{
  "id": "2107.04385",
  "version": "v1",
  "published": "2021-07-09T12:30:52.000Z",
  "updated": "2021-07-09T12:30:52.000Z",
  "title": "Thermodynamic formalism for invariant measures in iterated function systems with overlaps",
  "authors": [
    "Eugen Mihailescu"
  ],
  "comment": "To appear in Communications in Contemporary Mathematics, DOI: 10.1142/S0219199721500413. arXiv admin note: substantial text overlap with arXiv:1908.10050",
  "categories": [
    "math.DS",
    "math.CV",
    "math.MG",
    "math.PR"
  ],
  "abstract": "We study images of equilibrium (Gibbs) states for a class of non-invertible transformations associated to conformal iterated function systems with overlaps $\\mathcal S$. We prove exact dimensionality for these image measures, and find a dimension formula using their overlap numbers. In particular, we obtain a geometric formula for the dimension of self-conformal measures for iterated function systems with overlaps, in terms of the overlap numbers. This implies a necessary and sufficient condition for dimension drop. If $\\nu = \\pi_*\\mu$ is a self-conformal measure, then $HD(\\nu) < \\frac{h(\\mu)}{|\\chi(\\mu)|}$ if and only if the overlap number $o(\\mathcal S, \\mu) > 1$. Examples are also discussed.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-07-09T12:30:52.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37A35",
      "37D20",
      "37C45",
      "28A80"
    ],
    "keywords": [
      "invariant measures",
      "thermodynamic formalism",
      "overlap number",
      "self-conformal measure",
      "conformal iterated function systems"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}