the action coincides (mod 0) with the *big horospheric limit set* ${\Lambda}^{\mathsf{horB}}_G$ of the group $G$, i.e., the set of points ${\omega}\in{\partial}{\mathcal X}$ for which there exists $t=t({\omega})$ such that $\{g\in G:{\beta}_{\omega}(go,o)\ge t\}$ is infinite for a certain fixed reference point $o\in X$, or, in other words, the set of points ${\omega}\in{\partial}{\mathcal X}$ such that a *certain* horoball centered at ${\omega}$ contains infinitely many points from the orbit $Go$ ([Theorem \[th:main\]]{}).

This characterization of the conservative part of the boundary action was first established by Pommerenke [@Pommerenke76] for Fuchsian groups with respect to the visual stream on the boundary circle (although in a somewhat different terminology). Pommerenke’s argument uses analytic properties of the Blaschke products and does not immediately carry over to the higher dimensional situation. Sullivan [@Sullivan81] used a more direct geometrical approach and proved this characterization for Kleinian groups, again with respect to the visual stream. Actually he considered the *small horospheric limit set* ${\Lambda}^{\mathsf{horS}}_G$ (also called just *horospheric limit set*; it is defined by requiring that the intersection of *any* horoball centered at ${\omega}\in{\Lambda}^{\mathsf{horS}}_G$ with the orbit $Go$ be infinite) essentially showing that ${\Lambda}^{\mathsf{horB}}\setminus{\Lambda}^{\mathsf{horS}}$ is a null set. By extending Sullivan’s approach (with some technical complications) Tukia 