{
  "id": "0711.1662",
  "version": "v1",
  "published": "2007-11-11T16:41:47.000Z",
  "updated": "2007-11-11T16:41:47.000Z",
  "title": "Topological entropy and blocking cost for geodesics in riemannian manifolds",
  "authors": [
    "Eugene Gutkin"
  ],
  "comment": "13 pages",
  "journal": "Geometriae Dedicata 138 (2009), 13 -- 23",
  "doi": "10.1007/s10711-008-9296-3",
  "categories": [
    "math.DS",
    "math.DG"
  ],
  "abstract": "For a pair of points $x,y$ in a compact, riemannian manifold $M$ let $n_t(x,y)$ (resp. $s_t(x,y)$) be the number of geodesic segments with length $\\leq t$ joining these points (resp. the minimal number of point obstacles needed to block them). We study relationships between the growth rates of $n_t(x,y)$ and $s_t(x,y)$ as $t\\to\\infty$. We derive lower bounds on $s_t(x,y)$ in terms of the topological entropy $h(M)$ and its fundamental group. This strengthens the results of Burns-Gutkin \\cite{BG06} and Lafont-Schmidt \\cite{LS}. For instance, by \\cite{BG06,LS}, $h(M)>0$ implies that $s$ is unbounded; we show that $s$ grows exponentially, with the rate at least $h(M)/2$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2007-11-11T16:41:47.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37D40",
      "53C22"
    ],
    "keywords": [
      "riemannian manifold",
      "topological entropy",
      "blocking cost",
      "geodesic segments",
      "fundamental group"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007arXiv0711.1662G"
    }
  }
}