Limit sets of cyclic quaternionic Kleinian groups

In this paper, we consider the natural action of $\mathrm{SL}(3, \mathbb{H})$ on the quaternionic projective space $\mathbb{P}_{\mathbb{H}}^2$. Under this action, we investigate limit sets for cyclic subgroups of $\mathrm{SL}(3, \mathbb{H})$. We compute two types of limit sets, which were introduced by Kulkarni and Conze-Guivarc'h, respectively.