{
  "id": "1603.08556",
  "version": "v1",
  "published": "2016-03-28T20:38:07.000Z",
  "updated": "2016-03-28T20:38:07.000Z",
  "title": "Thermodynamics of the Katok Map",
  "authors": [
    "Yakov Pesin",
    "Samuel Senti",
    "Ke Zhang"
  ],
  "comment": "36 pages",
  "categories": [
    "math.DS",
    "math-ph",
    "math.MP",
    "math.PR"
  ],
  "abstract": "We effect thermodynamical formalism for the non-uniformly hyperbolic $C^\\infty$ map of the two dimensional torus known as the Katok map. It is a slowdown of a linear Anosov map near the origin and it is a local (but not small) perturbation. We prove the existence of equilibrium measures for any continuous potential function and obtain uniqueness of equilibrium measures associated to the geometric $t$-potential $\\varphi_t=-t\\log |df|_{E^u(x)}|$ for any $t\\in(t_0,\\infty)$, $t\\neq 1$ where $E^u(x)$ denotes the unstable direction. We show that $t_0$ tends to $-\\infty$ as the size of the perturbation tends to zero. Finally, we establish exponential decay of correlations as well as the Central Limit Theorem for the equilibrium measures associated to $\\varphi_t$ for all values of $t\\in (t_0, 1)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-03-28T20:38:07.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37D25",
      "37D35",
      "37D05",
      "37E10"
    ],
    "keywords": [
      "katok map",
      "equilibrium measures",
      "thermodynamics",
      "central limit theorem",
      "linear anosov map"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 36,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016arXiv160308556P"
    }
  }
}