Topological Entropy and Blocking Cost for Geodesics in Riemannian Manifolds

For a pair of points x, y in a compact, riemannian manifold M let nt(x, y) (resp. st(x, y)) be the number of geodesic segments with length ≤ t joining these points (resp. the minimal number of point obstacles needed to block them). We study relationships between the growth rates of nt(x, y) and st(x, y) as t → ∞. We derive lower bounds on st(x, y) in terms of the topological entropy h(M) and its fundamental group. This strengthens the results of Burns-Gutkin [2] and Lafont-Schmidt [13]. For instance, by [2, 13], h(M) > 0 implies that s is unbounded; we show that s grows exponentially, with the rate at least h(M)/2.