{
  "id": "math/0701579",
  "version": "v1",
  "published": "2007-01-20T22:55:41.000Z",
  "updated": "2007-01-20T22:55:41.000Z",
  "title": "Growth of the number of geodesics between points and insecurity for riemannian manifolds",
  "authors": [
    "Keith Burns",
    "Eugene Gutkin"
  ],
  "comment": "14 pages, 0 figures",
  "journal": "Discrete and Continuous Dynamical Systems A 21 (2008), 403 -- 413",
  "categories": [
    "math.DS",
    "math.DG"
  ],
  "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2007-01-20T22:55:41.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37D40"
    ],
    "keywords": [
      "finite number",
      "insecurity",
      "point obstacles",
      "uniformly secure manifold grows"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007math......1579B"
    }
  }
}