  We exhibit a class of Schottky subgroups of $\mathbf{PU}(1,n)$ ($n \geq 2$)
which we call well-positioned and show that the Hausdorff dimension of the
limit set $\Lambda_\Gamma$ associated with such a subgroup $\Gamma$, with
respect to the spherical metric on the boundary of complex hyperbolic
$n$-space, is equal to the growth exponent $\delta_\Gamma$.
  For general $\Gamma$ we establish (under rather mild hypotheses) a lower
bound involving the dimension of the Patterson-Sullivan measure along
boundaries of complex geodesics.
  Our main tool is a version of the celebrated Ledrappier-Young theorem.
