  Let $\Gamma$ be a non-elementary Gromov-hyperbolic group, and $\partial
\Gamma$ denote its Gromov boundary. We consider $\Gamma$-invariant proper
$\delta$-hyperbolic, quasi-convex metric $d$ on $\Gamma$, and the associated
Patterson-Sullivan measure class $[\nu]$ on $\partial^{(2)}\Gamma$, and its
square $[\nu\times\nu]$ on $\partial^{(2)}\Gamma$ -- the space of distinct
pairs of points on the boundary. We construct an analogue of a geodesic flow to
study ergodicity properties of the $\Gamma$-actions on $(\partial\Gamma,\nu)$
and on $(\partial^{(2)}\Gamma,[\nu\times\nu])$.
