{
  "id": "1604.06512",
  "version": "v1",
  "published": "2016-04-21T23:41:04.000Z",
  "updated": "2016-04-21T23:41:04.000Z",
  "title": "Ground States and Zero-Temperature Measures at the Boundary of Rotation Sets",
  "authors": [
    "Tamara Kucherenko",
    "Christian Wolf"
  ],
  "comment": "26 pages",
  "categories": [
    "math.DS"
  ],
  "abstract": "We consider a continuous dynamical system $f:X\\to X$ on a compact metric space $X$ equipped with an $m$-dimensional continuous potential $\\Phi=(\\phi_1,\\cdots,\\phi_m):X\\to \\bR^m$. We study the set of ground states $ GS(\\alpha)$ of the potential $\\alpha\\cdot \\Phi$ as a function of the direction vector $\\alpha\\in S^{m-1}$. %We also study the corresponding rotation vectors $\\rv(GS(\\alpha))$. We show that the structure of the ground state sets is naturally related to the geometry of the generalized rotation set of $\\Phi$. In particular, for each $\\alpha$ the set of rotation vectors of $ GS(\\alpha)$ forms a non-empty, compact and connected subset of a face $F_\\alpha(\\Phi)$ of the rotation set associated with $\\alpha$. Moreover, every ground state maximizes entropy among all invariant measures with rotation vectors in $F_\\alpha(\\Phi)$. We further establish the occurrence of several quite unexpected phenomena. Namely, we construct for any $m\\in\\bN$ examples with an exposed boundary point (i.e. $F_\\alpha(\\Phi)$ being a singleton) without a unique ground state. Further, we establish the possibility of a line segment face $F_\\alpha(\\Phi)$ with a unique but non-ergodic ground state. Finally, we establish the possibility that the set of rotation vectors of $GS(\\alpha)$ is a non-trivial line segment.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-04-21T23:41:04.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37D35",
      "37E45",
      "37B10",
      "37E45",
      "37L40"
    ],
    "keywords": [
      "rotation set",
      "rotation vectors",
      "zero-temperature measures",
      "ground state maximizes entropy",
      "line segment face"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 26,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016arXiv160406512K"
    }
  }
}