{
  "id": "1702.08815",
  "version": "v1",
  "published": "2017-02-28T15:12:59.000Z",
  "updated": "2017-02-28T15:12:59.000Z",
  "title": "Infinitely many periodic orbits just above the Mañé critical value on the 2-sphere",
  "authors": [
    "Gabriele Benedetti",
    "Marco Mazzucchelli"
  ],
  "comment": "18 pages, 1 figure",
  "categories": [
    "math.DS",
    "math.DG",
    "math.SG"
  ],
  "abstract": "We introduce a new critical value $c_\\infty(L)$ for Tonelli Lagrangians $L$ on the tangent bundle of the 2-sphere without minimizing measures supported on a point. We show that $c_\\infty(L)$ is strictly larger than the Ma\\~n\\'e critical value $c(L)$, and on every energy level $e\\in(c(L),c_\\infty(L))$ there exist infinitely many periodic orbits of the Lagrangian system of $L$, one of which is a local minimizer of the free-period action functional. This has applications to Finsler metrics of Randers type on the 2-sphere. We show that, under a suitable criticality assumption on a given Randers metric, after rescaling its magnetic part with a sufficiently large multiplicative constant, the new metric admits infinitely many closed geodesics, one of which is a waist. Examples of critical Randers metrics include the celebrated Katok metric.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-02-28T15:12:59.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37J45",
      "58E05"
    ],
    "keywords": [
      "critical value",
      "periodic orbits",
      "randers metric",
      "free-period action functional",
      "metric admits"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}