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

Keith Burns·Eugene Gutkin
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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 $T \to \infty$. 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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Keith Burns, Eugene Gutkin. 2007-01-20. Growth of the number of geodesics between points and insecurity for riemannian manifolds. https://arxiv.org/abs/math/0701579

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