Conical limit sets of hyperbolic subgroups

Given a hyperbolic subgroup $H$ of a hyperbolic group $G$ for which a Cannon-Thurston map $\hat i:\partial H \ra \partial G$ exists, we study the limit set $\Lambda_H$ of $H$ with respect to its action on $\partial G$. We prove that the set of conical limit points is exactly the subset of $\Lambda_H$ consisting of the points to which the Cannon-Thurston map $\hat i$ injects. Moreover, we show that when $H$ is not quasi-convex in $G$, there exists a non-conical limit point in $\Lambda_H$