Preprint D48/2007
Topological entropy and blocking cost for geodesics in riemannian manifolds

Eugene Gutkin

**Keywords: **
riemannian manifold | connecting geodesics | blocking threshold | counting of geodesics | topological entropy

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 $h(M)/2$.