{
  "id": "2101.01823",
  "version": "v1",
  "published": "2021-01-05T23:36:57.000Z",
  "updated": "2021-01-05T23:36:57.000Z",
  "title": "Phase transitions for the geodesic flow of a rank one surface with nonpositive curvature",
  "authors": [
    "Keith Burns",
    "Jérôme Buzzi",
    "Todd Fisher",
    "Noelle Sawyer"
  ],
  "comment": "9 pages, 1 figure",
  "categories": [
    "math.DS"
  ],
  "abstract": "We study the one parameter family of potential functions $q\\varphi^u$ associated with the geometric potential $\\varphi^u$ for the geodesic flow of a compact rank 1 surface of nonpositive curvature. For $q<1$ it is known that there is a unique equilibrium state associated with $q\\varphi^u$, and it has full support. For $q > 1$ it is known that an invariant measure is an equilibrium state if and only if it is supported on the singular set. We study the critical value $q=1$ and show that the ergodic equilibrium states are either the restriction to the regular set of the Liouville measure, or measures supported on the singular set. In particular, when~$q = 1$, there is a unique ergodic equilibrium state that gives positive measure to the regular set.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-01-05T23:36:57.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37B40"
    ],
    "keywords": [
      "geodesic flow",
      "nonpositive curvature",
      "phase transitions",
      "unique ergodic equilibrium state",
      "regular set"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 9,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}