  For a pair of points $x,y$ in a compact, riemannian manifold $M$ let
$n_t(x,y)$ (resp. $s_t(x,y)$) be the number of geodesic segments with length
$\leq t$ joining these points (resp. the minimal number of point obstacles
needed to block them). We study relationships between the growth rates of
$n_t(x,y)$ and $s_t(x,y)$ as $t\to\infty$. We derive lower bounds on $s_t(x,y)$
in terms of the topological entropy $h(M)$ and its fundamental group. This
strengthens the results of Burns-Gutkin \cite{BG06} and Lafont-Schmidt
\cite{LS}. For instance, by \cite{BG06,LS}, $h(M)>0$ implies that $s$ is
unbounded; we show that $s$ grows exponentially, with the rate at least
$h(M)/2$.
