Maximal bottom of spectrum or volume entropy rigidity in Alexandrov geometry

In \cite{LiWang2001complete1,LiWang2001complete2}, Li-Wang proved a splitting theorem for an n-dimensional Riemannian manifold with $Ric⩾-(n-1)$ and the bottom of spectrum ${\lambda }_0(M)=\frac{(n-1{)}^2}{4}$. For an n-dimensional compact manifold $M$ with $Ric⩾-(n-1)$ with the volume entropy $h(M)=n-1$, Ledrappier-Wang \cite{LeW2010volent} proved that the universal cover $\tilde{M}$ is isometric to the hyperbolic space ${\mathbb{H}}^n$. We will prove analogue theorems for Alexandrov spaces.