{
  "id": "2105.01841",
  "version": "v1",
  "published": "2021-05-05T02:28:47.000Z",
  "updated": "2021-05-05T02:28:47.000Z",
  "title": "Counting closed geodesics on rank one manifolds without focal points",
  "authors": [
    "Weisheng Wu"
  ],
  "categories": [
    "math.DS",
    "math.DG"
  ],
  "abstract": "In this article, we consider a closed rank one Riemannian manifold $M$ without focal points. Let $P(t)$ be the set of free-homotopy classes containing a closed geodesic on $M$ with length at most $t$, and $\\# P(t)$ its cardinality. We obtain the following Margulis-type asymptotic estimates: \\[\\lim_{t\\to \\infty}\\#P(t)/\\frac{e^{ht}}{ht}=1\\] where $h$ is the topological entropy of the geodesic flow. In the appendix, we also show that the unique measure of maximal entropy of the geodesic flow has the Bernoulli property.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-05-05T02:28:47.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "counting closed geodesics",
      "focal points",
      "geodesic flow",
      "margulis-type asymptotic estimates",
      "free-homotopy classes"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}