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

**Authors: Keith Burns
and Eugene Gutkin**
**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.

This article is available in pdf format.

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

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