points, the action is free. The complement ${\partial}{\mathcal X}\setminus{\Lambda}_G$ is obviously contained in the dissipative ($\equiv$ discontinual) part of the action. However, this is as much as can *a priori* be said about the ergodic properties of the boundary action. In particular, the action on ${\Lambda}_G$ need not be ergodic or conservative. There are numerous examples witnessing to this; see [@Grigorchuk-Kaimanovich-Nagnibeda07] for a detailed discussion of the simplest model case of the action of a subgroup of a free group on the boundary of the ambient group and for further references.

One can specialize the type of convergence in the definition of the limit set. For instance, the *radial limit set* ${\Lambda}^{\textsf{rad}}$ is the set of all the accumulation points of any fixed orbit $Go,\,o\in{\mathcal X}$, which stay inside a *tubular neighbourhood* of a certain geodesic ray in ${\mathcal X}$. Yet another type of the boundary convergence, which we are going to describe below, is provided by *horospheric neighborhoods*.

Denote by $${\beta}_z(x,y)
 = d(y,z) - d(x,z) \qquad x,y\in{\mathcal X}$$ the *distance cocycle* associated with a point $z\in{\mathcal X}$, and, following [@Kaimanovich04], put $$\label{eq:beta}
 {\beta}_{\omega}(x,y) = \limsup_{z\to{\omega}} {\beta}_z(x,y)
 \qquad \forall\,x,y\in{\mathcal X},\,{\omega}\in{\partial}{\mathcal X}\;.$$ If the space ${\mathcal X}$ is CAT($-1$) (e.g., a 