  For a non-vanishing gradient-like vector field on a compact manifold
$X^{n+1}$ with boundary, a discrete set of trajectories may be tangent to the
boundary with reduced multiplicity $n$, which is the maximum possible. (Among
them are trajectories that are tangent to $\partial X$ exactly $n$ times.) We
prove a lower bound on the number of such trajectories in terms of the
simplicial volume of $X$ by adapting methods of Gromov, in particular his
"amenable reduction lemma". We apply these bounds to vector fields on
hyperbolic manifolds.
