the set of all the points ${\omega}\in{\partial}{\mathcal X}$ such that a certain (resp., any) horoball centered at ${\omega}$ contains infinitely many points from a fixed orbit $Go,\,o\in{\mathcal X}$ (the resulting set does not depend on the choice of the orbit $Go$, see [Remark \[rem:ind\]]{} below).

\[rem:ind\] As it follows from [Proposition \[pr:quasicocycle\]]{}, for any fixed reference point $o\in{\mathcal X}$ the big (resp., small) horospheric limit sets can also be defined as the set of all the points ${\omega}\in{\partial}{\mathcal X}$ for which the set $$\{x\in Go: {\beta}_{\omega}(o,x) \le t \}$$ is infinite for a certain (resp., for any) $t\in{\mathbb R}$.

Usually our small horospheric limit set is called just the *horospheric limit set*, and in the context of Fuchsian groups its definition, along with the definition of the radial limit set, goes back to Hedlund [@Hedlund36]. Following [@Matsuzaki02] (in the Kleinian case) we call it *small* in order to better distinguish it from the *big* one, which, although apparently first explicitly introduced by Tukia [@Tukia97] (again just in the Kleinian case), essentially appears (for Fuchsian groups) already in Pommerenke’s paper [@Pommerenke76].

The horospheric limit sets ${\Lambda}^{\mathsf{horS}},{\Lambda}^{\mathsf{horB}}$ are obviously $G$-invariant, Borel, and contained in the full limit set ${\Lambda}$ (because the only 