{
  "id": "1010.4643",
  "version": "v3",
  "published": "2010-10-22T08:56:29.000Z",
  "updated": "2012-03-20T07:46:24.000Z",
  "title": "Renormalization, Thermodynamic Formalism and Quasi-Crystals in Subshifts",
  "authors": [
    "Henk Bruin",
    "Renaud Leplaideur"
  ],
  "comment": "The paper was withdrawn from publication due to an error found in some proof. This is a new version and resubmitted for publication. The occurance of phase transition is proved for a parameter a<1 and it is proved there is no phase transition for a>1. For the value a=1 it is still unkown",
  "categories": [
    "math.DS"
  ],
  "abstract": "We examine thermodynamic formalism for a class of renormalizable dynamical systems which in the symbolic space is generated by the Thue-Morse substitution, and in complex dynamics by the Feigenbaum-Coullet-Tresser map. The basic question answered is whether fixed points $V$ of a renormalization operator $\\CR$ acting on the space of potentials are such that the pressure function $\\gamma \\mapsto \\CP(-\\gamma V)$ exhibits phase transitions. This extends the work by Baraviera, Leplaideur and Lopes on the Manneville-Pomeau map, where such phase transitions were indeed detected. In this paper, however, the attractor of renormalization is a Cantor set (rather than a single fixed point), which admits various classes of fixed points of $\\CR$, some of which do and some of which do not exhibit phase transitions. In particular, we show it is possible to reach, as a ground state, a quasi-crystal before temperature zero by freezing a dynamical system.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2012-03-20T07:46:24.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "thermodynamic formalism",
      "phase transitions",
      "quasi-crystal",
      "dynamical system",
      "thue-morse substitution"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1010.4643B"
    }
  }
}