## 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 ≤ T between every pair of points in a uniformly secure manifold grows polynomially as T → ∞. By results of Gromov and Mañé, the fundamental group of such a manifold is virtually nilpotent, and the topological entropy of its geodesic flow is zero. Furthermore, if a uniformly secure manifold has no conjugate points, then it is flat. This follows from the virtual nilpotency of its fundamental group either via the theorems of Croke-Schroeder and Burago-Ivanov, or by more recent work of Lebedeva. 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.

Original language | English (US) |
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Pages (from-to) | 403-413 |

Number of pages | 11 |

Journal | Discrete and Continuous Dynamical Systems |

Volume | 21 |

Issue number | 2 |

State | Published - Jun 1 2008 |

## Keywords

- Connecting geodesics
- Entropy
- Security

## ASJC Scopus subject areas

- Analysis
- Discrete Mathematics and Combinatorics
- Applied Mathematics