant,where$Z_{g_0}$ is the centralizer of $g_0$ and $F_{g_0}$ is the fixed-point set of $g_0$ in $X$ (that is, $Z_{g_0}=\{h\in G\mathrm{\mid }{g}_{0}h=h{g}_0\}$ and $F_{g_0}=\{x\in X\mathrm{\mid }{g}_{0}x=x\}$). Then we show that each orbit $G\alpha$ is dense in the boundary $\mathrm{\partial }X$ (i.e.
$\mathrm{\partial }X$ is minimal) and the set $\{g^{\mathrm{\infty }}\mathrm{\mid }g\in G,o(g)=\mathrm{\infty }\}$ is also dense in the boundary $\mathrm{\partial }X$. We obtain an application for dense subsets on the boundary of a Coxeter system.