On equivariant homeomorphisms of boundaries of CAT(0) groups

In this paper, we investigate an equivariant homeomorphism of the boundaries $\mathrm{\partial }X$ and $\mathrm{\partial }Y$ of two proper CAT(0) spaces $X$ and $Y$ on which a CAT(0) group $G$ acts geometrically. We provide a sufficient condition to obtain a $G$-equivariant homeomorphism of the two boundaries $\mathrm{\partial }X$ and $\mathrm{\partial }Y$ as a continuous extension of the quasi-isometry $\phi:Gx_0\to Gy_0$definedby$\phi(gx_0)=gy_0$,where$x_0\in X$and$y_0\in Y$.