Let $\Gamma$ be a Kleinian Group (a discrete subgroup of $PSL_2(\mathbb{C})$) acting on $\mathbb{D}=\{z \in \mathbb{C} : |z| \leq 1\}$.
Let $\Lambda(\Gamma)$, $\Omega(\Gamma)$ be the Limit Set and the Domain of Discontinuity of $\Gamma$, respectively. Then we have $\Lambda(\Gamma) \subseteq \partial \mathbb{D}$ and $Int(\mathbb{D}) \subseteq \Omega(\Gamma)$.
Let $T \in \Gamma$ be a Parabolic or Hyperbolic element, the fixed points of $T$ belong to the limit set $\Lambda(\Gamma)$.
If $S \in \Gamma$ is an Elliptic element, its fixed points belong to $\Omega(\Gamma)$ or $Int(\mathbb{D})$?