Preprint D38/2007
Growth of the number of geodesics between points and insecurity for Riemannian manifolds

Eugene Gutkin | Burns, Keith

**Keywords: **
riemannian manifold | security | geodesics | entropy

A Riemannian manifold is said to be uniformly secure if there is a finite
number $s$ such that all geodesics connecting an arbitrary
pair of points in the manifold can be blocked by $s$ point obstacles.
We prove that the number of geodesics with length $\leq T$ between
every pair of points in a uniformly secure manifold grows
polynomially as $T \to \infty$. We derive from this that a
compact Riemannian manifold with no conjugate points whose geodesic flow has
positive topological entropy is totally insecure: the geodesics between any
pair of points cannot be blocked by a finite number of point obstacles.