natural *hyperbolic compactification* ${\overline}{\mathcal X}={\mathcal X}\cup{\partial}{\mathcal X}$, and the action of the isometry group ${\operatorname{Iso}}({\mathcal X})$ extends by continuity to a continuous *boundary action* on ${\partial}{\mathcal X}$. The boundary ${\partial}{\mathcal X}$ is a Polish space.

The *limit set* ${\Lambda}={\Lambda}_G\subset{\partial}{\mathcal X}$ of a discrete subgroup $G\subset{\operatorname{Iso}}({\mathcal X})$ (any such subgroup is at most countable) is the set of all the limit points of any given orbit $Go,\,o\in{\mathcal X}$, with respect to the hyperbolic compactification, so that the closure of the orbit $Go$ in the hyperbolic compactification is $Go\cup{\Lambda}_G$ (this definition does not depend on the choice of the basepoint $o$). The limit set is closed and $G$-invariant. Moreover, the action of $G$ on ${\Lambda}_G$ is *minimal* (there are no proper $G$-invariant closed subsets), whereas the action of $G$ on the complement ${\partial}{\mathcal X}\setminus{\Lambda}_G$ is *properly discontinuous* (no orbit has accumulation points) [@Gromov87; @Bourdon95].

The latter result provides a topological decomposition of the boundary action. On the other hand, the situation is more complicated from the measure-theoretical point of view. Let $m$ be a $G$-quasi-invariant measure on ${\partial}{\mathcal X}$. Without loss of generality we can assume that it is purely non-atomic. Then, since any element of $G$ fixes at most two boundary 