Growth of the number of geodesics between points and insecurity for riemannian manifolds

Abstract:
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 $T \to \infty$. By results of Gromov and Ma\~n\'e, the fundamental group of such a manifold is virtually nilpotent, and the topological entropy of its geodesic flow is zero. 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.

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The research in this article was supported by National Science Foundation grant DMS-0408704. Needless to say, any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.

Authors' addresses:
	Keith  Burns 
	Department of Mathematics
	Northwestern University
	Evanston, IL 60208-2730

        Eugene Gutkin
        IMPA
        Estrada Dona Castorina 110
        Rio de Janeiro 22460-320
        Brazil