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

TL;DR

This paper investigates the growth of geodesics in Riemannian manifolds, showing polynomial growth in uniformly secure cases and total insecurity in manifolds with positive topological entropy and no conjugate points.

Contribution

It establishes a link between security properties, geodesic growth, and dynamical complexity in Riemannian manifolds, introducing new results on insecurity.

Findings

Polynomial growth of geodesics in uniformly secure manifolds

Total insecurity in manifolds with positive topological entropy and no conjugate points

Finite blocking is impossible in certain complex 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 $T \to \infty$. 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.