  In \cite{LiWang2001complete1,LiWang2001complete2}, Li-Wang proved a splitting
theorem for an n-dimensional Riemannian manifold with $Ric\geqslant -(n-1)$ and
the bottom of spectrum $\lambda_0(M)=\frac{(n-1)^2}{4}$. For an n-dimensional
compact manifold $M$ with $Ric\geqslant-(n-1)$ 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.
