  In this article we prove that the Hausdorff dimension of geodesic directions
that are recurrent and diverge on average coincides with the entropy at
infinity of the geodesic flow for any complete, pinched negatively curved
Riemannian manifold. We derive an interesting consequence from this result, we
prove that the entropy of a {\sigma}-finite, ergodic and conservative infinite
invariant measure is bounded from above by the entropy at infinity of the
geodesic flow.
