Dense subsets of the boundary of a Coxeter system

In this paper, we investigate dense subsets of the boundary of a Coxeter system. We show that for a Coxeter system $(W,S)$, if $W^{\{s_0\}}$ is quasi-dense in $W$ and the order $o(s_0t_0)=\infty$ for some $s_0,t_0\in S$, then there exists a point $\alpha$ in the boundary $\partial\Sigma(W,S)$ of the Coxeter system $(W,S)$such that the orbit $W\alpha$ is dense in $\partial\Sigma(W,S)$. Here $W^{\{s_0\}}=\{w\in W\,\vert\,\ell(ws)<\ell(w) \text{for each} s\in S\setminus\{s_0\} \}\setminus \{1\}$. We also show that if the set $\bigcup\{W^{\{s\}}\,\vert\, s\in S \text{such that} o(st)=\infty \text{for some} t\in S\}$ is quasi-dense in $W$, then $\{w^\infty\,\vert\, w\in W \text{such that} o(w)=\infty\}$ is dense in $\partial\Sigma(W,S)$.