Topological entropy and blocking cost for geodesics in Riemannian manifolds

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 ≤ t joining these points (resp. the minimal number of point obstacles needed to block these geodesic segments). We study relationships between the growth rates of n _t (x, y) and s _t (x, y) as t → ∞. We obtain lower bounds on s _t (x, y) in terms of the topological entropy h(M) and the fundamental group π ₁(M). For instance, we show that if h(M) > 0 then s _t grows exponentially, with the rate at least h(M)/2. This strengthens earlier results on blocking of geodesics (Burns and Gutkin Discrete Contin Dyn Syst 21:403–413, 2008; Lafont and Schmidt Geom Topol 11:867–887, 2007), and puts them in a new perspective.