{
  "id": "2006.13988",
  "version": "v1",
  "published": "2020-06-24T18:46:30.000Z",
  "updated": "2020-06-24T18:46:30.000Z",
  "title": "Multiple phase transitions on compact symbolic systems",
  "authors": [
    "Tamara Kucherenko",
    "Antony Quas",
    "Christian Wolf"
  ],
  "categories": [
    "math.DS"
  ],
  "abstract": "Let $\\phi:X\\to R$ be a continuous potential associated with a symbolic dynamical system $T:X\\to X$ over a finite alphabet. Introducing a parameter $\\beta>0$ (interpreted as the inverse temperature) we study the regularity of the pressure function $\\beta\\mapsto P_{\\rm top}(\\beta\\phi)$. We say that $\\phi$ has a phase transition at $\\beta_0$ if the pressure function $P_{\\rm top}(\\beta\\phi)$ is not differentiable at $\\beta_0$. This is equivalent to the condition that the potential $\\beta_0\\phi$ has two (ergodic) equilibrium states with distinct entropies. In this paper we construct for any given countable set of positive real numbers $\\{\\beta_n\\}$ a potential $\\phi$ which has phase transitions precisely at $\\beta_n$. Taking $\\{\\beta_n\\}$ to be finite, we see that a continuous potential can have any finite number of phase transitions occurring at any set of predetermined points. Taking $\\{\\beta_n\\}$ to be infinite, we obtain a potential whose set of phase transitions is countably infinite.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-06-24T18:46:30.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37A60",
      "37B10",
      "37D35"
    ],
    "keywords": [
      "compact symbolic systems",
      "multiple phase transitions",
      "pressure function",
      "continuous potential",
      "symbolic dynamical system"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}