Cartan–Hadamard manifold with pinched sectional curvatures or a tree), then $\limsup$ in the above formula can be replaced just with the ordinary limit, and ${\beta}_{\omega}$ are the boundary *Busemann cocycles*. Although for a general Gromov hyperbolic space ${\beta}_{\omega}$ are not, generally speaking, cocycles, they still satisfy the cocycle identity with a uniformly bounded error (i.e., they are *quasi-cocycles*). Namely,

\[pr:quasicocycle\] There exists a constant $C\ge 0$ depending on the hyperbolicity constant ${\delta}$ of the space ${\mathcal X}$ only such that for any ${\omega}\in{\partial}{\mathcal X}$ the function ${\beta}_{\omega}$ has the following properties:

-   ${\beta}_{\omega}$ is “jointly Lipschitz”, i.e., $|{\beta}_{\omega}(x,y)|\le d(x,y)$ for all $x,y\in{\mathcal X}$, in particular, ${\beta}_{\omega}(x,x)\equiv 0\;;$

-   $0 \le  {\beta}_{\omega}(x,y) + {\beta}_{\omega}(y,z) + {\beta}_{\omega}(z,x) \le C$ for all $x,y,z\in{\mathcal X}\;.$

The quasi-cocycles ${\beta}_{\omega}$ are obviously invariant with respect to the isometries of ${\mathcal X}$, i.e., $${\beta}_{g{\omega}}(gx,gy) = {\beta}_{\omega}(x,y)
\qquad\forall\,x,y\in{\mathcal X},\,{\omega}\in{\partial}{\mathcal X},\,g\in{\operatorname{Iso}}({\mathcal X}) \;.$$

We shall define the *horoball* in ${\mathcal X}$ centered at a boundary point ${\omega}\in{\partial}{\mathcal X}$ and passing through a point $o\in{\mathcal X}$ as $${\operatorname{HBall}}_{\omega}(o) = \{x\in{\mathcal X}: {\beta}_{\omega}(o,x)\le 0 \} \;.$$

The *big* (resp., *small*) *horospheric limit set* ${\Lambda}^{\mathsf{horB}}={\Lambda}^{\mathsf{horB}}_G$ (resp., ${\Lambda}^{\mathsf{horS}}={\Lambda}^{\mathsf{horS}}_G$) of a discrete group $G$ of isometries of a Gromov hyperbolic space ${\mathcal X}$ is 